collab.html0100664003325600237220000000230707356157351012436 0ustar mliskovta68756.875: Student Collaborator Exchange

6.875 Student Collaboration Exchange

This page is for students to find other students to work with. We allow and encourage collaboration on the problem sets; we require only that you (1) write up your solutions on your own, and (2) that you name your collaborators. Also, please remember to state that you collaborated with nobody if you did not collaborate, so there will be no confusion. Email addresses default to @mit.edu.

Anyone wishing for their name and info to be added to this page, please email Moses at mliskov@theory.lcs.mit.edu. Email also if you want your entry removed or modified. Thanks, and happy collaborating!

Name Email address Location When
Nick Feamster feamster@lcs.mit.edu - -
Joshua Marron jmarron East Campus dorm weekday evenings
Eric Pan epan off campus / medical school weekday daytime or weekends
Marshall Tappen mtappen on campus weekdays

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Once you are done with your file, run ``latex ps1.tex'' from the Athena shell. If your TeX code is clean, the latex will exit back to a prompt. Once this is done, run ``dvips ps1.dvi'' which should print your file to the nearest printer. There will be residual files called ps1.log, ps1.aux, and ps1.dvi. All these can be deleted, but do not delete ps1.tex. \fi \documentclass[11pt]{article} \usepackage{amsfonts} \usepackage{latexsym} \setlength{\oddsidemargin}{.25in} \setlength{\evensidemargin}{.25in} \setlength{\textwidth}{6in} \setlength{\topmargin}{-0.4in} \setlength{\textheight}{8.5in} \newcommand{\handout}[5]{ \renewcommand{\thepage}{#1-\arabic{page}} \noindent \begin{center} \framebox{ \vbox{ \hbox to 5.78in { {\bf 6.875J/18.425J Cryptography and Cryptanalysis} \hfill #2 } \vspace{4mm} \hbox to 5.78in { {\Large \hfill #5 \hfill} } \vspace{2mm} \hbox to 5.78in { {\it #3 \hfill #4} } } } \end{center} \vspace*{4mm} } \newcommand{\ho}[5]{\handout{#1}{#2}{Instructor: #3}{Teaching Assistant: #4}{Handout #1: #5}} \newcommand{\al}{\alpha} \newcommand{\Z}{\mathbb Z} \begin{document} \handout{P1}{DUE: September 12, 2001}{Instructor: Anna Lysyanskaya} {Teaching Assistant: Moses Liskov} {*********** INSERT YOUR NAME HERE ***********: PS 1} I collaborated with *********** INSERT COLLABORATORS HERE *************. \section*{Problem 1} We define the Lily problem: given two integers $n$ and $S$, determine whether $S$ is relatively prime to $\varphi(n)$. Prove that if the Lily problem is hard, then RSA is hard to invert. \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 1 SOLUTION HERE *************** \section*{Problem 2} Assume one is using RSA with the same modulus $n$ but different relatively prime public exponents $e_1$ and $e_2$ to encrypt a message $m$. Prove that the adversary can reconstruct $m$ from the two encryptions (i.e. from $m^{e_1} \bmod\; n$ and $m^{e_2} \bmod\; n$). \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 2 SOLUTION HERE ************** \section*{Problem 3} Assume Alice, Bob and Carol have 3 randomly chosen moduli $n_A$, $n_B$ and $n_C$, s.t. $3$ is a common RSA exponent for all of them (i.e. $gcd(3,\varphi(n_X))=1$, $X=A,B,C$). A user $U$ encrypts the same message $m$ using their public keys. Show that an adversary can reconstruct $m$ from the $3$ encryptions (i.e. from $m^3 \bmod\; n_A$, $m^3 \bmod\; n_B$ and $m^3 \bmod\; n_C$). \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 3 SOLUTION HERE ************** \section*{Problem 4} Suppose $SQRT(a,p)$ is a polynomial-time algorithm that, if $p$ is prime and $a$ is a quadratic residue $a \in Z_p^*$, outputs (w.h.p.) some square root of $a$ mod $p$. \smallskip {\bf (a)} Prove that the following algorithm is a primality test (i.e. that on a prime input outputs PRIME with high probability, and on a composite input outputs COMPOSITE with probability at least $1/2$.) \begin{enumerate} \item If $n=2$, output PRIME. \item If $n$ is even or of the form $n=p^b$ for some $p, b > 1$, output COMPOSITE. \item Choose a random $r \in \{1, \ldots, n-1\}$. If $gcd(r,n) > 1$, output COMPOSITE. \item Compute $x \leftarrow SQRT(r^2 \bmod n, n)$. \item If $x = \pm r \bmod n$, output PRIME. Otherwise, output COMPOSITE. \end{enumerate} \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 4a SOLUTION HERE ************** \bigskip {\bf (b)} Suppose $p = 4m+3$ is a prime and that $a$ is a quadratic residue modulo $p$. Prove that $a^{m+1}$ is a square root of $a$ modulo $p$. \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 4b SOLUTION HERE ************** \bigskip {\bf (c)} Use (a) and (b) to devise a primality testing algorithm for integers equivalent to 3 modulo 4. \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 4c SOLUTION HERE ************** \section*{Problem 5} Suppose $p$ is a prime and $g$ is a generator modulo $p$. Experiment 1: Pick $x$ at random in $\{1, \ldots, p-1\}$. Output $g^x$. Experiment 2: Pick $x, y$ at random in $\{1, \ldots, p-1\}$. Output $g^{xy}$. Prove or disprove: Experiment 1 and Experiment 2 produce identically distributed outputs. \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 5 SOLUTION HERE ************** \end{document} handouts/HOlist0100664003325600237220000000044407374543215013264 0ustar mliskovta68751a: Course Information 1d: Problem Set 1 1e: Sign-up Sheet 2: Problem Set 2 3: Solution Set 1 4: Problem Set 3 5: Problem Set 4 6: Solution Set 2 7: Solution for PS2, problem 2 8: Solution Set 3 9: Problem Set 5 10: Solution Set 4 11: Problem Set 6 12: Problem Set 7 13: Solution Set 5 handouts/ho2.ps0100664003325600237220000050260507347757524013213 0ustar mliskovta6875%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: ho2.dvi %%Pages: 3 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips ho2 -o %+ /pub/www/classes/6.875/fall01/handouts/ho2.ps %DVIPSParameters: 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FFC00001FFC00001FFC00001FFC00001FFC00001FFC00001FFC00001FFC00001FFC01F01 FFC01F01FFC01F01FFC01F01FFC01F01FFC01F01FFC01F01FFC01F01FFC01F00FFE03E00 FFE03E007FE07E007FF8FC003FFFF8001FFFF00007FFE00000FF80203B7EB929>I121 D E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: Letter %%EndSetup %%Page: 1 1 1 0 bop 157 71 3586 4 v 157 574 4 504 v 246 164 a Fo(6.875J/18.425J)38 b(Cryptograph)m(y)d(and)g(Cryptanalysis)402 b Fn(Septem)m(b)s(er)29 b(12,)j(2001)1252 371 y Fm(Handout)37 b(2:)51 b(Problem)36 b(Set)j(2)246 531 y Fl(Instructor:)k(A)n(nna)33 b(Lysyanskaya)853 b(T)-7 b(e)i(aching)33 b(Assistant:)43 b(Moses)33 b(Liskov)p 3739 574 V 157 577 3586 4 v 291 812 a Fn(This)c(problem)g(set)h(will)e (b)s(e)i(due)g(on)g(W)-8 b(ednesda)m(y)g(,)32 b(Septem)m(b)s(er)e(26.) 291 1024 y(F)-8 b(or)46 b(reference,)j(here)c(is)f(the)h(de\014nition)e (of)i(a)g(GM-secure)h(cryptosystem)f(on)g(a)g(sequence)h(of)150 1137 y(message)23 b(spaces)f Fk(f)p Fj(M)894 1151 y Fi(n)941 1137 y Fk(g)p Fn(.)39 b(\()p Fj(G;)15 b(E)5 b(;)15 b(D)s Fn(\))23 b(is)e(a)h(GM-secure)h(cryptosystem)f(if)f(the)g(follo)m(wing) g(t)m(w)m(o)i(prop)s(erties)150 1249 y(hold:)291 1362 y(F)-8 b(aithfulness:)39 b(\()p Fj(G;)15 b(E)5 b(;)15 b(D)s Fn(\))33 b(has)d(faithfulness)e Fj(p)i Fn(\(for)g(0)c Fk(\024)f Fj(p)g Fk(\024)g Fn(1\))31 b(if)150 1573 y Fk(8)p Fj(k)s(;)15 b Fk(8)p Fj(m)25 b Fk(2)g Fj(M)621 1588 y Fi(k)663 1573 y Fj(;)15 b Fk(8)p Fn(\()p Fj(P)e(K)q(;)i(S)5 b(K)i Fn(\))27 b Fk(2)p Fn(Supp\()p Fj(G)p Fn(\(1)1636 1540 y Fi(k)1678 1573 y Fn(\)\))355 1686 y(Pr)o([)p Fj(c)f Fk( )f Fj(E)5 b Fn(\()p Fj(P)13 b(K)q(;)i(m)p Fn(\);)g Fj(a)28 b Fk( )d Fj(D)s Fn(\()p Fj(S)5 b(K)q(;)15 b(c)p Fn(\))27 b(:)e Fj(a)h Fn(=)f Fj(m)p Fn(])355 1799 y Fk(\025)g Fj(p)291 2010 y Fn(GM-securit)m(y:)42 b(\()p Fj(G;)15 b(E)5 b(;)15 b(D)s Fn(\))32 b(is)d(secure)i(if)150 2222 y Fk(8)25 b(f)p Fj(A)339 2236 y Fi(n)386 2222 y Fk(g)h(2)f Fj(P)13 b(P)g(T)38 b Fk(9)p Fj(f)c Fk(2)25 b Fj(neg)s(;)15 b(K)33 b Fk(2)25 b Fh(N)43 b Fn(suc)m(h)30 b(that)h Fk(8)p Fj(k)c(>)e(K)q(;)15 b Fk(8f)p Fj(m)2384 2236 y Fg(0)2425 2222 y Fj(;)g(m)2545 2236 y Fg(1)2584 2222 y Fk(g)26 b(2)f Fj(M)2829 2237 y Fi(k)355 2335 y Fn(Pr)o([\()p Fj(P)13 b(K)q(;)i(S)5 b(K)i Fn(\))27 b Fk( )e Fj(G)p Fn(\(1)1175 2302 y Fi(k)1219 2335 y Fn(\);)15 b Fj(b)26 b Fk( )f(f)p Fn(0)p Fj(;)15 b Fn(1)p Fk(g)p Fn(;)355 2447 y Fj(c)25 b Fk( )g Fj(E)5 b Fn(\()p Fj(P)13 b(K)q(;)i(m)911 2462 y Fi(b)947 2447 y Fn(\);)g Fj(b)1061 2414 y Ff(0)1111 2447 y Fk( )25 b Fj(A)1295 2462 y Fi(k)1338 2447 y Fn(\()p Fj(P)13 b(K)q(;)i(m)1642 2461 y Fg(0)1682 2447 y Fj(;)g(m)1802 2461 y Fg(1)1842 2447 y Fj(;)g(c)p Fn(\))26 b(:)g Fj(b)2072 2414 y Ff(0)2120 2447 y Fn(=)f Fj(b)p Fn(])h Fk(\024)f Fn(1)p Fj(=)p Fn(2)c(+)f Fj(f)10 b Fn(\()p Fj(k)s Fn(\))150 2734 y Fe(Problem)45 b(1)150 2937 y Fn(\(40)36 b(p)s(oin)m(ts.\))55 b(F)-8 b(or)36 b(eac)m(h)h(of)e(the)g(follo)m(wing)f(de\014nitions)f 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Fk(2)25 b Fh(N)43 b Fn(suc)m(h)30 b(that)h Fk(8)p Fj(k)d(>)d(K)q(;)355 4307 y Fn(Pr)o([\()p Fj(P)13 b(K)q(;)i(S)5 b(K)i Fn(\))27 b Fk( )e Fj(G)p Fn(\(1)1175 4274 y Fi(k)1219 4307 y Fn(\);)15 b Fj(b)26 b Fk( )f(f)p Fn(0)p Fj(;)15 b Fn(1)p Fk(g)p Fn(;)355 4420 y Fj(c)25 b Fk( )g Fj(E)5 b Fn(\()p Fj(P)13 b(K)q(;)i(b)p Fn(\);)g Fj(b)984 4387 y Ff(0)1035 4420 y Fk( )26 b Fj(A)1220 4435 y Fi(k)1262 4420 y Fn(\()p Fj(P)13 b(K)q(;)i(c)p Fn(\))28 b(:)d Fj(b)1677 4387 y Ff(0)1726 4420 y Fn(=)g Fj(b)p Fn(])g Fk(\024)g Fn(1)p Fj(=)p Fn(2)d(+)e(2)2300 4387 y Ff(\000)p Fi(k)291 4730 y Fo(\(b\))30 b Fl(A)n(lmost-al)5 b(l-p)-5 b(ar)g(ameter)36 b(se)-5 b(curity.)40 b Fn(Do)32 b Fl(al)5 b(l)30 b Fn(su\016cien)m(tly)g(large)g Fj(k)j Fn(ha)m(v)m(e)f(to)f(w)m(ork?)150 4941 y Fk(8)25 b(f)p Fj(A)339 4955 y Fi(n)386 4941 y Fk(g)h(2)f Fj(P)13 b(P)g(T)38 b Fk(9)p Fj(f)c Fk(2)25 b Fj(neg)s(;)15 b(K)33 b Fk(2)25 b Fh(N)43 b Fn(suc)m(h)30 b(that)h Fk(8)p Fj(k)c(>)e(K)37 b Fn(suc)m(h)31 b(that)f Fj(k)f Fk(6)p Fn(=)c(2)2823 4908 y Fi(i)2882 4941 y Fn(for)30 b(an)m(y)h Fj(i;)355 5054 y Fn(Pr)o([\()p Fj(P)13 b(K)q(;)i(S)5 b(K)i Fn(\))27 b Fk( )e Fj(G)p Fn(\(1)1175 5021 y Fi(k)1219 5054 y Fn(\);)15 b Fj(b)26 b Fk( )f(f)p Fn(0)p Fj(;)15 b Fn(1)p Fk(g)p Fn(;)355 5167 y Fj(c)25 b Fk( )g Fj(E)5 b Fn(\()p Fj(P)13 b(K)q(;)i(b)p Fn(\);)g Fj(b)984 5134 y Ff(0)1035 5167 y Fk( )26 b Fj(A)1220 5182 y Fi(k)1262 5167 y Fn(\()p Fj(P)13 b(K)q(;)i(c)p Fn(\))28 b(:)d Fj(b)1677 5134 y Ff(0)1726 5167 y Fn(=)g Fj(b)p Fn(])g Fk(\024)g Fn(1)p Fj(=)p Fn(2)d(+)e Fj(f)10 b Fn(\()p Fj(k)s Fn(\))1889 5416 y(2-1)p eop %%Page: 2 2 2 1 bop 291 159 a Fo(\(c\))27 b Fl(A)n(l)5 b(l-p)-5 b(ar)g(ameter)32 b(se)-5 b(curity.)39 b Fn(Do)28 b(w)m(e)g(need)f(the)g(assertion)g (that)h Fj(k)i Fn(has)d(to)h(b)s(e)f(su\016cien)m(tly)f(large?)150 334 y Fk(8)f(f)p Fj(A)339 348 y Fi(n)386 334 y Fk(g)h(2)f Fj(P)13 b(P)g(T)38 b Fk(9)p Fj(f)c Fk(2)25 b Fj(neg)34 b Fn(suc)m(h)c(that)h Fk(8)p Fj(k)d Fk(2)c Fh(N)7 b Fj(;)355 447 y Fn(Pr)o([\()p Fj(P)13 b(K)q(;)i(S)5 b(K)i Fn(\))27 b Fk( )e Fj(G)p Fn(\(1)1175 414 y Fi(k)1219 447 y Fn(\);)15 b Fj(b)26 b Fk( )f(f)p Fn(0)p Fj(;)15 b Fn(1)p Fk(g)p Fn(;)355 559 y Fj(c)25 b Fk( )g Fj(E)5 b Fn(\()p Fj(P)13 b(K)q(;)i(b)p Fn(\);)g Fj(b)984 526 y Ff(0)1035 559 y Fk( )26 b Fj(A)1220 574 y Fi(k)1262 559 y Fn(\()p Fj(P)13 b(K)q(;)i(c)p Fn(\))28 b(:)d Fj(b)1677 526 y Ff(0)1726 559 y Fn(=)g Fj(b)p Fn(])g Fk(\024)g Fn(1)p Fj(=)p Fn(2)d(+)e Fj(f)10 b Fn(\()p Fj(k)s Fn(\))291 809 y Fo(\(d\))21 b Fl(A)n(l)5 b(l-message)24 b(se)-5 b(curity.)38 b Fn(Do)22 b(w)m(e)g(need)g(to)g(pic)m(k)f(the)h(message)h(randomly)d(during)f (the)j(proto)s(col?)150 984 y Fk(8)j(f)p Fj(A)339 998 y Fi(n)386 984 y Fk(g)h(2)f Fj(P)13 b(P)g(T)38 b Fk(9)p Fj(f)c Fk(2)25 b Fj(neg)s(;)15 b(K)33 b Fk(2)25 b Fh(N)63 b Fk(8)p Fj(k)28 b(>)d(K)32 b Fk(8)p Fj(b)24 b Fk(2)h(f)p Fn(0)p Fj(;)15 b Fn(1)p Fk(g)p Fj(;)355 1097 y(P)e(r)s Fn([\()p Fj(P)g(K)q(;)i(S)5 b(K)i Fn(\))26 b Fk( )f Fj(G)p Fn(\(1)1192 1064 y Fi(k)1236 1097 y Fn(\);)355 1210 y Fj(c)g Fk( )g Fj(E)5 b Fn(\()p Fj(P)13 b(K)q(;)i(b)p Fn(\);)g Fj(b)984 1177 y Ff(0)1035 1210 y Fk( )26 b Fj(A)1220 1225 y Fi(k)1262 1210 y Fn(\()p Fj(P)13 b(K)q(;)i(c)p Fn(\))28 b(:)d Fj(b)1677 1177 y Ff(0)1726 1210 y Fn(=)g Fj(b)p Fn(])g Fk(\024)g Fn(1)p Fj(=)p Fn(2)d(+)e Fj(f)10 b Fn(\()p Fj(k)s Fn(\))291 1460 y Fo(\(e\))36 b Fl(Ciphertext)k(game)f (se)-5 b(curity.)61 b Fn(Can)36 b(w)m(e)i(de\014ne)e(securit)m(y)h (di\013eren)m(tly:)52 b(sa)m(y)38 b(a)f(cryptosystem)150 1573 y(is)44 b(secure)g(if)g(the)g(c)m(hance)i(an)e(adv)m(ersary)h (determines)f(a)h(message)g(giv)m(en)g(its)f(encryption)f(is)h(only)150 1686 y(negligibly)33 b(b)s(etter)k(than)e(the)i(probabilit)m(y)c(that)k (the)f(adv)m(ersary)h(determines)e(the)h(message)h(giv)m(en)g(a)150 1799 y(random)30 b(encryption)f(of)h(some)h(kind?)150 1974 y Fk(8)25 b(f)p Fj(A)339 1988 y Fi(n)386 1974 y Fk(g)h(2)f Fj(P)13 b(P)g(T)38 b Fk(9)p Fj(f)c Fk(2)25 b Fj(neg)s(;)15 b(K)33 b Fk(2)25 b Fh(N)63 b Fk(8)p Fj(k)28 b(>)d(K)q(;)355 2087 y Fn(Pr)o([\()p Fj(P)13 b(K)q(;)i(S)5 b(K)i Fn(\))27 b Fk( )e Fj(G)p Fn(\(1)1175 2054 y Fi(k)1219 2087 y Fn(\);)15 b Fj(b)1333 2101 y Fg(0)1373 2087 y Fj(;)g(b)1452 2101 y Fg(1)1517 2087 y Fk( )26 b(f)p Fn(0)p Fj(;)15 b Fn(1)p Fk(g)355 2199 y Fj(c)25 b Fk( )g Fj(E)5 b Fn(\()p Fj(P)13 b(K)q(;)i(b)870 2213 y Fg(0)912 2199 y Fn(\);)g Fj(b)1026 2166 y Ff(0)1075 2199 y Fk( )25 b Fj(A)1259 2214 y Fi(k)1302 2199 y Fn(\()p Fj(P)13 b(K)q(;)i(c)p Fn(\))27 b(:)f Fj(b)1717 2166 y Ff(0)1765 2199 y Fn(=)f Fj(b)1900 2213 y Fg(0)1940 2199 y Fn(])p Fk(\000)355 2312 y Fn(Pr)o([\()p Fj(P)13 b(K)q(;)i(S)5 b(K)i Fn(\))27 b Fk( )e Fj(G)p Fn(\(1)1175 2279 y Fi(k)1219 2312 y Fn(\);)15 b Fj(b)1333 2326 y Fg(0)1373 2312 y Fj(;)g(b)1452 2326 y Fg(1)1517 2312 y Fk( )26 b(f)p Fn(0)p Fj(;)15 b Fn(1)p Fk(g)355 2425 y Fj(c)25 b Fk( )g Fj(E)5 b Fn(\()p Fj(P)13 b(K)q(;)i(b)870 2439 y Fg(1)912 2425 y Fn(\);)g Fj(b)1026 2392 y Ff(0)1075 2425 y Fk( )25 b Fj(A)1259 2440 y Fi(k)1302 2425 y Fn(\()p 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handouts/ps2.tex0100664003325600237220000001350307354456364013373 0ustar mliskovta6875\iffalse NOTE: If you do not collaborate with anyone, simply remove the '**** INSERT COLLABORATORS HERE****' and replace it with 'nobody.' Don't erase the collaboration statement! Thanks. INSTRUCTIONS: (if this is not ps1, use the right file name) Clip out the ********* INSERT HERE ********* bits below and insert appropriate TeX code. Once you are done with your file, run ``latex ps1.tex'' from the Athena shell. If your TeX code is clean, the latex will exit back to a prompt. Once this is done, run ``dvips ps1.dvi'' which should print your file to the nearest printer. There will be residual files called ps1.log, ps1.aux, and ps1.dvi. All these can be deleted, but do not delete ps1.tex. \fi \documentclass[11pt]{article} \usepackage{amsfonts} \usepackage{latexsym} \setlength{\oddsidemargin}{.25in} \setlength{\evensidemargin}{.25in} \setlength{\textwidth}{6in} \setlength{\topmargin}{-0.4in} \setlength{\textheight}{8.5in} \newcommand{\handout}[5]{ \renewcommand{\thepage}{#1-\arabic{page}} \noindent \begin{center} \framebox{ \vbox{ \hbox to 5.78in { {\bf 6.875J/18.425J Cryptography and Cryptanalysis} \hfill #2 } \vspace{4mm} \hbox to 5.78in { {\Large \hfill #5 \hfill} } \vspace{2mm} \hbox to 5.78in { {\it #3 \hfill #4} } } } \end{center} \vspace*{4mm} } \newcommand{\al}{\alpha} \newcommand{\Z}{\mathbb Z} \begin{document} \handout{P2}{DUE: September 26, 2001}{Instructor: Anna Lysyanskaya} {Teaching Assistant: Moses Liskov} {*********** INSERT YOUR NAME HERE ***********: PS 2} I collaborated with ******** INSERT COLLABORATORS OR `nobody' HERE **********. \section*{Problem 1} (40 points.) For each of the following definitions of security for a cryptosystem, compare them to GM-security. Some of the definitions may be too strong, that is, unattainable. If the definition is unattainable, prove it. Otherwise, for each definition you should prove or disprove that \begin{enumerate} \item If $(G, E, D)$ is secure under that definition then it is GM-secure, and \item If $(G, E, D)$ is GM-secure then it is secure under the given definition. \end{enumerate} \bigskip {\bf (a)} {\em $2^{-k}$ security.} Do we need to be so general about the negligible function? \begin{tabbing} XXX\=\kill $\forall \; \{A_n\} \in PPT \; \exists K \in {\Bbb N} $ such that $ \forall k > K,$\\ \>$\Pr[(PK, SK) \leftarrow G(1^k); b \leftarrow \{0,1\};$\\ \>$c \leftarrow E(PK, b); b' \leftarrow A_k(PK, c): b'=b] \leq 1/2 + 2^{-k}$ \end{tabbing} \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 1a SOLUTION HERE *************** \bigskip {\bf (b)} {\em Almost-all-parameter security.} Do {\em all} sufficiently large $k$ have to work? \begin{tabbing} XXX\=\kill $\forall \; \{A_n\} \in PPT \; \exists f \in neg, K \in {\Bbb N} $ such that $ \forall k > K $ such that $ k \neq 2^i$ for any $i,$\\ \>$\Pr[(PK, SK) \leftarrow G(1^k); b \leftarrow \{0,1\};$\\ \>$c \leftarrow E(PK, b); b' \leftarrow A_k(PK, c): b'=b] \leq 1/2 + f(k)$ \end{tabbing} \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 1b SOLUTION HERE *************** \bigskip {\bf (c)} {\em All-parameter security.} Do we need the assertion that $k$ has to be sufficiently large? \begin{tabbing} XXX\=\kill $\forall \; \{A_n\} \in PPT \; \exists f \in neg $ such that $ \forall k \in {\Bbb N},$\\ \>$\Pr[(PK, SK) \leftarrow G(1^k); b \leftarrow \{0,1\};$\\ \>$c \leftarrow E(PK, b); b' \leftarrow A_k(PK, c): b'=b] \leq 1/2 + f(k)$ \end{tabbing} \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 1c SOLUTION HERE *************** \bigskip {\bf (d)} {\em All-message security.} Do we need to pick the message randomly during the protocol? \begin{tabbing} XXX\=\kill $\forall \; \{A_n\} \in PPT \; \exists f \in neg, K \in {\Bbb N} \;\; \forall k > K \; \forall b \in \{0,1\},$\\ \>$Pr[(PK, SK) \leftarrow G(1^k);$\\ \>$c \leftarrow E(PK, b); b' \leftarrow A_k(PK, c): b'=b] \leq 1/2 + f(k)$ \end{tabbing} \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 1d SOLUTION HERE *************** \bigskip {\bf (e)} {\em Ciphertext game security.} Can we define security differently: say a cryptosystem is secure if the chance an adversary determines a message given its encryption is only negligibly better than the probability that the adversary determines the message given a random encryption of some kind? \begin{tabbing} XXX\=\kill $\forall \; \{A_n\} \in PPT \; \exists f \in neg, K \in {\Bbb N} \; \; \forall k > K,$\\ \>$\Pr[ $\=$ (PK, SK) \leftarrow G(1^k); b_0, b_1 \leftarrow \{0,1\}$\\ \>$c \leftarrow E(PK, b_0); b' \leftarrow A_k(PK, c): b'=b_0] -$\\ \>$\Pr[ $\>$ (PK, SK) \leftarrow G(1^k); b_0, b_1 \leftarrow \{0,1\}$\\ \>$c \leftarrow E(PK, b_1); b' \leftarrow A_k(PK, c): b'=b_0] \leq f(k)$ \end{tabbing} \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 1e SOLUTION HERE *************** \section*{Problem 2} (20 points.) Prove that if for any $0 < q \leq 1/2$, the Goldwasser-Micali cryptosystem is $q$-GM-secure, then it is GM-secure. \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 2 SOLUTION HERE *************** \section*{Problem 3} (20 points.) Is $q$-GM-security equivalent to GM-security? Prove or disprove each: For any $0 < q \leq 1/2$, $q$-GM-security implies GM-security; for any $0 < q \leq 1/2$, GM-security implies $q$-GM-security. \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 3 SOLUTION HERE *************** \section*{Problem 4} (20 points.) Prove that if there exists a GM-secure cryptosystem $(G, E, D)$ then $P \neq NP$. 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stream 0 0 0 27 46 98 d1 46 0 0 71 0 27 cm BI /IM true /W 46 /H 71 /BPC 1 /F /CCF /DP <> ID &£4ˆGPÊ ôxƒá·Ó‡q >ÃÞMPwíÿ÷‡|0^7‡þAyþù§£æ=oAðý>þÿ÷ÿÿ¿ûþðÿ‡ÿ‡ýÿ~ýÿ¾ÿ}ùâýßòj®ýßÿ‡¿ÿ¶÷þø{uá¯cá ÖC EI endstream endobj 124 0 obj <> stream 46 0 0 0 0 0 d1 endstream endobj 125 0 obj <> stream 0 0 0 -1 46 77 d1 46 0 0 78 0 -1 cm BI /IM true /W 46 /H 78 /BPC 1 /F /CCF /DP <> ID &¹Ànx/âÁ´zzß¼š®ßÿjÿÃÿß~ïÿ÷¼>ÿ‡ÿÿŽëõÚø_Ao^. ¿Ã.ÃækÙïÛÿvvCç‡ý°ƒÃ·~ü6Ûøw»lpcàÖá…ãßþÿßþýÿáÿ‡ÿ¿÷ù Wÿ" EI endstream endobj 126 0 obj <> stream 44 0 0 0 0 0 d1 endstream endobj 127 0 obj <> stream 55 0 0 0 0 0 d1 endstream endobj 128 0 obj <> stream 0 0 0 2 69 75 d1 69 0 0 73 0 2 cm BI /IM true /W 69 /H 73 /BPC 1 /F /CCF /DP <> ID &¹( Ïûä1¦ wÿ¿ûÿûÿÿïÿýÿïßþÿßþÿÃÿß*ò ï»ÿ½ÿ°ïÿ½ÿü?÷öîÿÿðÛÿýß·ÿü7]‡ ðá‚â?~  EI endstream endobj 129 0 obj <> stream 0 0 0 27 39 77 d1 39 0 0 50 0 27 cm BI /IM true /W 39 /H 50 /BPC 1 /F /CCF /DP <> ID &¡ÉÁ‚„x`õ‡„ˆziúá½&þõa÷ïÃðÿÿûÿÿä/”l4zm†üûí÷|?Çmÿo·ßƒh-ƒøapÅ`ÂÁ‚€€ EI endstream endobj 130 0 obj <> stream 43 0 0 0 0 0 d1 endstream endobj 131 0 obj <> stream 0 0 0 -1 49 77 d1 49 0 0 78 0 -1 cm BI /IM true /W 49 /H 78 /BPC 1 /F /CCF /DP <> ID &¹À0|'âÂÑÆ·á¹5]ÿûÛÿû½ÿ½ÿþ8wøwÿîBšÞýûþÿÝÿïûÿpÿ÷ü?ößÿø~ÿïïÿn3@׈4Åxan/ÿ÷þÿ÷ïÿü?ýÿ¿ÈJ¿ûü€€ EI endstream endobj 132 0 obj <> stream 0 0 0 2 27 77 d1 27 0 0 75 0 2 cm BI /IM true /W 27 /H 75 /BPC 1 /F /CCF /DP <> ID &¡ø z<Þ·íÿïì?û¿ü?ýÞïßþÿxáþÿßíþóÅïäÕ_»ÿþÿößýýºá®Ç†°ÂÇÿÿÿòó?ïÿ  EI endstream endobj 133 0 obj <> stream 0 0 0 27 48 98 d1 48 0 0 71 0 27 cm BI /IM true /W 48 /H 71 /BPC 1 /F /CCF /DP <> ID &¢”l±˜Aèø€úàÛí¿¿oÃýß ÿ‹ûÿûäŸ„à‡£3ô ú°õÿáýõýøßÿûÿÝÿ¿ïÿwþÿ‡ÿnÿí¿ÞØýûwmoÃö .Ö›5ƒC†@ EI endstream endobj 134 0 obj <> stream 91 0 0 0 0 0 d1 endstream endobj 135 0 obj <> stream 84 0 0 0 0 0 d1 endstream endobj 136 0 obj <> stream 0 0 0 0 105 75 d1 105 0 0 75 0 0 cm BI /IM true /W 105 /H 75 /BPC 1 /F /CCF /DP <> ID &¨¨6@ÀŽ¿ÃþüŒŒ†Äa­­«¼?ýûÛ÷ßþè?÷ûûšnïïÿí‡ÿø}ßì?uëÃ{wÿáûÿwÿûÛ½öÿÿoØP÷ð¿}¿ößÿ︿¿ÿ°Ãþø~·ïOÿ¿íýíßÿ‡áÿ»ÿýá»ßoÿýmû§¿ÿ}¿öTÿûâá÷ÿÙµýŽú¼ÙPÿ½Ãü@ EI endstream endobj 137 0 obj 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BI /IM true /W 36 /H 51 /BPC 1 /F /CCF /DP <> ID 0œ?†N»ð€ã¸@Ý7 êÿW×ëÒþM@¾–A„ùX%¥®–p\êÄœ È1uÂë¯õù5Úÿ¿ÚðÂì0¸cÁšî À@ EI endstream endobj 146 0 obj <> stream 31 0 0 0 0 0 d1 endstream endobj 147 0 obj <> stream 0 0 0 23 49 74 d1 49 0 0 51 0 23 cm BI /IM true /W 49 /H 51 /BPC 1 /F /CCF /DP <> ID &¡œ 3ü†`[Â0ð§ÒoI¿õ½&ÿÿü,>ÿŽÿÿÿÿ&«_ÿÿ .û×ý®õ¶–õ†Ð[ip`» %†!`Á`Á@@ EI endstream endobj 148 0 obj <> stream 0 0 0 -4 26 72 d1 26 0 0 76 0 -4 cm BI /IM true /W 26 /H 76 /BPC 1 /F /CCF /DP <> ID òj)@íÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿë…þMDÀ@ EI endstream endobj 149 0 obj <> stream 0 0 0 25 74 74 d1 74 0 0 49 0 25 cm BI /IM true /W 74 /H 49 /BPC 1 /F /CCF /DP <> ID & ÜØ76 ÿÚëµÿ ð×ýo°ºßÌך.þÚúýÿ_V¯ïúõ¯µð¯ß _øx^—·ñÿú†¯­ÿý¯^úß^µõ¿¯…ïÒO^ÿôÂa8ƒ&\ÿÿ EI endstream endobj 150 0 obj <> stream 0 0 0 -4 55 74 d1 55 0 0 78 0 -4 cm BI /IM true /W 55 /H 78 /BPC 1 /F /CCF /DP <> ID & A@†‚ÿk àŒÅÖøA:@ÍE„ÖkWÿKÔ?_úÿÿÿÿÿÿÿÿɯÿïÿý¿ýýþÂí¯Ãøl.Ú8¼0_+Ãä4Çÿÿÿÿÿÿÿÿÿü.¹ Ôÿÿù j@@ EI endstream endobj 151 0 obj <> stream 0 0 0 24 56 74 d1 56 0 0 50 0 24 cm BI /IM true /W 56 /H 50 /BPC 1 /F /CCF /DP <> ID &¡  SA^¹¬7„h0|-ôé< »×Q^úÿøÿ÷ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿýWü(XÈtÿÿ&¢Hl¤@ EI endstream endobj 152 0 obj <> stream 0 0 0 24 56 72 d1 56 0 0 48 0 24 cm BI /IM true /W 56 /H 48 /BPC 1 /F /CCF /DP <> ID $ßþME( å´×ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿßÿýÿ÷úèÑ_øPÜ(Øa. pÅ`Ôš‰ œ EI endstream endobj 153 0 obj <> stream 0 0 0 -3 106 75 d1 106 0 0 78 0 -3 cm BI /IM true /W 106 /H 78 /BPC 1 /F /CCF /DP <> ID & Òp4̓Moþ×ýaö¿ÿ­öýoÿû_ð·ÃE Ê¿ÿA{~Kÿõø}Z¿ÿï^¿a~¼>¯Wÿõïí ¿ÿ ÷Ô5zÿýíý¯áë×áõïö¿½zýý…úðþõë÷׿Úþð¼/ß^þÅú÷ÂßÃ_þ–ÃÿÿÖ¯ÂýÕëþ˜Aá˜@âK‡ @àÿÿü@ EI endstream endobj 154 0 obj <> stream 97 0 0 0 0 0 d1 endstream endobj 155 0 obj <> stream 102 0 0 0 0 0 d1 endstream endobj 156 0 obj <> stream 0 0 0 23 50 74 d1 50 0 0 51 0 23 cm BI /IM true /W 50 /H 51 /BPC 1 /F /CCF /DP <> ID &¡Ê‹ø `àÓÂÃOAæŸH6“zxOØñ ÿÿ÷ÿðüšÿÞBšßÿoïØ?oÁü0~ üx>AEYŠþE¿þŸéÿ¯ÖÕp’õØ0AaˆXad0ˆ€ EI endstream endobj 157 0 obj <> stream 0 0 0 25 52 95 d1 52 0 0 70 0 25 cm BI /IM true /W 52 /H 70 /BPC 1 /F /CCF /DP <> ID 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<0|<<0ð|0ðÁðð`ðÃÁðÃÃÃÁƒÃà >àð]a„ „ X.X \, XA`¸A`p°a‚á‚„p§À¬šá@@ EI endstream endobj 228 0 obj <> stream 0 0 0 40 35 95 d1 35 0 0 55 0 40 cm BI /IM true /W 35 /H 55 /BPC 1 /F /CCF /DP <> ID &¡Êü ð˜xXz7ÓôúÞ‚ ÿþ·ÿOÿñÃÿÿÿÿÿÿÿ“_ ÿÿÿµÞ¿Þ¿á…ÛKý´°Ú ,0  EI endstream endobj 229 0 obj <> stream 0 0 0 40 28 93 d1 28 0 0 53 0 40 cm BI /IM true /W 28 /H 53 /BPC 1 /F /CCF /DP <> ID &«ü„¦Âÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ™Øù5!{÷ÃÀ@ EI endstream endobj 230 0 obj <> stream 98 0 0 0 0 0 d1 endstream endobj 231 0 obj <> stream 139 0 0 0 0 0 d1 endstream endobj 232 0 obj <> stream 0 0 0 -4 42 74 d1 42 0 0 78 0 -4 cm BI /IM true /W 42 /H 78 /BPC 1 /F /CCF /DP <> ID &¡ GÂ:†z8aáoH=&ýøMêøz·×ïÝÿíÿÿ°ÿáÿ{þþÿ¿Ýÿ¿ÿ¿¶¿ïÝxußð°Úóû q¼Ñ ðan/ÿ÷þÿ÷ïÿïÿü?É×ÿßä EI endstream endobj 233 0 obj <> stream 0 0 0 39 31 96 d1 31 0 0 57 0 39 cm BI /IM true /W 31 /H 57 /BPC 1 /F /CCF /DP <> ID &¢ŸþŽzÞ“Ó~þñ†ûý‡ÿ·÷öMßûÿû¿ðÿﶾ»û®ìBæixaxïýÿáÿ¿ýÿ‡ùaÿä'À@ EI endstream endobj 234 0 obj <> stream 0 0 0 -12 19 30 d1 19 0 0 42 0 -12 cm BI /IM true /W 19 /H 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0 33 cm BI /IM true /W 71 /H 68 /BPC 1 /F /CCF /DP <> ID &¡˜u Ïn“Ar<"œ@| | A½&õ½&o§Âß õ¿õ¿ÿéõ‡ÿÿëÿÿÿÿÿÿÿÿÞ¿ÿýáÿk½½Ã\=o]®ØAv»il4°x[D« BüX2¬Hfk EI endstream endobj 242 0 obj <> stream 0 0 0 -1 79 101 d1 79 0 0 102 0 -1 cm BI /IM true /W 79 /H 102 /BPC 1 /F /CCF /DP <> ID &¡ªu ¿žAu„´†>Ñ oæh Áñ7÷Ó}7ÓÿÓïÿ ÿÿÿ~ÿÿÿßÿÿÿÿÿÿÿÿþ¿ÿ×ÿ¯þ¿ð—ý¯p¾»K´¸5á‚\ꎂ…ï ÁŠðÁá‚ò ¯ÿÿÿÿÿÿÿÿÿÿÿÿár”ÿÿ䈀 EI endstream endobj 243 0 obj <> stream 0 0 0 -1 35 99 d1 35 0 0 100 0 -1 cm BI /IM true /W 35 /H 100 /BPC 1 /F /CCF /DP <> ID &«ÿþBkÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ®Rÿÿü‚ EI endstream endobj 244 0 obj <> stream 0 0 0 33 64 101 d1 64 0 0 68 0 33 cm BI /IM true /W 64 /H 68 /BPC 1 /F /CCF /DP <> ID &¡˜j  ¡^‡‚ è§A‚0HÐA½oIô›è?øI‡_ôöëý~¿ÿÿü™xü§ÿÿßÿ…ßáwÿÞ¸®ÿiaýëm.×l.ÃAaë AlWàÁ`ÂÈeë EI endstream endobj 245 0 obj <> stream 0 0 0 34 125 99 d1 125 0 0 65 0 34 cm BI /IM true /W 125 /H 65 /BPC 1 /F /CCF /DP <> ID &¨ŠEDÿÿÿÿù Y ʈfUÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÝÿÿÿýßÿÿÜ?»ùÕᦗ¼ÕÿooðÝ7XXa`×)C°ÿ` ’ {Ðzö)ˆ^ ¿ü0@ÂÈ!ÐÐ €€ EI 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267 0 obj <> stream 0 0 0 27 54 97 d1 54 0 0 70 0 27 cm BI /IM true /W 54 /H 70 /BPC 1 /F /CCF /DP <> ID !§Éªätÿß¿ýÿ¿ü?ðÿÉ~ÿ„´lÁñßL=„~ý7íÿ½ûø}áÿÛÿ‡oþ÷þïÿßîÿ‘zÿÿîÿ°ÿ÷Ö¿á…îÿëyõûØio·ö°×Šb¶‚Á‚ (€ EI endstream endobj 268 0 obj <> stream 0 0 0 -5 86 72 d1 86 0 0 77 0 -5 cm BI /IM true /W 86 /H 77 /BPC 1 /F /CCF /DP <> ID $Ù&ÿÿɨÉA $ ¿ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ ÿúÄGüšˆh gñÿÿÿÿÿÿÿÿÿ Þü( ýP}ß¿ÿ÷~ÿÝûm¾Ã 0aaݬ6ŽA„°Á‚A† qǃ C-3¢  EI endstream endobj 269 0 obj <> stream 0 0 0 -4 52 72 d1 52 0 0 76 0 -4 cm BI /IM true /W 52 /H 76 /BPC 1 /F /CCF /DP <> ID *Ïÿ&£(ÎŽP__úýxKÿA}zúý}zðÂ]ý »KþëŠþh¾ö°ý¿¿oÁ¿aûïàÿø?—ÿÿþ?ÿÿÿÿÿÿÿÿÿø]cÿ“Q0 EI endstream endobj 270 0 obj <> stream 0 0 0 -2 54 99 d1 54 0 0 101 0 -2 cm BI /IM true /W 54 /H 101 /BPC 1 /F /CCF /DP <> ID &¹€e&»þ÷áïûÞÿ†ÿ½ø{þß÷¿á¿ï~ÿ·ýïøoû߇¿íÿ{þþ÷áïûÞÿ†ÿ½ø{þß÷¿á¿ï~ÿ·ýïøoû߇¿íÿxj  EI endstream endobj 271 0 obj <> stream 0 0 0 -21 22 35 d1 22 0 0 56 0 -21 cm BI /IM true /W 22 /H 56 /BPC 1 /F /CCF /DP <> ID &¢Á=ÿ¨7÷ïï°ïûßáöþýþ¿†÷÷¿¹5Aþø¿oïØ­…ƒ ÿÿ ½þžþÚ€€ EI endstream endobj 272 0 obj <> stream 0 0 0 -1 28 74 d1 28 0 0 75 0 -1 cm BI /IM true /W 28 /H 75 /BPC 1 /F /CCF /DP <> ID &¡O„øAáƒÖOíø¿~ïÿû»¿÷þ‡5Ãßûýÿ¿Ûýç‡ïäÕC÷ü;ÿß¿Ûï÷ï ƒ] a…ÿÿÿä6^?ïÿ  EI endstream endobj 279 0 obj <> stream 0 0 0 31 48 84 d1 48 0 0 53 0 31 cm BI /IM true /W 48 /H 53 /BPC 1 /F /CCF /DP <> ID &¡°è ò ƒÂ‚zxE8ß„o¦¾“t¾úêA¤×ÿð£ÿÿÿ“^CgxAðÓôÿûýûûøm…¶üÔ/b¶°k CfŸ EI endstream endobj 280 0 obj <> stream 0 0 0 24 45 74 d1 45 0 0 50 0 24 cm BI /IM true /W 45 /H 50 /BPC 1 /F /CCF /DP <> ID &£8  ÷ûÿ÷¿ÿ‡ÿáþÿýþÿýþÿóÄ÷ù5[ü†3ÿ¸a‡ ßýßï¼ò¾öû„˜`ûáøÁŠØA¬ Á@@ EI endstream endobj 281 0 obj <> stream 0 0 0 31 50 84 d1 50 0 0 53 0 31 cm BI /IM true /W 50 /H 53 /BPC 1 /F /CCF /DP <> ID &¡°êò Á èÖ(|"M á&õô›×ÃÒù³Zÿ ãÿåL7ó¨òkÿõíÿ÷®Âÿ¶»ÖAm¥Á‚ìVÖ a‚Èh×à EI endstream endobj 282 0 obj <> stream 0 0 0 -5 71 75 d1 71 0 0 80 0 -5 cm BI /IM true /W 71 /H 80 /BPC 1 /F /CCF /DP <> ID &¡  $ø@ðDëz'ÃÒO¤Â7­ûzL?~Ÿ[ûðúMÿï÷ÿ°ÿ÷ÿïÿü¿ÿï÷ÿï÷ÿáîCo;÷ÿÃýßûûûwÃÿÛðþÝí¯û~Á§‡öýµípÁÎ ðß¾ Dø1ÿpÂ!®Œ@ EI endstream endobj 283 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stream 0 0 0 38 36 81 d1 36 0 0 43 0 38 cm BI /IM true /W 36 /H 43 /BPC 1 /F /CCF /DP <> ID & ¹ _õa<ô‚f“}>¯Ô:ü< ¾Åÿÿ&«ßù;ðÂ{ýûû~áðØk`Èÿ ,‚瀀 EI endstream endobj 307 0 obj <> stream 0 0 0 39 46 81 d1 46 0 0 42 0 39 cm BI /IM true /W 46 /H 42 /BPC 1 /F /CCF /DP <> ID & ¹9™ø çPÞ™Ö:²Ai¬/ïÿÿûÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿªÆD×ÿþMG ¼à EI endstream endobj 308 0 obj <> stream 0 0 0 18 20 79 d1 20 0 0 61 0 18 cm BI /IM true /W 20 /H 61 /BPC 1 /F /CCF /DP <> ID òjdçkÿÿÿÿÿÿÿÿÿÿÿÿÿõþMKÿÿä»Oôÿÿö¶  EI endstream endobj 309 0 obj <> stream 0 0 0 24 29 81 d1 29 0 0 57 0 24 cm BI /IM true /W 29 /H 57 /BPC 1 /F /CCF /DP <> ID & ¹òø xFmëxA?Oþ•÷ÿÿÿÿÿÿÿÿÿÿÿÿÿþ$ºòjÿ†h}ÿÿ¿þÿÿÿ EI endstream endobj 310 0 obj <> stream 24 0 0 0 0 0 d1 endstream endobj 311 0 obj <> stream 0 0 0 40 45 99 d1 45 0 0 59 0 40 cm BI /IM true /W 45 /H 59 /BPC 1 /F /CCF /DP <> ID &¦f _Aàû‹aï‡ÛþäÕ'Æ÷þÿ~ðŸéÿþŸéÿ3]aðƒÿ…¾Ÿýo§ÖÿéÿÒoýaÿÂþoïTõ åêÿÿø€ EI endstream endobj 312 0 obj <> stream 0 0 0 39 73 79 d1 73 0 0 40 0 39 cm BI /IM true /W 73 /H 40 /BPC 1 /F 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Don't erase the collaboration statement! Thanks. INSTRUCTIONS: (if this is not ps1, use the right file name) Clip out the ********* INSERT HERE ********* bits below and insert appropriate TeX code. Once you are done with your file, run ``latex ps1.tex'' from the Athena shell. If your TeX code is clean, the latex will exit back to a prompt. Once this is done, run ``dvips ps1.dvi'' which should print your file to the nearest printer. There will be residual files called ps1.log, ps1.aux, and ps1.dvi. All these can be deleted, but do not delete ps1.tex. \fi \documentclass[11pt]{article} \usepackage{amsfonts} \usepackage{latexsym} \setlength{\oddsidemargin}{.25in} \setlength{\evensidemargin}{.25in} \setlength{\textwidth}{6in} \setlength{\topmargin}{-0.4in} \setlength{\textheight}{8.5in} \newcommand{\handout}[5]{ \renewcommand{\thepage}{#1-\arabic{page}} \noindent \begin{center} \framebox{ \vbox{ \hbox to 5.78in { {\bf 6.875J/18.425J Cryptography and Cryptanalysis} \hfill #2 } \vspace{4mm} \hbox to 5.78in { {\Large \hfill #5 \hfill} } \vspace{2mm} \hbox to 5.78in { {\it #3 \hfill #4} } } } \end{center} \vspace*{4mm} } \newcommand{\al}{\alpha} \newcommand{\Z}{\mathbb Z} \newcommand{\N}{\mathbb N} \begin{document} \handout{P4}{DUE: October 15, 2001}{Instructor: Anna Lysyanskaya} {Teaching Assistant: Moses Liskov} {*********** INSERT YOUR NAME HERE ***********: PS 3} I collaborated with ******** INSERT COLLABORATORS OR `nobody' HERE **********. \section*{Problem 1: Sampling Domain Security} Consider the following definition of security, which we call sampling domain security (SD security). Let $(G, E, D)$ be a public-key cryptosystem, and let $\mathcal{M} = \{M_n\}$ be a sequence of message spaces, each with a uniform distribution. $(G, E, D)$ is SD secure if there exists a probabilistic polynomial time algorithm $S$ (for ``sampling'') such that \begin{tabbing} $\forall \{A_n\}~\exists f \in neg$ such that $\forall k \in \N$\\ $\Pr[$\=$(PK, SK) \leftarrow G(1^k); m \leftarrow M_k; c_0 \leftarrow E(PK, m); c_1 \leftarrow S(1^k, PK);$\\ \>$b \leftarrow \{0, 1\}; b' \leftarrow A_k(PK, c_b, m): b'=b] \leq 1/2 + f(k)$ \end{tabbing} \bigskip {\bf (a)} If $M_n = \{0,1\}$ for every $n$, then GM-security implies SD security. \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 1a SOLUTION HERE *************** \bigskip {\bf (b)} If $M_n = \{0,1\}$ for every $n$, then SD security implies GM-security. \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 1b SOLUTION HERE *************** \bigskip {\bf (c)} If $M_n = \{0,1\}^n$ for every $n$, then GM-security implies SD security. \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 1c SOLUTION HERE *************** \bigskip {\bf (d)} If $M_n = \{0,1\}^n$ for every $n$, then SD security implies GM-security. \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 1d SOLUTION HERE *************** \section*{Problem 2: Random Message Security} Consider the following definition of security, which we call random message security (RM security). We say that a cryptosystem $(G, E, D)$ on a sequence of message spaces $\mathcal{M} = \{M_n\}$ is RM secure if \begin{tabbing} $\forall \{A_n\}~\exists f \in neg$ such that $\forall k \in \N$\\ $\Pr[$\=$(PK, SK) \leftarrow G(1^k); (\alpha, m) \leftarrow A_k(PK, 1^k); r \leftarrow M_k;$\\ \>$c_0 \leftarrow E(PK, m); c_1 \leftarrow E(PK, r); b \leftarrow \{0,1\};$\\ \>$b' \leftarrow A_k(PK, c_b, m, r): b'=b] \leq 1/2 + f(k)$ \end{tabbing} Prove or disprove that for all polynomials $p$, for the sequence of message spaces $\{M_n = \{0,1\}^{p(n)}\}$, RM security is equivalent to GM security. \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 2 SOLUTION HERE ************** \section*{Problem 3: Security of ElGamal-A} Prove that under the decisional Diffie-Hellman assumption, the ElGamal-A cryptosystem is GM-secure. ***************** INSERT PROBLEM 3 SOLUTION HERE ************** \end{document} handouts/ho4.ps0100664003325600237220000051306607354707332013206 0ustar 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b(to)j(GM-securit)m(y)-8 b(.)1889 5416 y(4-2)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF handouts/ps3.tex0100664003325600237220000001021207354710037013354 0ustar mliskovta6875\iffalse NOTE: If you do not collaborate with anyone, simply remove the '**** INSERT COLLABORATORS HERE****' and replace it with 'nobody.' Don't erase the collaboration statement! Thanks. INSTRUCTIONS: (if this is not ps1, use the right file name) Clip out the ********* INSERT HERE ********* bits below and insert appropriate TeX code. Once you are done with your file, run ``latex ps1.tex'' from the Athena shell. If your TeX code is clean, the latex will exit back to a prompt. Once this is done, run ``dvips ps1.dvi'' which should print your file to the nearest printer. There will be residual files called ps1.log, ps1.aux, and ps1.dvi. All these can be deleted, but do not delete ps1.tex. \fi \documentclass[11pt]{article} \usepackage{amsfonts} \usepackage{latexsym} \setlength{\oddsidemargin}{.25in} \setlength{\evensidemargin}{.25in} \setlength{\textwidth}{6in} \setlength{\topmargin}{-0.4in} \setlength{\textheight}{8.5in} \newcommand{\handout}[5]{ \renewcommand{\thepage}{#1-\arabic{page}} \noindent \begin{center} \framebox{ \vbox{ \hbox to 5.78in { {\bf 6.875J/18.425J Cryptography and Cryptanalysis} \hfill #2 } \vspace{4mm} \hbox to 5.78in { {\Large \hfill #5 \hfill} } \vspace{2mm} \hbox to 5.78in { {\it #3 \hfill #4} } } } \end{center} \vspace*{4mm} } \newcommand{\al}{\alpha} \newcommand{\Z}{\mathbb Z} \newcommand{\N}{\mathbb N} \begin{document} \handout{P3}{DUE: October 3, 2001}{Instructor: Anna Lysyanskaya} {Teaching Assistant: Moses Liskov} {*********** INSERT YOUR NAME HERE ***********: PS 3} I collaborated with ******** INSERT COLLABORATORS OR `nobody' HERE **********. \section*{Problem 1: Random-bit hybrid argument} Recall the theorem from lecture, that if $(G', E', D')$ is a GM-secure cryptosystem operating on 1-bit messages, then $(G, E, D)$ is a GM-secure cryptosystem operating on $l$-bit messages where $l$ is some polynomial on $k$, and where $(G, E, D)$ just uses $(G', E', D')$ to encrypt a longer message bit-by-bit. That is, \begin{tabbing} XXX\=\kill $G(1^k)$:\\ \>1. Return output of $G'(1^k)$\\ $E(PK, m = b_1 \circ b_2 \circ \ldots \circ b_l)$: \\ \>1. For each $1 \leq i \leq l$, compute $c_i = E'(PK, m_i)$\\ \>2. Return $c_1 \circ c_2 \circ \ldots \circ c_l$\\ $D(SK, c = c_1 \circ \ldots c_l)$: \\ \>1. For each $1 \leq i \leq l$, compute $b_i = D'(SK, c_i)$\\ \>2. Return $m = b_1 \circ \ldots \circ b_l.$ \end{tabbing} Consider the following reduction $R'$. \begin{tabbing} XXX\=\kill $R'(m_0, m_1, A, PK, c, k)$:\\ \>1. Pick $i$ at random from the range $[0, l]$.\\ \>2. For $1 \leq j \leq i$, compute $c'_j = E'(PK, b'_j)$\\ \>3. Set $c'_{i+1} = c$\\ \>4. For $i+1 \leq j \leq l$, compute $c'_j = E'(PK, b_j)$\\ \>5. Return $b' = A(PK, c'_1 \circ \ldots \circ c'_l, m_0, m_1)$ \end{tabbing} Prove that if $A$ is an adversary that breaks the GM-security of $(G, E, D)$ on messages $m_0, m_1$, then $R'$ provides an adversary that breaks the GM-security of $(G', E', D')$. \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 1 SOLUTION HERE *************** \section*{Problem 2: Moses Security} Recall that the Hamming weight of a binary string is the number of 1s in that string. Let $\mathcal{H}_k^i$ be the set of all strings of length exactly $k$ and of Hamming weight exactly $i$. We say that a cryptosystem $(G, E, D)$ is $i$-Moses secure if \begin{tabbing} $\forall \{A_n\}~\exists f \in neg$ such that $\forall k > i \in \N~\forall m_0, m_1 \in \{0,1\}^k$ such that $m_0 \oplus m_1 \in \mathcal{H}_k^i$\\ $\Pr[$\=$(PK, SK) \leftarrow G(1^k); b \leftarrow \{0, 1\}; c \leftarrow E(PK, m_b);$ \\ \>$b' \leftarrow A_k(PK, c, m_0, m_1): b' = b] \leq 1/2 + f(k)$ \end{tabbing} \bigskip{\bf (a)} Prove or disprove: For all odd $i$, $i$-Moses security is equivalent to GM-security. \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 2a SOLUTION HERE *************** \bigskip{\bf (b)} Prove or disprove: For all even $i$, $i$-Moses security is equivalent to GM-security. \bigskip {\bf Solution:} \bigskip \noindent ***************** INSERT PROBLEM 2b SOLUTION HERE *************** \end{document} handouts/ho5.ps0100664003325600237220000062555507354707764013227 0ustar mliskovta6875%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software 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3183 y Fj(0)1587 3169 y Fm( )25 b Fn(E)5 b Fq(\()p Fn(P)13 b(K)q(;)i(m)p Fq(\);)g Fn(c)2193 3183 y Fj(1)2260 3169 y Fm( )26 b Fn(S)5 b Fq(\(1)2518 3136 y Fl(k)2561 3169 y Fn(;)15 b(P)e(K)7 b Fq(\);)273 3282 y Fn(b)25 b Fm( )g(f)p Fq(0)p Fn(;)15 b Fq(1)p Fm(g)p Fq(;)g Fn(b)752 3249 y Fi(0)803 3282 y Fm( )26 b Fn(A)988 3297 y Fl(k)1030 3282 y Fq(\()p Fn(P)13 b(K)q(;)i(c)1293 3297 y Fl(b)1329 3282 y Fn(;)g(m)p Fq(\))26 b(:)g Fn(b)1600 3249 y Fi(0)1648 3282 y Fq(=)f Fn(b)p Fq(])h Fm(\024)f Fq(1)p Fn(=)p Fq(2)c(+)f Fn(f)10 b Fq(\()p Fn(k)s Fq(\))291 3495 y(The)28 b(in)m(tuition)f(b)s(ehind)e (this)j(de\014nition)e(is)i(that)h(a)g(cryptosystem)g(is)e(secure)i(if) e(no)i(adv)m(ersary)g(can)150 3607 y(distinguish)j(the)37 b(encryption)e(of)h(a)h(random)e(message)j(from)d(a)i(random)f(sample)f (from)h(the)g(domain)150 3720 y(sampled)29 b(b)m(y)h Fn(S)5 b Fq(.)41 b(Pro)m(v)m(e)32 b(or)e(dispro)m(v)m(e)g(eac)m(h)h(of) g(the)f(follo)m(wing.)291 3933 y Fr(\(a\))g Fq(If)f Fn(M)632 3947 y Fl(n)705 3933 y Fq(=)c Fm(f)p Fq(0)p Fn(;)15 b Fq(1)p Fm(g)32 b Fq(for)f(ev)m(ery)g Fn(n)p Fq(,)f(then)g(GM-securit)m (y)h(implies)d(SD)i(securit)m(y)-8 b(.)291 4145 y Fr(\(b\))30 b Fq(If)g Fn(M)640 4159 y Fl(n)712 4145 y Fq(=)25 b Fm(f)p Fq(0)p Fn(;)15 b Fq(1)p Fm(g)33 b Fq(for)d(ev)m(ery)h Fn(n)p Fq(,)f(then)g(SD)g(securit)m(y)h(implies)c(GM-securit)m(y)-8 b(.)291 4358 y Fr(\(c\))30 b Fq(If)g Fn(M)628 4372 y Fl(n)701 4358 y Fq(=)25 b Fm(f)p Fq(0)p Fn(;)15 b Fq(1)p Fm(g)1017 4325 y Fl(n)1096 4358 y Fq(for)30 b(ev)m(ery)h Fn(n)p Fq(,)f(then)g(GM-securit)m(y)i(implies)27 b(SD)j(securit)m(y)-8 b(.)291 4571 y Fr(\(d\))30 b Fq(If)g Fn(M)640 4585 y Fl(n)712 4571 y Fq(=)25 b Fm(f)p Fq(0)p Fn(;)15 b Fq(1)p Fm(g)1028 4538 y Fl(n)1108 4571 y Fq(for)30 b(ev)m(ery)h Fn(n)p Fq(,)f(then)g(SD)g(securit)m(y)g(implies)e(GM-securit)m(y)-8 b(.)1889 5416 y(5-1)p eop %%Page: 2 2 2 1 bop 150 159 a Fh(Problem)45 b(2:)61 b(Random)45 b(Message)h (Securit)l(y)150 361 y Fq(Consider)32 b(the)j(follo)m(wing)e (de\014nition)e(of)k(securit)m(y)-8 b(,)35 b(whic)m(h)e(w)m(e)i(call)f (random)f(message)i(securit)m(y)f(\(RM)150 474 y(securit)m(y\).)k(W)-8 b(e)21 b(sa)m(y)g(that)g(a)g(cryptosystem)g(\()p Fn(G;)15 b(E)5 b(;)15 b(D)s Fq(\))22 b(on)e(a)h(sequence)g(of)f(message)i (spaces)f Fm(M)k Fq(=)g Fm(f)p Fn(M)3657 488 y Fl(n)3705 474 y Fm(g)150 587 y Fq(is)k(RM)i(secure)f(if)150 800 y Fm(8f)p Fn(A)314 814 y Fl(n)361 800 y Fm(g)h(9)p Fn(f)j Fm(2)25 b Fn(neg)34 b Fq(suc)m(h)c(that)h Fm(8)p Fn(k)d Fm(2)c Fk(N)150 913 y Fq(Pr)o([\()p Fn(P)13 b(K)q(;)i(S)5 b(K)i Fq(\))27 b Fm( )f Fn(G)p Fq(\(1)971 880 y Fl(k)1015 913 y Fq(\);)15 b(\()p Fn(\013;)g(m)p Fq(\))27 b Fm( )e Fn(A)1549 928 y Fl(k)1592 913 y Fq(\()p Fn(P)13 b(K)q(;)i Fq(1)1861 880 y Fl(k)1906 913 y Fq(\);)g Fn(r)28 b Fm( )d Fn(M)2254 928 y Fl(k)2297 913 y Fq(;)273 1026 y Fn(c)312 1040 y Fj(0)377 1026 y Fm( )g Fn(E)5 b Fq(\()p Fn(P)13 b(K)q(;)i(m)p Fq(\);)g Fn(c)983 1040 y Fj(1)1050 1026 y Fm( )25 b Fn(E)5 b Fq(\()p Fn(P)13 b(K)q(;)i(r)s Fq(\);)g Fn(b)28 b Fm( )d(f)p Fq(0)p Fn(;)15 b Fq(1)p Fm(g)p Fq(;)273 1139 y Fn(b)312 1106 y Fi(0)360 1139 y Fm( )25 b Fn(A)544 1154 y Fl(k)587 1139 y Fq(\()p Fn(P)13 b(K)q(;)i(c)850 1154 y Fl(b)886 1139 y Fn(;)g(m;)g(r)s Fq(\))26 b(:)g Fn(b)1241 1106 y Fi(0)1289 1139 y Fq(=)f Fn(b)p Fq(])h Fm(\024)f Fq(1)p Fn(=)p Fq(2)c(+)f Fn(f)10 b Fq(\()p Fn(k)s Fq(\))291 1351 y(The)35 b(in)m(tuition)f(b)s(ehind)f(this)h (de\014nition)g(is)h(that)h(for)f(an)m(y)h(\014xed)g(message)g Fn(m)g Fq(it)f(is)g(infeasible)e(to)150 1464 y(distinguish)27 b(b)s(et)m(w)m(een)k(an)f(encryption)f(of)i Fn(m)f Fq(and)g(an)g (encryption)f(of)i(some)f(random)g(message)i Fn(r)s Fq(.)291 1577 y(Pro)m(v)m(e)f(or)e(dispro)m(v)m(e)f(that)i(for)f(all)f(p)s (olynomials)f Fn(p)p Fq(,)i(for)g(the)g(sequence)h(of)f(message)i (spaces)e Fm(f)p Fn(M)3606 1591 y Fl(n)3679 1577 y Fq(=)150 1690 y Fm(f)p Fq(0)p Fn(;)15 b Fq(1)p Fm(g)370 1657 y Fl(p)p Fj(\()p Fl(n)p Fj(\))510 1690 y Fm(g)p Fq(,)31 b(RM)f(securit)m(y)g(is)g(equiv)-5 b(alen)m(t)30 b(to)h(GM)g(securit)m (y)-8 b(.)150 1976 y Fh(Problem)45 b(3:)61 b(Securit)l(y)45 b(of)g(ElGamal-A)150 2179 y Fq(Assume)26 b(that)h(one)f(can,)i(in)d (probabilistic)e(p)s(olynomial)h(time,)j(compute,)h(on)e(input)e(1)3137 2146 y Fl(k)3180 2179 y Fq(,)k(a)f Fn(k)s Fq(-bit)e(prime)150 2292 y Fn(p)32 b Fq(suc)m(h)g(that)i Fn(p)28 b Fq(=)h(2)p Fn(q)c Fq(+)c(1)33 b(where)f Fn(q)j Fq(is)d(also)g(prime.)46 b(Consider)31 b(the)h(follo)m(wing)f(cryptosystem,)j(whic)m(h)150 2405 y(is)29 b(a)i(v)-5 b(arian)m(t)31 b(of)f(the)h(ElGamal)e (cryptosystem)i(\(to)h(b)s(e)d(attributed)h(later\),)h(call)f(it)f (ElGamal-A.)291 2518 y(Alice)24 b(selects)h(at)g(random)f(\(1\))h(a)g Fn(k)s Fq(-bit)f(prime)f Fn(p)i Fq(=)g(2)p Fn(q)12 b Fq(+)c(1)25 b(where)f Fn(q)j Fq(is)d(also)g(prime,)h(\(2\))g(an)f (elemen)m(t)150 2631 y Fn(g)35 b Fq(in)30 b Fn(Z)404 2598 y Fi(\003)397 2653 y Fl(p)474 2631 y Fq(of)i(order)f Fn(q)s Fq(,)g(and)g(\(3\))i(an)e Fn(x)g Fq(in)f Fk(Z)1626 2598 y Fi(\003)1626 2653 y Fl(q)1661 2631 y Fq(.)44 b(Then,)31 b(she)g(sets)h(her)f(public)d(k)m(ey)k(to)h(b)s(e)d(\()p Fn(p;)15 b(g)s(;)g(h)p Fq(\),)34 b(where)150 2744 y Fn(h)26 b Fq(=)f Fn(g)370 2711 y Fl(x)439 2744 y Fq(mo)s(d)g Fn(p)p Fq(,)30 b(and)g(her)g(secret)h(k)m(ey)g(to)g(b)s(e)f Fn(x)p Fq(.)291 2857 y(Alice's)22 b(public)d(k)m(ey)k(restricts)f(the)g (message)h(space)g(to)g(consist)e(of)i(all)e(elemen)m(ts)h(in)f Fn(Z)3197 2824 y Fi(\003)3190 2879 y Fl(p)3258 2857 y Fq(ha)m(ving)g(order)150 2970 y Fn(q)s Fq(.)52 b(T)-8 b(o)35 b(send)e(suc)m(h)h(a)h(message)g Fn(m)f Fq(to)h(Alice,)g(one)f (pic)m(ks)g(an)g Fn(r)g Fm( )e Fk(Z)2546 2984 y Fl(q)2613 2970 y Fq(and)i(computes)g(his)f(ciphertext)150 3083 y Fn(c)26 b Fq(=)f(\()p Fn(g)392 3050 y Fl(r)456 3083 y Fq(mo)s(d)f Fn(p;)15 b(h)793 3050 y Fl(r)832 3083 y Fn(m)25 b Fq(mo)s(d)f Fn(p)p Fq(\).)291 3195 y(T)-8 b(o)34 b(decrypt)f Fn(c)d Fq(=)g(\()p Fn(a;)15 b(b)p Fq(\),)36 b(Alice)c(computes)i Fn(b=)p Fq(\()p Fn(a)1997 3162 y Fl(x)2042 3195 y Fq(\))25 b(mo)s(d)g Fn(p)p Fq(.)49 b(\(Exercise:)e(v)m (erify)32 b(that)i(this)e(indeed)150 3308 y(pro)s(duces)d Fn(m)p Fq(.\))291 3421 y(Call)k(the)h(pro)s(cedure)f(for)h(generating)h Fn(p)f Fq(and)g Fn(g)j Fq(describ)s(ed)32 b(ab)s(o)m(v)m(e)k Fm(D)s(L)o Fq(.)53 b(The)33 b Fo(de)-5 b(cisional)38 b(Di\016e-)150 3534 y(Hel)5 b(lman)31 b Fq(assumption)e(is)g(as)i (follo)m(ws:)150 3747 y Fm(8f)p Fn(A)314 3761 y Fl(n)361 3747 y Fm(g)g(9)p Fn(f)j Fm(2)25 b Fn(neg)34 b Fq(suc)m(h)c(that)h Fm(8)p Fn(k)d Fm(2)c Fk(N)150 3860 y Fq(Pr)o([\()p Fn(p;)15 b(g)s Fq(\))27 b Fm( )f(D)s(L)o Fq(\(1)833 3827 y Fl(n)881 3860 y Fq(\);)15 b Fn(x)26 b Fm( )f Fk(Z)1215 3827 y Fi(\003)1215 3882 y Fl(q)1250 3860 y Fq(;)15 b Fn(y)28 b Fm( )e Fk(Z)1545 3827 y Fi(\003)1545 3882 y Fl(q)273 3974 y Fn(b)f Fm( )g(f)p Fq(0)p Fn(;)15 b Fq(1)p Fm(g)p Fq(;)g Fn(z)755 3988 y Fj(0)823 3974 y Fm( )25 b Fn(xy)j Fq(mo)s(d)c Fn(q)s Fq(;)15 b Fn(z)1389 3988 y Fj(1)1455 3974 y Fm( )25 b Fk(Z)1636 3941 y Fi(\003)1636 3997 y Fl(q)1671 3974 y Fq(;)15 b Fn(b)1750 3941 y Fi(0)1799 3974 y Fm( )25 b Fn(A)1983 3989 y Fl(k)2026 3974 y Fq(\()p Fn(p;)15 b(g)s(;)g(g)2279 3941 y Fl(x)2325 3974 y Fn(;)g(g)2411 3941 y Fl(y)2453 3974 y Fn(;)g(g)2539 3941 y Fl(z)2572 3953 y Fg(b)2609 3974 y Fq(\))25 b(:)h Fn(b)2759 3941 y Fi(0)2807 3974 y Fq(=)f Fn(b)p Fq(])h Fm(\024)f Fq(1)p Fn(=)p Fq(2)c(+)f Fn(f)10 b Fq(\()p Fn(k)s Fq(\))291 4187 y(where)30 b Fm(f)p Fn(A)667 4201 y Fl(n)715 4187 y Fm(g)g Fq(is)g(PPTF.)1156 4154 y Fj(1)291 4299 y Fq(Pro)m(v)m(e)35 b(that)e(under)f(the)i(decisional)d(Di\016e-Hellman)h(assumption,)h (the)h(ElGamal-A)f(cryptosys-)150 4412 y(tem)e(is)e(GM-secure.)p 150 4808 1440 4 v 254 4861 a Ff(1)288 4893 y Fe(In)22 b(other)g(w)n(ords,)j(it)d(is)h(hard,)g(giv)n(en)f(four)h(elemen)n(ts)f (of)h Fd(Z)1968 4861 y Fc(\003)1968 4906 y Fg(p)1998 4893 y Fe(,)h(\()p Fb(g)s(;)12 b(g)2183 4901 y Ff(1)2217 4893 y Fb(;)h(g)2288 4901 y Ff(2)2323 4893 y Fb(;)g(g)2394 4901 y Ff(3)2428 4893 y Fe(\),)23 b(eac)n(h)g(of)g(order)g Fb(q)s 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Fk(A)2251 512 y Fh(k)2327 497 y Fn(is)g(go)s(o)s(d,) i(w)m(e)f(use)g(it)f(rep)s(eatedly)g(to)i(get)150 610 y(a)g(high)e(lev)m(el)i(of)g(certain)m(t)m(y)g(ab)s(out)g(the)g(answ)m (er.)53 b(If)34 b Fk(A)2079 625 y Fh(k)2157 610 y Fn(is)g(not)h(go)s(o) s(d,)g(w)m(e)h(basically)d(are)i(getting)g(no)150 723 y(help)27 b(from)g Fk(A)625 738 y Fh(k)696 723 y Fn(so)h(w)m(e)g(migh)m (t)g(as)g(w)m(ell)f(just)g(guess)h(the)g(most)h(lik)m(ely)d(correct)j (answ)m(er,)g(namely)e(1)h(\(since)150 836 y Fk(q)g Fi(\024)d Fn(1)p Fk(=)p Fn(2\).)42 b(Let)31 b Fk(S)771 850 y Fh(q)r Ff(\000)p Fh(GM)994 836 y Fn(\()p Fk(k)s(;)15 b(A)1187 803 y Ff(0)1211 836 y Fn(\))31 b(b)s(e)f(the)g(ev)m(en)m(t)i(that)473 1153 y(\(\()p Fk(n;)15 b(x)p Fn(\))p Fk(;)g(S)5 b(K)i Fn(\))26 b Fi( )f Fk(G)p Fn(\(1)1238 1116 y Fh(k)1282 1153 y Fn(\);)15 b Fk(b)26 b Fi( )g(f)p Fk(q)i Fn(:)e(0)p Fk(;)15 b Fn(1)p Fi(g)p Fn(;)g Fk(c)27 b Fi( )f Fk(E)5 b Fn(\()p Fk(n;)15 b(x;)g(b)p Fn(\);)g Fk(b)2549 1116 y Ff(0)2599 1153 y Fi( )25 b 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Fk(k)s Fn(\))26 b(:)g Fk(\015)5 b Fn(\()p Fk(\036;)15 b(n)p Fn(\)].)40 b(No)m(w,)150 1927 y(let's)34 b(assume)f(for)h(the)f(momen)m(t)i(that)f (if)f Fk(n)g Fn(is)g(go)s(o)s(d,)h(w)m(e)g(alw)m(a)m(ys)h(use)e(steps)h (3)g(and)f(4)h(to)g(output)f(our)150 2040 y(answ)m(er)d(and)g(if)f Fk(n)h Fn(is)g(not)g(go)s(o)s(d,)h(w)m(e)g(output)f(1.)41 b(Then,)951 2266 y Fk(P)d Fn(=)25 b Fk(P)1201 2280 y Fh(\015)1261 2266 y Fn(Pr)o([)p Fk(S)1439 2280 y Fh(q)r Ff(\000)p Fh(GM)1662 2266 y Fn(\()p Fk(k)s(;)15 b(A)1855 2228 y Ff(0)1879 2266 y Fn(\))p Fi(j)p Fk(\015)5 b Fn(\()p Fk(\036;)15 b(n)p Fn(\)])22 b(+)e(\(1)h Fi(\000)f Fk(P)2598 2280 y Fh(\015)2643 2266 y Fn(\)\(1)h Fi(\000)f Fk(q)s Fn(\))291 2433 y(where)30 b Fk(\036)g Fn(is)g(the)g(same)h(in)e Fk(\015)5 b Fn(\()p Fk(\036;)15 b(n)p Fn(\))32 b(and)d(in)g(the)i (algorithm)e(run)g(b)m(y)i Fk(A)2738 2400 y Ff(0)2761 2433 y Fn(.)150 2645 y Fo(Theorem)j(1)46 b Fl(If)g Fk(P)859 2659 y Fh(\015)954 2645 y Fi(\025)51 b Fn(1)p Fk(=p)p Fn(\()p Fk(k)s Fn(\))d Fl(for)f(some)g(p)-5 b(olynomial)50 b Fk(p)p Fl(,)g(then)d(for)g(lar)-5 b(ge)47 b(enough)g Fk(g)s Fn(\()p Fk(k)s Fn(\))p Fl(,)52 b Fk(P)63 b Fi(\025)150 2758 y Fn(1)21 b Fi(\000)f Fk(q)j Fn(+)d(1)p Fk(=p)598 2725 y Ff(0)621 2758 y Fn(\()p Fk(k)s Fn(\))34 b Fl(for)f(some)g(nonne) -5 b(gligible)33 b(function)g Fn(1)p Fk(=p)2185 2725 y Ff(0)2242 2758 y Fl(and)g(su\016ciently)g(lar)-5 b(ge)34 b Fk(k)s Fl(.)291 2971 y(Pr)-5 b(o)g(of.)45 b Fn(The)31 b(basic)g(idea)g(here)h(is)e(that)i(if)f(there)g(are)h(a)g(p)s (olynomial)d(fraction)i(of)h(go)s(o)s(d)f Fk(n)p Fn(,)h(and)e(w)m(e)150 3084 y(alw)m(a)m(ys)d(get)g(the)f(adv)-5 b(an)m(tage)29 b(the)d(adv)m(ersary)g(pro)m(vides)g(on)g(those)g Fk(n)p Fn(,)h(then)f(w)m(e)h(will)c(ha)m(v)m(e)28 b(enough)e(of)g(an)150 3196 y(adv)-5 b(an)m(tage)36 b(to)e(break)f Fk(q)s Fn(-GM)i(securit)m (y)-8 b(.)50 b(T)-8 b(o)34 b(pro)m(v)m(e)h(this,)e(w)m(e)h(pro)m(v)m(e) h(that)f(Pr)o([)p Fk(S)2945 3210 y Fh(q)r Ff(\000)p Fh(GM)3168 3196 y Fn(\()p Fk(k)s(;)15 b(A)3361 3164 y Ff(0)3385 3196 y Fn(\))p Fi(j)p Fk(\015)5 b Fn(\()p Fk(n)p Fn(\)])32 b Fi(\025)150 3309 y Fn(1)15 b Fi(\000)g Fn(2)341 3276 y Ff(\000)p Fh(k)439 3309 y Fn(.)40 b(If)28 b Fk(n)f Fn(is)g(go)s(o)s(d,)h(and)g(w)m(e)g(use)f(steps)h(3)g(and)g(4)g(to)g (output)g(our)f(answ)m(er,)i(w)m(e)f(ha)m(v)m(e)h(a)f(probabilit)m(y) 150 3422 y(of)j(ab)s(out)f(\(1)21 b Fi(\000)f Fn(1)p Fk(=\036)p Fn(\()p Fk(k)s Fn(\)\))1005 3389 y Fh( )r Fg(\()p Fh(k)r Fg(\))1153 3422 y Fn(,)30 b(whic)m(h)g(for)g(su\016cien) m(tly)f(large)h Fk( )k Fn(is)29 b(at)j(least)e(1)21 b Fi(\000)f Fn(2)2995 3389 y Ff(\000)p Fh(k)3093 3422 y Fn(.)291 3535 y(No)m(w,)1081 3880 y Fk(P)96 b Fi(\025)1459 3818 y Fn(1)p 1398 3859 167 4 v 1398 3942 a Fk(p)p Fn(\()p Fk(k)s Fn(\))1575 3880 y(\(1)21 b Fi(\000)f Fn(2)1812 3842 y Ff(\000)p Fh(k)1910 3880 y Fn(\))g(+)2056 3751 y Fd(\022)2123 3880 y Fn(1)h Fi(\000)2350 3818 y Fn(1)p 2290 3859 V 2290 3942 a Fk(p)p Fn(\()p Fk(k)s Fn(\))2466 3751 y Fd(\023)2548 3880 y Fn(\(1)g Fi(\000)f Fk(q)s Fn(\))1235 4140 y(=)1398 4078 y(1)h Fi(\000)f Fn(2)1600 4045 y Ff(\000)p Fh(k)p 1398 4119 300 4 v 1465 4202 a Fk(p)p Fn(\()p Fk(k)s Fn(\))1728 4140 y(+)g(1)h Fi(\000)e Fk(q)k Fi(\000)2140 4078 y Fn(1)e Fi(\000)f Fk(q)p 2140 4119 201 4 v 2157 4202 a(p)p Fn(\()p Fk(k)s Fn(\))1235 4399 y(=)82 b(1)21 b Fi(\000)f Fk(q)j Fn(+)1710 4337 y Fk(q)g Fi(\000)d Fn(2)1910 4304 y Ff(\000)p Fh(k)p 1710 4378 298 4 v 1775 4461 a Fk(p)p Fn(\()p Fk(k)s Fn(\))1235 4637 y(=)82 b(1)21 b Fi(\000)f Fk(q)j Fn(+)1782 4576 y(1)p 1710 4616 190 4 v 1710 4699 a Fk(p)1756 4673 y Ff(0)1779 4699 y Fn(\()p Fk(k)s Fn(\))1910 4637 y Fk(;)291 4901 y Fn(where)30 b Fk(p)600 4868 y Ff(0)623 4901 y Fn(\()p Fk(k)s Fn(\))c(=)f Fk(p)p Fn(\()p Fk(k)s Fn(\))p Fk(=)p Fn(\()p Fk(q)f Fi(\000)19 b Fn(2)1311 4868 y Ff(\000)p Fh(k)1409 4901 y Fn(\).)41 b(F)-8 b(or)30 b(su\016cien)m(tly)f(large)h Fk(k)s Fn(,)h Fk(q)d(>)d Fn(2)2665 4868 y Ff(\000)p Fh(k)2763 4901 y Fn(,)30 b(so)g(for)g(suc)m(h)g Fk(k)s Fn(,)g(1)p Fk(=p)3514 4868 y Ff(0)3538 4901 y Fn(\()p Fk(k)s Fn(\))h(is)150 5014 y(a)i(p)s(ositiv)m(e)f(function)f(whic)m(h)h(is)f(nonnegligible,)g (since)h(it)g(approac)m(hes)h Fk(p)p Fn(\()p Fk(k)s Fn(\))p Fk(=q)k Fn(as)c Fk(k)j Fn(go)s(es)d(to)g(in\014nit)m(y)-8 b(.)150 5127 y Fc(2)1889 5416 y Fn(7-2)p eop %%Page: 3 3 3 2 bop 150 159 a Fo(Theorem)34 b(2)46 b Fk(P)750 173 y Fh(\015)820 159 y Fi(\025)25 b Fn(1)p Fk(=p)p Fn(\()p Fk(k)s Fn(\))34 b Fl(for)f(some)g(p)-5 b(olynomial)36 b Fk(p)p Fl(.)291 350 y(Pr)-5 b(o)g(of.)42 b Fn(De\014ne)30 b Fk(S)908 364 y Fh(QR)1021 350 y Fn(\()p Fk(k)s(;)15 b(A)p Fn(\))32 b(to)f(b)s(e)f(the)h(ev)m(en)m(t)873 575 y Fk(n)25 b Fi( )g Fk(R)q(S)5 b(AM)10 b Fn(\()p Fk(k)s Fn(\);)15 b Fk(c)27 b Fi( )e Fk(J)1758 589 y Fg(1)1798 575 y Fn(\()p Fk(n)p Fn(\);)15 b Fk(b)26 b Fi( )f Fk(A)p Fn(\()p Fk(n;)15 b(c)p Fn(\))27 b(:)e Fk(b)g Fn(=)g Fk(QC)7 b Fn(\()p Fk(n;)15 b(c)p Fn(\))p Fk(:)291 731 y Fn(No)m(w,)31 b(w)m(e)g(kno)m(w)g(that)g(Pr)o([)p Fk(S)1270 745 y Fh(QR)1383 731 y Fn(\()p Fk(k)s(;)15 b(A)p Fn(\)])27 b Fi(\025)e Fn(1)p Fk(=)p Fn(2)d(+)e(1)p Fk(=\036)p Fn(\()p Fk(k)s Fn(\))32 b(for)e(some)h(p)s(olynomial)d Fk(\036)p Fn(.)291 844 y(W)-8 b(e)32 b(kno)m(w)e(that)488 1070 y(Pr)o([)p Fk(S)666 1084 y Fh(QR)780 1070 y Fn(\()p Fk(k)s(;)15 b(A)p Fn(\)])27 b(=)e(Pr)o([)p Fk(S)1334 1084 y Fh(QR)1447 1070 y Fn(\()p Fk(k)s(;)15 b(A)p Fn(\))p Fi(j)p Fk(\015)5 b Fn(\()p Fk(\036;)15 b(n)p Fn(\)])p Fk(P)2054 1084 y Fh(\015)2122 1070 y Fn(+)20 b(Pr[)p Fk(S)2392 1084 y Fh(QR)2505 1070 y Fn(\()p Fk(k)s(;)15 b(A)p Fn(\))p Fi(j)p 2758 991 272 4 v Fk(\015)5 b Fn(\()p Fk(\036;)15 b(n)p Fn(\))r(]\(1)22 b Fi(\000)e Fk(P)3307 1084 y Fh(\015)3351 1070 y Fn(\))p Fk(:)291 1226 y Fn(F)-8 b(urthermore,)35 b(w)m(e)f(kno)m(w)g(that)h(Pr)o([)p Fk(S)1605 1240 y Fh(QR)1718 1226 y Fn(\()p Fk(k)s(;)15 b(A)p Fn(\))p Fi(j)p Fk(\015)5 b Fn(\()p Fk(\036;)15 b(n)p Fn(\)])34 b Fi(\024)d Fn(1,)k(b)s(ecause)f(it)g(is)e(a)j(probabilit)m(y)-8 b(,)33 b(and)150 1348 y(w)m(e)39 b(kno)m(w)g(that)h(Pr)o([)p Fk(S)922 1362 y Fh(QR)1036 1348 y Fn(\()p Fk(k)s(;)15 b(A)p Fn(\))p Fi(j)p 1289 1269 V Fk(\015)5 b Fn(\()p Fk(\036;)15 b(n)p Fn(\))r(])40 b Fk(<)f Fn(1)p Fk(=)p Fn(2)28 b(+)d(1)p Fk(=)p Fn(2)p Fk(\036)p Fn(\()p Fk(k)s Fn(\))41 b(b)m(y)e(de\014nition)e(of)i(\\go)s(o)s(d.")67 b(Th)m(us,)40 b(w)m(e)150 1461 y(kno)m(w)30 b(that)712 1687 y Fk(P)770 1701 y Fh(\015)835 1687 y Fn(+)20 b(\(1)p Fk(=)p Fn(2)i(+)e(1)p Fk(=)p Fn(2)p Fk(\036)p Fn(\()p Fk(k)s Fn(\)\)\(1)k Fi(\000)19 b Fk(P)1805 1701 y Fh(\015)1850 1687 y Fn(\))26 b Fk(>)f Fn(Pr)o([)p Fk(S)2185 1701 y Fh(QR)2299 1687 y Fn(\()p Fk(k)s(;)15 b(A)p Fn(\)])27 b Fi(\025)d Fn(1)p Fk(=)p Fn(2)e(+)e(1)p Fk(=\036)p Fn(\()p Fk(k)s Fn(\))291 1843 y(and)30 b(th)m(us,)g(w)m(e)h(can)g(deduce)617 2138 y Fk(P)675 2152 y Fh(\015)720 2138 y Fn(\(1)21 b Fi(\000)f Fn(1)p Fk(=)p Fn(2)h Fi(\000)f Fn(1)p Fk(=)p Fn(2)p Fk(\036)p Fn(\()p Fk(k)s Fn(\)\))j(+)d(1)p Fk(=)p Fn(2)i(+)e(1)p Fk(=)p Fn(2)p Fk(\036)p Fn(\()p Fk(k)s Fn(\))85 b Fi(\025)e Fn(1)p Fk(=)p Fn(2)21 b(+)f(1)p Fk(=\036)p Fn(\()p Fk(k)s Fn(\))1444 2276 y Fk(P)1502 2290 y Fh(\015)1546 2276 y Fn(\(1)p Fk(=)p Fn(2)i Fi(\000)e Fn(1)p Fk(=)p Fn(2)p Fk(\036)p Fn(\()p Fk(k)s Fn(\)\))86 b Fi(\025)d Fn(1)p Fk(=)p Fn(2)p Fk(\036)p Fn(\()p Fk(k)s Fn(\))2073 2456 y Fk(P)2131 2470 y Fh(\015)2259 2456 y Fi(\025)2825 2395 y Fn(1)p 2423 2435 851 4 v 2423 2519 a(2)p Fk(\036)p Fn(\()p Fk(k)s Fn(\)\(1)p Fk(=)p Fn(2)23 b Fi(\000)d Fn(1)p Fk(=)p Fn(2)p Fk(\036)p Fn(\()p Fk(k)s Fn(\)\))2259 2695 y(=)2566 2633 y(1)p 2423 2674 332 4 v 2423 2757 a Fk(\036)p Fn(\()p Fk(k)s Fn(\))h Fi(\000)f Fn(1)291 2936 y(So)30 b(since)g Fk(\036)h Fn(is)e(a)i(p)s(olynomial,)d Fk(P)1445 2950 y Fh(\015)1515 2936 y Fi(\025)1667 2901 y Fg(1)p 1621 2916 129 4 v 1621 2969 a Fh(p)p Fg(\()p Fh(k)r Fg(\))1790 2936 y Fn(where)i Fk(p)p Fn(\()p Fk(k)s Fn(\))c(=)f Fk(\036)p Fn(\()p Fk(k)s Fn(\))c Fi(\000)f Fn(1)31 b(is)e(a)i(p)s(olynomial.)345 b Fc(2)291 3049 y Fn(So,)33 b(w)m(e)g(ha)m(v)m(e)g(sho)m(wn)f(that)h(if)e(w)m(e)i(alw)m (a)m(ys)g(use)f(steps)g(3)g(and)g(4)h(when)e Fk(n)h Fn(is)f(go)s(o)s (d,)i(and)f(w)m(e)g(alw)m(a)m(ys)150 3162 y(use)k(step)f(2)i(for)e(our) h(output)f(when)g Fk(n)g Fn(is)g(not)h(go)s(o)s(d,)i(then)d(w)m(e)h (are)h(successful)d(with)h(probabilit)m(y)e(at)150 3275 y(least)f(1)22 b Fi(\000)f Fk(q)j Fn(+)d(1)p Fk(=p)816 3242 y Ff(0)840 3275 y Fn(\()p Fk(k)s Fn(\).)46 b(No)m(w,)34 b(I)d(claim)g(\(without)h(giving)f(a)h(pro)s(of)7 b(\))32 b(that)g(the)g(probabilit)m(y)e(that)i Fk(n)g Fn(is)150 3388 y(go)s(o)s(d)c(but)g(w)m(e)g(use)g(step)h(2)f(is)g(appro)m (ximately)f(equal)h(to)h(the)g(probabilit)m(y)c(that)k Fk(n)f Fn(is)f(not)i(go)s(o)s(d)f(but)f(w)m(e)150 3501 y(use)f(steps)h(3)g(and)f(4,)i(whic)m(h)d(is)h(appro)m(ximately)g (equal)g(to)h(1)13 b Fi(\000)g Fn(2)2340 3468 y Ff(\000)p Fh(k)2464 3501 y Fn(for)26 b(su\016cien)m(tly)g(large)g Fk( )s Fn(\()p Fk(k)s Fn(\).)41 b(Giv)m(en)150 3614 y(this,)30 b(w)m(e)g(need)g(to)i(pro)m(v)m(e)f(only)e(the)i(\014nal)e(claim.)150 3805 y Fo(Theorem)34 b(3)46 b Fl(This)d(r)-5 b(e)g(duction)45 b(is)e(suc)-5 b(c)g(essful)44 b(with)g(pr)-5 b(ob)g(ability)46 b(at)e(le)-5 b(ast)44 b Fn(1)29 b Fi(\000)e Fk(q)k Fn(+)d(1)p Fk(=p)3319 3772 y Ff(00)3362 3805 y Fn(\()p Fk(k)s Fn(\))44 b Fl(wher)-5 b(e)150 3918 y Fn(1)p Fk(=p)286 3885 y Ff(00)329 3918 y Fn(\()p Fk(k)s Fn(\))34 b Fl(is)e(a)h(nonne)-5 b(gligible)33 b(function.)291 4109 y(Pr)-5 b(o)g(of.)42 b Fn(W)-8 b(e)31 b(kno)m(w)g(that)677 4404 y Fk(P)96 b Fi(\024)83 b Fk(P)1043 4418 y Fh(\015)1088 4404 y Fn(\(Pr[)p Fk(S)1302 4418 y Fh(q)r Ff(\000)p Fh(GM)1525 4404 y Fn(\()p Fk(k)s(;)15 b(A)1718 4367 y Ff(0)1742 4404 y Fn(\))p Fi(j)p Fk(\015)5 b Fn(\()p Fk(\036;)15 b(n)p Fn(\)]\(1)23 b Fi(\000)d Fn(2)2337 4367 y Ff(\000)p Fh(k)2435 4404 y Fn(\))g(+)g(\(1)h Fi(\000)f Fk(q)s Fn(\)2)2897 4367 y Ff(\000)p Fh(k)2995 4404 y Fn(\))h(+)985 4551 y(\(1)g Fi(\000)f Fk(P)1235 4565 y Fh(\015)1280 4551 y Fn(\(\(1)h Fi(\000)f Fk(q)s Fn(\)\(1)h Fi(\000)f Fn(2)1823 4513 y Ff(\000)p Fh(k)1921 4551 y Fn(\))h(+)f(Pr)o([)p Fk(S)2246 4565 y Fh(q)r Ff(\000)p Fh(GM)2469 4551 y Fn(\()p Fk(k)s(;)15 b(A)2662 4513 y Ff(0)2686 4551 y Fn(\))p Fi(j)p 2746 4472 272 4 v Fk(\015)5 b Fn(\()p Fk(\036;)15 b(n)p Fn(\))r(]2)3089 4513 y Ff(\000)p Fh(k)3187 4551 y Fn(\))291 4734 y(No)m(w,)983 5029 y Fk(P)96 b Fi(\025)83 b Fk(P)1349 5043 y Fh(\015)1393 5029 y Fn(\(1)21 b Fi(\000)f Fn(2)1630 4992 y Ff(\000)p Fh(k)1728 5029 y Fn(\))1763 4992 y Fg(2)1823 5029 y Fn(+)g(\(1)h Fi(\000)f Fk(P)2164 5043 y Fh(\015)2209 5029 y Fn(\)\(1)h Fi(\000)f Fn(2)2481 4992 y Ff(\000)p Fh(k)2579 5029 y Fn(\)\(1)i Fi(\000)e Fk(q)s Fn(\))1137 5167 y(=)83 b(\(1)21 b Fi(\000)f Fn(2)1528 5130 y Ff(\000)p Fh(k)1626 5167 y Fn(\)\()p Fk(P)1754 5181 y Fh(\015)1799 5167 y Fn(\(1)h Fi(\000)f Fn(2)2036 5130 y Ff(\000)p Fh(k)2134 5167 y Fn(\))h(+)f(\(1)h Fi(\000)e Fk(P)2530 5181 y Fh(\015)2575 5167 y Fn(\)\(1)j Fi(\000)d Fk(q)s Fn(\)\))1889 5416 y(7-3)p eop 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0 -1 44 75 d1 44 0 0 76 0 -1 cm BI /IM true /W 44 /H 76 /BPC 1 /F /CCF /DP <> ID & ¹8kø ðXxF`ïI¾Ÿ ÃÐMÿ¥~ýzø?¿Ûÿÿÿß× ÿ×þ¿ÖG×kÝv½„ /®o‚ðÄþE?ÿÿÿÿÿÿùÑÄê áðø?ï¾ÿ“‡œ ž ¸€€ EI endstream endobj 68 0 obj <> stream 81 0 0 0 0 0 d1 endstream endobj 69 0 obj <> stream 0 0 0 60 14 94 d1 14 0 0 34 0 60 cm BI /IM true /W 14 /H 34 /BPC 1 /F /CCF /DP <> ID &±wû÷·Þûðýþÿýßÿåç1õõ&«ƒ EI endstream endobj 70 0 obj <> stream 0 0 0 -1 44 72 d1 44 0 0 73 0 -1 cm BI /IM true /W 44 /H 73 /BPC 1 /F /CCF /DP <> ID &¨/ÿÛßýŸ ®}ý¿aýÝ·†ÿÛì2 Íw··‡ÛØ}÷·¾Ø}÷·¿{¿Ã¿ù8oðôúÿÿÿú]V͆W¿ÒïÜ.ï­´a. b°ÂÁ‚Á‚€€ EI endstream endobj 71 0 obj <> stream 63 0 0 0 0 0 d1 endstream endobj 72 0 obj <> stream 0 0 0 -1 47 75 d1 47 0 0 76 0 -1 cm BI /IM true /W 47 /H 76 /BPC 1 /F /CCF /DP <> ID & A@ðFc ƒzÞ“z7Óáaé7þ·Óÿþ·ÿúßOÿÿÿ?ÿÿÿÿÿÿÿÿòkáÿÿÿþõÚÿþõÿûi½v¸x[il0—k°Â[ƒƒ EI endstream endobj 73 0 obj <> stream 53 0 0 0 0 0 d1 endstream endobj 74 0 obj <> stream 54 0 0 0 0 0 d1 endstream endobj 75 0 obj <> stream 0 0 0 -1 36 72 d1 36 0 0 73 0 -1 cm BI 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DŸ§Å ¸ZLV¾“ôýxWÐAÿ×ÿ¯ýÿëÿ_ÿÿÿÿÿÿÿÿÿ¿ÿÿßÿ¿‡ïÿ†Âúíûk¿¶¸`Á^×ÙÐEáˆ/¼0^@Â?ñÿÿÿÿÿÿÿÿÿÿÿÿÿÿ …Èkÿÿüƒ)à EI endstream endobj 82 0 obj <> stream 78 0 0 0 0 0 d1 endstream endobj 83 0 obj <> stream 0 0 0 35 61 101 d1 61 0 0 66 0 35 cm BI /IM true /W 61 /H 66 /BPC 1 /F /CCF /DP <> ID & `þC`Œƒ 4 @ô7„xAþ·¤ßO…¾ëý ßÿëÿÓÿ­ÿÿÿÿÿÿÿýëÿÿÿm/ÿÞ» ¼/ðÃKþ×zÞ¶ÒÛAp`¸a„¶%‡…ƒXa~@Á EI endstream endobj 84 0 obj <> stream 0 0 0 36 71 101 d1 71 0 0 65 0 36 cm BI /IM true /W 71 /H 65 /BPC 1 /F /CCF /DP <> ID &¡”PNù ‚$¸o>áÂàƒ~çAÒn VëÿãÂOïÿÿøÿ÷ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿõ\(.Qr ÿÿÿÈTÌ?ð EI endstream endobj 85 0 obj <> stream 0 0 0 10 45 101 d1 45 0 0 91 0 10 cm BI /IM true /W 45 /H 91 /BPC 1 /F /CCF /DP <> ID & `Ð/ÁÁ§¢‡¤ßO¤ï ôúßðŸÿÿÿÿÿÿÿÿþAA¯ÿÿÿÿÿÿÿÿÿÿÿÿÿø‘¯òjø3¨(ø}÷ßü?ïÿûÿÿïÿÿÿð EI endstream endobj 86 0 obj <> stream 76 0 0 0 0 0 d1 endstream endobj 87 0 obj <> stream 0 0 0 4 47 99 d1 47 0 0 95 0 4 cm BI /IM true /W 47 /H 95 /BPC 1 /F /CCF /DP <> ID ù5#àÂØ_ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿó€™ ðÇ&§ü‡¯ ¸x>ø?ï¼@ EI endstream endobj 88 0 obj <> stream 112 0 0 0 0 0 d1 endstream endobj 89 0 obj <> stream 0 0 0 -1 69 101 d1 69 0 0 102 0 -1 cm BI /IM true /W 69 /H 102 /BPC 1 /F /CCF /DP <> ID &¡¤PþBÙ á„„°Ð" £éûAÒoŽúo¦úý‡Âþü?ß¿ÿÿÿ¿ßÿÿÿÿÿÿÿúÿ×ÿþ¿õÿëþKáv½¯i}rtÒöK᯲xb¼0¼0^AÈÿÇÿÿÿÿÿÿÿÿÿÿÿÿÿõÂÇÿɨ\@ EI endstream endobj 90 0 obj <> stream 0 0 0 38 16 99 d1 16 0 0 61 0 38 cm BI /IM true /W 16 /H 61 /BPC 1 /F /CCF /DP <> ID &¤pàƒÐzÿÿµ°°a~?ÿÿÿÿÿÿÿ’ÏÁ ôÿÿÿka`€€ EI endstream endobj 91 0 obj <> stream 0 0 0 0 83 99 d1 83 0 0 99 0 0 cm BI /IM true /W 83 /H 99 /BPC 1 /F /CCF /DP <> ID &¨”ßÿü‡¯kkÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùS ÿ‡È``Ãò ø0~Ûöþü?·ÿ°ÿûÿßÿ¿ÿÿÿÿÿÿ×þ¿ýzü/¯_ éz_^Z‚ Lq™[Î …ÊX7ÊÈex€ EI endstream endobj 92 0 obj <> stream 94 0 0 0 0 0 d1 endstream endobj 93 0 obj <> stream 0 0 0 36 48 99 d1 48 0 0 63 0 36 cm BI /IM true /W 48 /H 63 /BPC 1 /F /CCF /DP <> ID "€ŸäÔC¨†¸_ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿü?ÿþÿš/ä1Ÿ`þŸo÷÷ë…oðÃ{°°Ôš„ƒü@ EI endstream endobj 94 0 obj <> stream 0 0 0 -1 32 99 d1 32 0 0 100 0 -1 cm BI /IM true /W 32 /H 100 /BPC 1 /F /CCF /DP <> ID &«ÿò&¸aÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿë…Ê/ÿÿù ˆ€ EI endstream endobj 95 0 obj <> stream 79 0 0 0 0 0 d1 endstream endobj 96 0 obj <> stream 0 0 0 35 55 101 d1 55 0 0 66 0 35 cm BI /IM true /W 55 /H 66 /BPC 1 /F /CCF /DP <> ID &¡”P|†Á ðƒÑÐ@ðD0\>è Þ}oJ~Ÿ…z‡ÖõÕÿ¢F¿_þ£ÿÿ•€ß’Kɯ]ÿýëþý×ýë¿Âì8Xm~¶×m.×a„°`Â[ÁaˆX0_.D@ EI endstream endobj 97 0 obj <> stream 39 0 0 0 0 0 d1 endstream endobj 98 0 obj <> stream 0 0 0 36 112 99 d1 112 0 0 63 0 36 cm BI /IM true /W 112 /H 63 /BPC 1 /F /CCF /DP <> ID &¨ FÅÿÿÿü‰©/®A‚ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿwÿÿ÷ÿæ‹ÿwý¦ÞOëá÷ûïêI´°“ ,¡Öû4 þÄ&+Á„^@Ád!Ã!†X/ø€ EI endstream endobj 99 0 obj <> stream 0 0 0 -2 63 102 d1 63 0 0 104 0 -2 cm BI /IM true /W 63 /H 104 /BPC 1 /F /CCF /DP <> ID 0`Ý 'àƒ°ÃðAôPáèq ¿¦é¸]¨Mý{Âûÿ¯úÿɨ|-/ý-p\/ !¬_!¤£!š.%Ð+ø%‚…ÒÁÂÐ-0Qô@Ã2Š@jÈ0ÕÍaµ/Ð\.ºŒƒmü/ëÖ¿ýäÕ ßëþÿkë†÷ïí‚í¯k°`Š  }áxbŸ ÿà È7!à EI endstream endobj 100 0 obj <> stream 74 0 0 0 0 0 d1 endstream endobj 101 0 obj <> stream 0 0 0 0 49 75 d1 49 0 0 75 0 0 cm BI /IM true /W 49 /H 75 /BPC 1 /F /CCF /DP <> ID &¨ÿ!Ã]ÿïßïÿßïÿÃýÿûü?ÿ{ÿ÷ûÿ÷ûÿðÿþÿÿÞÿýþÿýþÿü>ò ~  EI endstream endobj 102 0 obj <> stream 0 0 0 27 54 77 d1 54 0 0 50 0 27 cm BI /IM true /W 54 /H 50 /BPC 1 /F /CCF /DP <> ID &¤l2Ï„þaÚ8Ãõ½ðßïÿÞÞÿ÷ýïÿ¼8}ßþïÿä)¯¿~÷¿ïÿÞÿ‡þxß¼?äÕmþÿþÿÃþþÿïÞýíÂß`×µ†Šb¶…ƒ @ EI endstream endobj 103 0 obj <> stream 45 0 0 0 0 0 d1 endstream endobj 104 0 obj <> stream 0 0 0 27 38 77 d1 38 0 0 50 0 27 cm BI /IM true /W 38 /H 50 /BPC 1 /F /CCF /DP <> ID &¡ À’¯ð ð· ˆ?ôÃá<|pïù5Nü?…ÚðaxúëÏ®AÄazÒÒü ¸\è à¸^A/¤ÿð÷ÿíî÷|:Û^\…†P EI endstream endobj 105 0 obj <> stream 0 0 0 6 31 77 d1 31 0 0 71 0 6 cm BI /IM true /W 31 /H 71 /BPC 1 /F /CCF /DP <> ID &¢þ0õ¿aêþþûÿÛï÷ûÝÿïÿÇïÿïÿü?ýûÿßûÿÞ$brkä<ÿ¿ü?ðÿ÷ïÿïÿ}…ð EI endstream endobj 106 0 obj <> stream 47 0 0 0 0 0 d1 endstream endobj 107 0 obj <> stream 0 0 0 27 43 77 d1 43 0 0 50 0 27 cm BI /IM true /W 43 /H 50 /BPC 1 /F /CCF /DP <> ID &¤lþ}þÿýïÿßïÿßáÿø¿ÿ¿üñ½þMVÿ ÷ÿÝ„÷áûÿ}çÕ÷·ßl>ÖÅ1[L,"8€ EI endstream endobj 108 0 obj <> stream 35 0 0 0 0 0 d1 endstream endobj 109 0 obj <> stream 0 0 0 27 51 77 d1 51 0 0 50 0 27 cm BI /IM true /W 51 /H 50 /BPC 1 /F /CCF /DP <> ID & ¹ sãþ0žòŽO[Xz ½¿?¬?¿…{ÿ÷»ÿ÷áÿû‡ÿ‡È†½ÿáßûþýûßýþûóžÿäÕ]û»ÿÇÿßÿmÿý÷Û¯†¾Çá ÂÃ` EI endstream endobj 110 0 obj <> stream 0 0 0 27 40 77 d1 40 0 0 50 0 27 cm BI /IM true /W 40 /H 50 /BPC 1 /F /CCF /DP <> ID & „àÇÈ•èðÁà°ô§A¿¤öõ¿a÷ïAÃÿÿÿ¿ßÿ¿ßÿoû!‰ïxaÿïß¾ßl=‡ ¯ v+ @ EI endstream endobj 111 0 obj <> stream 0 0 0 27 44 77 d1 44 0 0 50 0 27 cm BI /IM true /W 44 /H 50 /BPC 1 /F /CCF /DP <> ID & †€ßäJ° ôƒ¦ü>“Vþý¿øO¨o‡ÿ¿ý¿ûþ÷ýÿïÿ{ïû÷߆¿·÷í¥·áëa¥·ðÂáˆ,Áš EI endstream endobj 112 0 obj <> stream 37 0 0 0 0 0 d1 endstream endobj 113 0 obj <> stream 0 0 0 28 23 75 d1 23 0 0 47 0 28 cm BI /IM true /W 23 /H 47 /BPC 1 /F /CCF /DP <> ID &±ÀÅÓïÿ÷ÿß ,ÿÿÿÿü‚gàŸz¿ïkðû EI endstream endobj 114 0 obj <> stream 0 0 0 -3 70 75 d1 70 0 0 78 0 -3 cm BI /IM true /W 70 /H 78 /BPC 1 /F /CCF /DP <> ID &¨ƒ;ÿù3CÃAö¼=wïíÿïì?þü?þßßþß¿þÇÿ†¹ÀÑßïßïßáûÿá¿¿ý¿ýµÿ‡ï_á¿¿ý¿ý¿þ·ÿ‡öÿø÷ý¯ßÿ¿ïÿ¾ð EI endstream endobj 115 0 obj <> stream 82 0 0 0 0 0 d1 endstream endobj 116 0 obj <> stream 0 0 0 27 46 77 d1 46 0 0 50 0 27 cm BI /IM true /W 46 /H 50 /BPC 1 /F /CCF /DP <> ID &¡ ÁÏ ü Á‚>ÃG§¬5½ oo¦?Ýõ„÷ÿ¨wþÛ÷ÿáÿ¿ä!¯ûþ÷ýÿïûßðÿíÿý·ÿì;ÿ¿mvï …Øk†&Í`Ðá… EI endstream endobj 117 0 obj <> stream 0 0 0 0 63 75 d1 63 0 0 75 0 0 cm BI /IM true /W 63 /H 75 /BPC 1 /F /CCF /DP <> ID &¨ª„ßþBAŠ°ä6؇¸~ßßvýÿîü>ÿ¿ß÷Þÿ÷ïøpÿax÷¿ÿ¿ÿ¿ÿ÷ÿïðÿýïÿßïÿßïÿÃï!œoûÿ€€ EI endstream endobj 118 0 obj <> stream 90 0 0 0 0 0 d1 endstream endobj 119 0 obj <> stream 0 0 0 27 46 98 d1 46 0 0 71 0 27 cm BI /IM true /W 46 /H 71 /BPC 1 /F /CCF /DP <> ID &£4ˆGPÊ ôxƒá·Ó‡q >ÃÞMPwíÿ÷‡|0^7‡þAyþù§£æ=oAðý>þÿ÷ÿÿ¿ûþðÿ‡ÿ‡ýÿ~ýÿ¾ÿ}ùâýßòj®ýßÿ‡¿ÿ¶÷þø{uá¯cá ÖC EI endstream endobj 120 0 obj <> stream 46 0 0 0 0 0 d1 endstream endobj 121 0 obj <> stream 0 0 0 -1 46 77 d1 46 0 0 78 0 -1 cm BI /IM true /W 46 /H 78 /BPC 1 /F /CCF /DP <> ID &¹Ànx/âÁ´zzß¼š®ßÿjÿÃÿß~ïÿ÷¼>ÿ‡ÿÿŽëõÚø_Ao^. ¿Ã.ÃækÙïÛÿvvCç‡ý°ƒÃ·~ü6Ûøw»lpcàÖá…ãßþÿßþýÿáÿ‡ÿ¿÷ù Wÿ" EI endstream endobj 122 0 obj <> stream 44 0 0 0 0 0 d1 endstream endobj 123 0 obj <> stream 55 0 0 0 0 0 d1 endstream endobj 124 0 obj <> stream 0 0 0 2 69 75 d1 69 0 0 73 0 2 cm BI /IM true /W 69 /H 73 /BPC 1 /F /CCF /DP <> ID &¹( Ïûä1¦ wÿ¿ûÿûÿÿïÿýÿïßþÿßþÿÃÿß*ò ï»ÿ½ÿ°ïÿ½ÿü?÷öîÿÿðÛÿýß·ÿü7]‡ ðá‚â?~  EI endstream endobj 125 0 obj <> stream 0 0 0 27 39 77 d1 39 0 0 50 0 27 cm BI /IM true /W 39 /H 50 /BPC 1 /F /CCF /DP <> ID &¡ÉÁ‚„x`õ‡„ˆziúá½&þõa÷ïÃðÿÿûÿÿä/”l4zm†üûí÷|?Çmÿo·ßƒh-ƒøapÅ`ÂÁ‚€€ EI endstream endobj 126 0 obj <> stream 43 0 0 0 0 0 d1 endstream endobj 127 0 obj <> stream 56 0 0 0 0 0 d1 endstream endobj 128 0 obj <> stream 0 0 0 -1 49 77 d1 49 0 0 78 0 -1 cm BI /IM true /W 49 /H 78 /BPC 1 /F /CCF /DP <> ID &¹À0|'âÂÑÆ·á¹5]ÿûÛÿû½ÿ½ÿþ8wøwÿîBšÞýûþÿÝÿïûÿpÿ÷ü?ößÿø~ÿïïÿn3@׈4Åxan/ÿ÷þÿ÷ïÿü?ýÿ¿ÈJ¿ûü€€ EI endstream endobj 129 0 obj <> stream 0 0 0 2 27 77 d1 27 0 0 75 0 2 cm BI /IM true /W 27 /H 75 /BPC 1 /F /CCF /DP <> ID &¡ø z<Þ·íÿïì?û¿ü?ýÞïßþÿxáþÿßíþóÅïäÕ_»ÿþÿößýýºá®Ç†°ÂÇÿÿÿòó?ïÿ  EI endstream endobj 130 0 obj <> stream 0 0 0 27 48 98 d1 48 0 0 71 0 27 cm BI /IM true /W 48 /H 71 /BPC 1 /F /CCF /DP <> ID &¢”l±˜Aèø€úàÛí¿¿oÃýß ÿ‹ûÿûäŸ„à‡£3ô ú°õÿáýõýøßÿûÿÝÿ¿ïÿwþÿ‡ÿnÿí¿ÞØýûwmoÃö .Ö›5ƒC†@ EI endstream endobj 131 0 obj <> stream 91 0 0 0 0 0 d1 endstream endobj 132 0 obj <> stream 84 0 0 0 0 0 d1 endstream endobj 133 0 obj <> stream 0 0 0 0 105 75 d1 105 0 0 75 0 0 cm BI /IM true /W 105 /H 75 /BPC 1 /F /CCF /DP <> ID &¨¨6@ÀŽ¿ÃþüŒŒ†Äa­­«¼?ýûÛ÷ßþè?÷ûûšnïïÿí‡ÿø}ßì?uëÃ{wÿáûÿwÿûÛ½öÿÿoØP÷ð¿}¿ößÿ︿¿ÿ°Ãþø~·ïOÿ¿íýíßÿ‡áÿ»ÿýá»ßoÿýmû§¿ÿ}¿öTÿûâá÷ÿÙµýŽú¼ÙPÿ½Ãü@ EI endstream endobj 134 0 obj <> stream 80 0 0 0 0 0 d1 endstream endobj 135 0 obj <> stream 103 0 0 0 0 0 d1 endstream endobj 136 0 obj <> stream 72 0 0 0 0 0 d1 endstream endobj 137 0 obj <> stream 32 0 0 0 0 0 d1 endstream endobj 138 0 obj <> stream 0 0 0 27 45 77 d1 45 0 0 50 0 27 cm BI /IM true /W 45 /H 50 /BPC 1 /F /CCF /DP <> ID & Á˜hù×ÌÃÖúô~ý7…‡ÿ·áÿöÿûï÷Ãþßü?áïû÷ïý÷ø}ùâýßòj¯÷þ8_þôÿÛuÿ·UÃ_cð׆a@@ EI endstream endobj 139 0 obj <> stream 0 0 0 0 72 73 d1 72 0 0 73 0 0 cm BI /IM true /W 72 /H 73 /BPC 1 /F /CCF /DP <> ID & ¹* ü†n»Xkÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿç.C*ÿÿÿä×xKÿÿÿÿÿ½ÿû׿ÿáávð‚àÂ`¼Gÿð EI endstream endobj 140 0 obj <> stream 0 0 0 -3 56 73 d1 56 0 0 76 0 -3 cm BI /IM true /W 56 /H 76 /BPC 1 /F /CCF /DP <> ID $ßþME( å´×ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿßÿýÿ÷úðëþi¸^à |øb¼ðÉÀŽ?ÿÿÿÿÿÿÿÿÿ®?ù5 EI endstream endobj 141 0 obj <> stream 0 0 0 -2 25 73 d1 25 0 0 75 0 -2 cm BI /IM true /W 25 /H 75 /BPC 1 /F /CCF /DP <> ID òj)AZÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ×ð¹êÿü‰Lÿÿü‰˜ úÿû_ƒð EI endstream endobj 142 0 obj <> stream 0 0 0 24 36 75 d1 36 0 0 51 0 24 cm BI /IM true /W 36 /H 51 /BPC 1 /F /CCF /DP <> ID 0œ?†N»ð€ã¸@Ý7 êÿW×ëÒþM@¾–A„ùX%¥®–p\êÄœ È1uÂë¯õù5Úÿ¿ÚðÂì0¸cÁšî À@ EI endstream endobj 143 0 obj <> stream 31 0 0 0 0 0 d1 endstream endobj 144 0 obj <> stream 0 0 0 23 49 74 d1 49 0 0 51 0 23 cm BI /IM true /W 49 /H 51 /BPC 1 /F /CCF /DP <> ID &¡œ 3ü†`[Â0ð§ÒoI¿õ½&ÿÿü,>ÿŽÿÿÿÿ&«_ÿÿ .û×ý®õ¶–õ†Ð[ip`» %†!`Á`Á@@ EI endstream endobj 145 0 obj <> stream 0 0 0 -4 26 72 d1 26 0 0 76 0 -4 cm BI /IM true /W 26 /H 76 /BPC 1 /F /CCF /DP <> ID òj)@íÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿë…þMDÀ@ EI endstream endobj 146 0 obj <> stream 0 0 0 25 74 74 d1 74 0 0 49 0 25 cm BI /IM true /W 74 /H 49 /BPC 1 /F /CCF /DP <> ID & ÜØ76 ÿÚëµÿ ð×ýo°ºßÌך.þÚúýÿ_V¯ïúõ¯µð¯ß _øx^—·ñÿú†¯­ÿý¯^úß^µõ¿¯…ïÒO^ÿôÂa8ƒ&\ÿÿ EI endstream endobj 147 0 obj <> stream 0 0 0 -4 55 74 d1 55 0 0 78 0 -4 cm BI /IM true /W 55 /H 78 /BPC 1 /F /CCF /DP <> ID & A@†‚ÿk àŒÅÖøA:@ÍE„ÖkWÿKÔ?_úÿÿÿÿÿÿÿÿɯÿïÿý¿ýýþÂí¯Ãøl.Ú8¼0_+Ãä4Çÿÿÿÿÿÿÿÿÿü.¹ Ôÿÿù j@@ EI endstream endobj 148 0 obj <> stream 85 0 0 0 0 0 d1 endstream endobj 149 0 obj <> stream 0 0 0 24 56 74 d1 56 0 0 50 0 24 cm BI /IM true /W 56 /H 50 /BPC 1 /F /CCF /DP <> ID &¡  SA^¹¬7„h0|-ôé< »×Q^úÿøÿ÷ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿýWü(XÈtÿÿ&¢Hl¤@ EI endstream endobj 150 0 obj <> stream 0 0 0 24 56 72 d1 56 0 0 48 0 24 cm BI /IM true /W 56 /H 48 /BPC 1 /F /CCF /DP <> ID $ßþME( å´×ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿßÿýÿ÷úèÑ_øPÜ(Øa. pÅ`Ôš‰ œ EI endstream endobj 151 0 obj <> stream 0 0 0 -3 106 75 d1 106 0 0 78 0 -3 cm BI /IM true /W 106 /H 78 /BPC 1 /F /CCF /DP <> ID & Òp4̓Moþ×ýaö¿ÿ­öýoÿû_ð·ÃE Ê¿ÿA{~Kÿõø}Z¿ÿï^¿a~¼>¯Wÿõïí ¿ÿ ÷Ô5zÿýíý¯áë×áõïö¿½zýý…úðþõë÷׿Úþð¼/ß^þÅú÷ÂßÃ_þ–ÃÿÿÖ¯ÂýÕëþ˜Aá˜@âK‡ @àÿÿü@ EI endstream endobj 152 0 obj <> stream 97 0 0 0 0 0 d1 endstream endobj 153 0 obj <> stream 102 0 0 0 0 0 d1 endstream endobj 154 0 obj <> stream 0 0 0 23 50 74 d1 50 0 0 51 0 23 cm BI /IM true /W 50 /H 51 /BPC 1 /F /CCF /DP <> ID &¡Ê‹ø `àÓÂÃOAæŸH6“zxOØñ ÿÿ÷ÿðüšÿÞBšßÿoïØ?oÁü0~ üx>AEYŠþE¿þŸéÿ¯ÖÕp’õØ0AaˆXad0ˆ€ EI endstream endobj 155 0 obj <> stream 0 0 0 25 52 95 d1 52 0 0 70 0 25 cm BI /IM true /W 52 /H 70 /BPC 1 /F /CCF /DP <> ID &¤fЃÐxGÛè†ÜmÝïÃßäÕ ßø0¯ïöÿЧÿúøOš/úßý>°úõ¿ú­ôÿáoþë}?úÃÿ§Öúþ·þ¿ ÿ¦Ä§þ  EI endstream endobj 156 0 obj <> stream 0 0 0 60 13 72 d1 13 0 0 12 0 60 cm BI /IM true /W 13 /H 12 /BPC 1 /F /CCF /DP <> ID &¬ã‚ñ&©~ @ EI endstream endobj 157 0 obj <> stream 0 0 0 0 97 99 d1 97 0 0 99 0 0 cm BI /IM true /W 97 /H 99 /BPC 1 /F /CCF /DP <> ID &¨ª¯ÿÿÈ.#ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿùY2 ?àùaýù¢þ~þ߇ûö·ÿþÿÿÿßÿÿÿÿÿýÿþ¿_ýz ×ÿ½/¯ y °™[Âë†óºŒî7À@ EI endstream endobj 158 0 obj <> stream 0 0 0 34 58 99 d1 58 0 0 65 0 34 cm BI /IM true /W 58 /H 65 /BPC 1 /F /CCF /DP <> ID &¨Š†Ïÿÿšÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ÷ÿÿü2 ŸÓÿÂ}þƒÎ‰þÿýþß ”ö/p׆+Úÿრ EI endstream endobj 159 0 obj <> stream 110 0 0 0 0 0 d1 endstream endobj 160 0 obj <> stream 0 0 0 33 71 101 d1 71 0 0 68 0 33 cm BI /IM true /W 71 /H 68 /BPC 1 /F /CCF /DP <> ID &¡˜u Ïn“Ar<"œ@| | A½&õ½&o§Âß õ¿õ¿ÿéõ‡ÿÿëÿÿÿÿÿÿÿÿÞ¿ÿýáÿk½½Ã\=o]®ØAv»il4°x[D« BüX2¬Hfk EI endstream endobj 161 0 obj <> stream 0 0 0 -1 79 101 d1 79 0 0 102 0 -1 cm BI /IM true /W 79 /H 102 /BPC 1 /F /CCF /DP <> ID &¡ªu ¿žAu„´†>Ñ oæh Áñ7÷Ó}7ÓÿÓïÿ ÿÿÿ~ÿÿÿßÿÿÿÿÿÿÿÿþ¿ÿ×ÿ¯þ¿ð—ý¯p¾»K´¸5á‚\ꎂ…ï ÁŠðÁá‚ò ¯ÿÿÿÿÿÿÿÿÿÿÿÿár”ÿÿ䈀 EI endstream endobj 162 0 obj <> stream 0 0 0 -1 35 99 d1 35 0 0 100 0 -1 cm BI /IM true /W 35 /H 100 /BPC 1 /F /CCF /DP <> ID &«ÿþBkÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ®Rÿÿü‚ EI endstream endobj 163 0 obj <> stream 0 0 0 33 64 101 d1 64 0 0 68 0 33 cm BI /IM true /W 64 /H 68 /BPC 1 /F /CCF /DP <> ID &¡˜j  ¡^‡‚ è§A‚0HÐA½oIô›è?øI‡_ôöëý~¿ÿÿü™xü§ÿÿßÿ…ßáwÿÞ¸®ÿiaýëm.×l.ÃAaë AlWàÁ`ÂÈeë EI endstream endobj 164 0 obj <> stream 0 0 0 34 125 99 d1 125 0 0 65 0 34 cm BI /IM true /W 125 /H 65 /BPC 1 /F /CCF /DP <> ID &¨ŠEDÿÿÿÿù Y ʈfUÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÝÿÿÿýßÿÿÜ?»ùÕᦗ¼ÕÿooðÝ7XXa`×)C°ÿ` ’ {Ðzö)ˆ^ ¿ü0@ÂÈ!ÐÐ €€ EI endstream endobj 165 0 obj <> stream 0 0 0 5 58 99 d1 58 0 0 94 0 5 cm BI /IM true /W 58 /H 94 /BPC 1 /F /CCF /DP <> ID &¿ÿü†Ê¿ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿ+¾u?ÿä`y ü†ŠòÃÃïƒþÔ@ EI endstream endobj 166 0 obj <> stream 0 0 0 -5 56 72 d1 56 0 0 77 0 -5 cm BI /IM true /W 56 /H 77 /BPC 1 /F /CCF /DP <> ID $ßþMFH †ƒ_ÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿáõˆÿ“Q gÇÿÿÿÿÿüƒ{ðAú¿þýþý·Ø`Âõ†K 0K !–ˆ€ EI endstream endobj 167 0 obj <> stream 0 0 0 -3 60 72 d1 60 0 0 75 0 -3 cm BI /IM true /W 60 /H 75 /BPC 1 /F /CCF /DP <> ID üšˆt8a.G‡öþßßÿÿÿßÿÿûÿÿÿÿä5Íÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿÿþ8”€_ü@ EI endstream endobj 168 0 obj <> stream 92 0 0 0 0 0 d1 endstream endobj 169 0 obj <> stream 0 0 0 25 13 72 d1 13 0 0 47 0 25 cm BI /IM true /W 13 /H 47 /BPC 1 /F /CCF /DP <> ID &¬ã‚ñ&©~ ?ÿÿÿÿþ^~Åù5Kÿ  EI endstream endobj 170 0 obj <> stream 96 0 0 0 0 0 d1 endstream endobj 171 0 obj <> stream 0 0 0 23 50 96 d1 50 0 0 73 0 23 cm BI /IM true /W 50 /H 73 /BPC 1 /F /CCF /DP <> ID & ÁÔêðƒÁƒÂ' áðƒé7 ƒxX}ÄÍÃ1È1@Ö Á@@ EI endstream endobj 172 0 obj <> stream 0 0 0 26 52 75 d1 52 0 0 49 0 26 cm BI /IM true /W 52 /H 49 /BPC 1 /F /CCF /DP <> ID & \ØúÿOÿOÿ óEÿ[éÿÖý?úA¿õ¿úð‚oýoþŸý&úßýÿ[ÿ _ßú`ħþ  EI endstream endobj 173 0 obj <> stream 38 0 0 0 0 0 d1 endstream endobj 174 0 obj <> stream 0 0 0 25 60 75 d1 60 0 0 50 0 25 cm BI /IM true /W 60 /H 50 /BPC 1 /F /CCF /DP <> ID &¢ Ã0¿Á0ƒì#óõ‡½ÿ ÿ÷¾oïþø{ÿïpû¿ýßþáÈ9¯þ¾÷ÿ¿Þÿ¿óÃ~ø|š®ýßÿ‡ÿoîÿ¿xÏzÛÛûH6—†»b¶a`Á  EI endstream endobj 175 0 obj <> stream 0 0 0 -5 65 75 d1 65 0 0 80 0 -5 cm BI /IM true /W 65 /H 80 /BPC 1 /F /CCF /DP <> ID &¨À0H­¢?ì }£@ø,>&`¸{éøMôßÞá‡ÓÿÓ{ÿ~ÿáßþÿ¿÷ñõýuý,!¤# У!”“ò Àð–º p¹& (ƒ Ï¢0³¨japºò 7ÿ_ÿïÿßþ÷ÿ~v?öÂíý®Úx`×yƒýÃòˆÀ@ EI endstream endobj 176 0 obj <> stream 99 0 0 0 0 0 d1 endstream endobj 177 0 obj <> stream 83 0 0 0 0 0 d1 endstream endobj 178 0 obj <> stream 0 0 0 24 61 96 d1 61 0 0 72 0 24 cm BI /IM true /W 61 /H 72 /BPC 1 /F /CCF /DP <> ID & æÃnŸÿÿoÿ÷þÿ÷ÿÃüÔwÁ„ƒæ±z'¤ú †Âðô úÃ};éa¾·¯õ°ü;¿_í†ÿÿoï÷þÿîû÷†ýýýÿûkÛ|> stream 0 0 0 -10 25 99 d1 25 0 0 109 0 -10 cm BI /IM true /W 25 /H 109 /BPC 1 /F /CCF /DP <> ID & ß_KK ¥¯ K_ÂZ_Ö¿ ¿ô¿ ý~½~¿ÿÖ¿ÿÿÿ ¿ÿÿÿÿÿÿÿÿÿÿöÿÿÿïßÿïýïýþöÿöïßíá¿ßo‡¾ÞooÜ@ EI endstream endobj 180 0 obj <> stream 0 0 0 -9 25 100 d1 25 0 0 109 0 -9 cm BI /IM true /W 25 /H 109 /BPC 1 /F /CCF /DP <> ID &¨ÀlßííðöûÃí÷ûo÷ïðßþßðÿïïÿ÷ïÿÿÿðßÿÿÿÿÿÿÿÿ ÿÿÿZÿÿ_¯_¯ð¿Kÿ Zþ–‚þ´°½ip´´¿¨€ EI endstream endobj 181 0 obj <> stream 0 0 0 -3 65 72 d1 65 0 0 75 0 -3 cm BI /IM true /W 65 /H 75 /BPC 1 /F /CCF /DP <> ID #`×ù5ê °Áÿÿÿÿÿÿÿÿÿÿÿÿÿò, ~“†€È#ÃßÃ~ßßÿ~ÿü?ÿÿÿÿÿø_ú_ÿé}xKÐ^2kP ¤4` EI endstream endobj 182 0 obj <> stream 73 0 0 0 0 0 d1 endstream endobj 183 0 obj <> stream 0 0 0 -4 39 73 d1 39 0 0 77 0 -4 cm BI 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36 35 89 d1 35 0 0 53 0 36 cm BI /IM true /W 35 /H 53 /BPC 1 /F /CCF /DP <> ID òk½÷Ù°ÑØo߆áÛ}†{xaï°öûÛÃ{ðöûÛýûÃðüýü[ÿ^°¡I®¾Â÷[k½m¥°aãÁ…†N  EI endstream endobj 198 0 obj <> stream 93 0 0 0 0 0 d1 endstream endobj 199 0 obj <> stream 0 0 0 24 90 74 d1 90 0 0 50 0 24 cm BI /IM true /W 90 /H 50 /BPC 1 /F /CCF /DP <> ID &¢ £a˜fþ ‚aÝ„~¬=ú ûá¿ÿ»ßíÿ‡ï‡ÿ~á÷wÿîïÿßAÍ{ÿÃîÿß~ýÿß»ÿ½ÿÝÿç†ïÃ÷Çɪáß»¿ÿÃÿíßÝß÷~ðçÐÏzzÛÛÛûH Òáó2Ã]Š V   ô@ EI endstream endobj 200 0 obj <> stream 77 0 0 0 0 0 d1 endstream endobj 201 0 obj <> stream 88 0 0 0 0 0 d1 endstream endobj 202 0 obj <> stream 126 0 0 0 0 0 d1 endstream endobj 203 0 obj <> stream 75 0 0 0 0 0 d1 endstream endobj 204 0 obj <> stream 24 0 0 0 0 0 d1 endstream endobj 205 0 obj <> stream 127 0 0 0 0 0 d1 endstream endobj 206 0 obj <> stream 0 0 0 -3 33 35 d1 33 0 0 38 0 -3 cm BI /IM true /W 33 /H 38 /BPC 1 /F /CCF /DP <> ID &¡ þ¨H#ä·¤Cÿ ¿ÒoÛÿqƒÿ|šþS‚d ¸|ÐaîCÿ¾Øoß»ßÛ|?¶»¸k†<X`  EI endstream endobj 207 0 obj <> stream 0 0 0 3 22 43 d1 22 0 0 40 0 3 cm BI /IM 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254 4836 a Fg(1)288 4868 y Ff(T)-6 b(o)25 b(giv)n(e)h(a)f(coun)n(terexample,)f(y)n(ou)g(ma)n(y)g(need)g (to)h(assume)g(the)f(existence)h(of)h(some)f(other)f Fe(F)3033 4836 y Fd(0)3056 4868 y Ff(,)h Fe(G)3162 4836 y Fd(0)3184 4868 y Ff(,)h(or)f Fe(P)3384 4836 y Fd(0)3406 4868 y Ff(,)g(and)g(then)150 4959 y(describ)r(e)i(ho)n(w)g Fe(F)11 b Ff(,)28 b Fe(G)p Ff(,)f(or)g Fe(P)38 b Ff(could)27 b(b)r(e)f(constructed)h(from)g(them)e(suc)n(h)h(that)h Fe(F)11 b Ff(,)27 b Fe(G)p Ff(,)g(or)h Fe(P)37 b Ff(is)27 b(still)h(a)f(O)n(WF)g(\(or)g(PR)n(G)g(or)150 5051 y(O)n(WP\))e(but)g (the)g(answ)n(er)i(to)f(the)f(question)g(is)i(no)e(for)i(that)e Fe(F)11 b Ff(,)26 b Fe(G)p Ff(,)g(and)f Fe(P)11 b Ff(.)254 5110 y Fg(2)288 5142 y Fe(f)26 b Fc(\016)17 b Fe(g)28 b Ff(refers)f(to)f(the)f Fb(c)l(omp)l(osition)i Ff(of)f(functions)g Fe(f)34 b Ff(and)25 b Fe(g)s Ff(,)h(that)f(is)h Fe(f)g Fc(\016)17 b Fe(g)s Ff(\()p Fe(x)p Ff(\))j(=)h Fe(f)8 b Ff(\()p Fe(g)s Ff(\()p Fe(x)p Ff(\)\))1889 5416 y Fp(9-1)p eop %%Page: 2 2 2 1 bop 150 159 a Fm(Problem)45 b(3:)61 b(O)l(WP)45 b Fa(\))f Fm(PR)l(G)150 361 y Fp(\(20)35 b(p)s(oin)m(ts\))f(Let)h Fl(P)44 b Fp(:)32 b Fk(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)1169 328 y Fj(\003)1243 361 y Fk(!)31 b(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)1585 328 y Fj(\003)1661 361 y Fp(b)s(e)34 b(a)g(O)m(WP)-8 b(,)36 b(and)d(let)h Fl(b)h Fp(b)s(e)e(a)i(hard-core)f (predicate)g(of)h Fl(P)13 b Fp(.)150 474 y(Consider)28 b(the)h(follo)m(wing)f(construction,)i(due)f(to)h(Blum)e(and)h(Micali,) g(of)h(a)g(PR)m(G.)g(T)-8 b(o)30 b(generate)h(a)f(k)m(ey)150 587 y Fl(P)13 b(K)7 b Fp(,)30 b(just)g(use)g(the)h(k)m(ey)g(generation) g(algorithm)e(for)h Fl(P)13 b Fp(.)41 b(Let)31 b Fl(s)f Fp(denote)g(the)h(input,)e(where)h Fk(j)p Fl(s)p Fk(j)25 b Fp(=)g Fl(k)s Fp(.)291 700 y(Let)31 b Fl(x)506 714 y Fi(0)571 700 y Fp(=)25 b Fl(s)p Fp(,)30 b(and)g(let)g Fl(x)1125 714 y Fh(i)1178 700 y Fp(=)25 b Fl(P)13 b Fp(\()p Fl(x)1432 714 y Fh(i)p Fj(\000)p Fi(1)1551 700 y Fp(\))31 b(for)f(all)f Fl(i)d(>)e Fp(0.)42 b(De\014ne)1230 926 y Fl(G)p Fp(\()p Fl(s)p Fp(\))26 b(=)f Fl(b)p Fp(\()p Fl(s)g Fp(=)g Fl(x)1826 940 y Fi(0)1866 926 y Fp(\))20 b Fk(\016)h Fl(b)p Fp(\()p Fl(x)2113 940 y Fi(1)2153 926 y Fp(\))f Fk(\016)h Fl(:)15 b(:)g(:)21 b Fk(\016)g Fl(b)p Fp(\()p Fl(x)2592 941 y Fh(k)2635 926 y Fp(\))291 1192 y Fq(\(a\))30 b Fp(Pro)m(v)m(e)h(that)g Fl(G)g Fp(is)e Fn(next-bit)k(unpr)-5 b(e)g(dictable)p Fp(.)42 b(That)30 b(is,)g(that)150 1405 y Fk(8f)p Fl(A)p Fk(g)359 1419 y Fh(n)437 1405 y Fk(9)p Fl(f)k Fk(2)25 b Fl(neg)34 b Fk(8)p Fl(k)150 1518 y Fp(Pr)o([\()p Fl(\013;)15 b(j)5 b Fp(\))28 b Fk( )d Fl(A)694 1533 y Fh(k)737 1518 y Fp(;)15 b Fl(s)25 b Fk( )h(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)1182 1485 y Fh(k)1227 1518 y Fp(;)273 1631 y Fl(r)28 b Fk( )d Fl(G)p Fp(\()p Fl(s)p Fp(\);)15 b Fl(b)26 b Fk( )f Fl(A)931 1646 y Fh(k)974 1631 y Fp(\()p Fl(\013;)15 b(r)1148 1645 y Fi(1)1209 1631 y Fk(\016)21 b Fl(:)15 b(:)g(:)21 b Fk(\016)g Fl(r)1508 1645 y Fh(j)1544 1631 y Fp(\))26 b(:)f Fl(b)h Fp(=)f Fl(r)1857 1645 y Fh(j)t Fi(+1)1983 1631 y Fp(])h Fk(\024)f Fp(1)p Fl(=)p Fp(2)c(+)f Fl(f)10 b Fp(\()p Fl(k)s Fp(\))291 1843 y(where)30 b Fl(r)e Fp(=)d Fl(r)760 1857 y Fi(1)819 1843 y Fk(\016)c Fl(r)926 1857 y Fi(2)986 1843 y Fk(\016)f Fl(:)15 b(:)g(:)22 b Fk(\016)e Fl(r)1284 1858 y Fh(k)r Fi(+1)1417 1843 y Fp(.)291 2056 y Fq(\(b\))40 b Fp(Pro)m(v)m(e)j(that)e(for)g Fn(any)h Fl(G)p Fp(,)i(if)c Fl(G)h Fp(is)g(next-bit)f(unpredictable)f(then)h(it) h(is)f(a)h(pseudorandom)150 2169 y(generator)32 b(\(in)d(the)i(sense)f (w)m(e)h(used)e(in)g(lecture)i(-)f(that)h Fl(G)g Fp(is)e (indistinguishable)c(from)30 b(random\).)150 2455 y Fm(Problem)45 b(4:)61 b(Pumping)44 b(Pseudorandomness)150 2658 y Fp(\(20)34 b(p)s(oin)m(ts\))d(Supp)s(ose)g(y)m(ou)h(are)h(giv)m(en)g(a)f(PR)m(G)h Fl(G)g Fp(suc)m(h)f(that)h Fk(j)p Fl(G)p Fp(\()p Fl(x)p Fp(\))p Fk(j)d Fp(=)e Fk(j)p Fl(x)p Fk(j)22 b Fp(+)g(1,)33 b(and)f(a)h(p)s(olynomial)150 2771 y Fl(p)p Fp(.)43 b(Construct)31 b(a)h Fl(G)838 2738 y Fj(0)893 2771 y Fp(suc)m(h)f(that)h Fk(j)p Fl(G)1393 2738 y Fj(0)1417 2771 y Fp(\()p Fl(x)p Fp(\))p Fk(j)c Fp(=)e Fl(p)p Fp(\()p Fk(j)p Fl(x)p Fk(j)p Fp(\),)33 b(and)e(pro)m(v)m(e)h(that)g Fl(G)2658 2738 y Fj(0)2713 2771 y Fp(is)e(also)h(a)h(PR)m(G.)g(Y)-8 b(our)32 b(pro)s(of)150 2884 y(should)21 b(b)s(e)i(along)g(similar)e (lines)h(to)i(the)f(pro)s(of)g(y)m(ou)g(use)g(in)f(problem)g(3.)39 b(A)23 b(certain)g(lev)m(el)g(of)h(informalit)m(y)150 2997 y(is)29 b(acceptable.)150 3283 y Fm(Problem)45 b(5:)61 b(Predictable)46 b(O)l(WFs)150 3486 y Fp(\(5)31 b(p)s(oin)m(ts\))540 3453 y Fi(3)609 3486 y Fp(Let)g Fl(F)39 b Fp(:)26 b Fk(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)1140 3453 y Fj(\003)1207 3486 y Fk(!)25 b(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)1543 3453 y Fj(\003)1615 3486 y Fp(b)s(e)30 b(a)h(length-preserving)d(one-w) m(a)m(y)33 b(function.)39 b(Let)31 b Fl(b)3475 3500 y Fh(i)3503 3486 y Fp(\()p Fl(x)p Fp(\))h(b)s(e)150 3599 y(a)h(predicate)f(whic)m(h)f(returns)h(the)g Fl(i)1392 3566 y Fh(th)1495 3599 y Fp(bit)g(of)g Fl(x)p Fp(.)47 b(Construct)32 b(a)h(length)f(preserving)f(function)g Fl(F)3522 3566 y Fj(0)3545 3599 y Fp(,)j(and)150 3712 y(pro)m(v)m(e)e(that)g Fl(F)665 3679 y Fj(0)719 3712 y Fp(is)f(\(1\))h(a)g(one-w)m(a)m(y)g(function,)f(and)f(\(2\))j(for)e (an)m(y)g(1)c Fk(\024)g Fl(i)g Fk(\024)f Fl(k)s Fp(,)32 b(it)f(is)f(easy)i(for)f(random)f Fl(x)150 3825 y Fp(of)h(length)e Fl(k)34 b Fp(to)d(compute)g Fl(b)1131 3839 y Fh(i)1159 3825 y Fp(\()p Fl(x)p Fp(\),)g(giv)m(en)f(only)g Fl(F)1845 3792 y Fj(0)1868 3825 y Fp(\()p Fl(x)p Fp(\))h(as)g(input.)p 150 4990 1440 4 v 254 5044 a Fg(3)288 5076 y Ff(Consider)d(this)e(a)h (c)n(hallenge)g(problem.)36 b(A)n(t)26 b(only)g(5)h(p)r(oin)n(ts)f(it)h (should)f(not)g(seriously)i(impact)d(y)n(our)h(grade)h(whether)150 5167 y(y)n(ou)e(do)h(this)f(problem)g(or)h(not.)1889 5416 y Fp(9-2)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF handouts/ps5.tex0100664003325600237220000001511707362623452013372 0ustar mliskovta6875\iffalse NOTE: If you do not collaborate with anyone, simply remove the '**** INSERT COLLABORATORS HERE****' and replace it with 'nobody.' Don't erase the collaboration statement! Thanks. INSTRUCTIONS: (if this is not ps1, use the right file name) Clip out the ********* INSERT HERE ********* bits below and insert appropriate TeX code. Once you are done with your file, run ``latex ps1.tex'' from the Athena shell. If your TeX code is clean, the latex will exit back to a prompt. Once this is done, run ``dvips ps1.dvi'' which should print your file to the nearest printer. There will be residual files called ps1.log, ps1.aux, and ps1.dvi. All these can be deleted, but do not delete ps1.tex. \fi \documentclass[11pt]{article} \usepackage{amsfonts} \usepackage{latexsym} \setlength{\oddsidemargin}{.25in} \setlength{\evensidemargin}{.25in} \setlength{\textwidth}{6in} \setlength{\topmargin}{-0.4in} \setlength{\textheight}{8.5in} \newcommand{\handout}[5]{ \renewcommand{\thepage}{#1-\arabic{page}} \noindent \begin{center} \framebox{ \vbox{ \hbox to 5.78in { {\bf 6.875J/18.425J Cryptography and Cryptanalysis} \hfill #2 } \vspace{4mm} \hbox to 5.78in { {\Large \hfill #5 \hfill} } \vspace{2mm} \hbox to 5.78in { {\it #3 \hfill #4} } } } \end{center} \vspace*{4mm} } \newcommand{\al}{\alpha} \newcommand{\Z}{\mathbb Z} \newcommand{\N}{\mathbb N} \begin{document} \handout{P5}{DUE: October 24, 2001}{Instructor: Anna Lysyanskaya} {Teaching Assistant: Moses Liskov} {*********** INSERT YOUR NAME HERE ***********: PS 5} I collaborated with ******** INSERT COLLABORATORS OR `nobody' HERE **********. \bigskip \section*{Problem 1: Fun with primitives} (35 points) Let $F: \{0, 1\}^* \rightarrow \{0, 1\}^*$ be a one-way function. Let $G: \{0, 1\}^* \rightarrow \{0, 1\}^*$ be a pseudorandom generator. Finally, let $P: \{0, 1\}^* \rightarrow \{0, 1\}^*$ be a one-way permutation. (Refer to lecture notes from lecture 10 for rigorous definitions of these primitives.) For each of the following, answer either ``yes'' or ``no.'' If you answer ``yes'' give an intuitive argument to support your answer. If you answer ``no'' give a counterexample.\footnote{To give a counterexample, you may need to assume the existence of some other $F'$, $G'$, or $P'$, and then describe how $F$, $G$, or $P$ could be constructed from them such that $F$, $G$, or $P$ is still a OWF (or PRG or OWP) but the answer to the question is no for that $F$, $G$, and $P$.} For at least {\em one} of your ``yes'' answers, if you give any, give an actual proof (5 points) of your answer. \bigskip {\bf (a)} Is $G \circ G$ a PRG? \bigskip{\bf Solution:} ***************** INSERT PROBLEM 1a SOLUTION HERE *************** \bigskip {\bf (b)} Is $F \circ F$ a OWF? \bigskip{\bf Solution:} ***************** INSERT PROBLEM 1b SOLUTION HERE *************** \bigskip {\bf (c)} Is $P \circ P$ a OWP? \bigskip{\bf Solution:} ***************** INSERT PROBLEM 1c SOLUTION HERE *************** \bigskip {\bf (d)} Is $P \circ F$ a OWP? \bigskip{\bf Solution:} ***************** INSERT PROBLEM 1d SOLUTION HERE *************** \bigskip {\bf (e)} Is $G \circ P$ a PRG? \bigskip{\bf Solution:} ***************** INSERT PROBLEM 1e SOLUTION HERE *************** \bigskip {\bf (f)} Is $G \circ F$ a PRG? \bigskip{\bf Solution:} ***************** INSERT PROBLEM 1f SOLUTION HERE *************** \section*{Problem 2: Hard-core bits} (20 points) Let $b: \{0, 1\}^* \rightarrow \{0, 1\}$ be an efficiently computable, nontrivial predicate (with respect to sampling with the uniform distribution). \bigskip {\bf (a)} Prove that if OWP's exist, there is a OWP $P: \{0, 1\}^* \rightarrow \{0, 1\}^*$ such that $b$ is a hard-core predicate of $P$. Here you may assume that $b$ is of the form $b_r(x) = r \cdot x \bmod 2$, where $|r| = |x|$. \bigskip{\bf Solution:} ***************** INSERT PROBLEM 2a SOLUTION HERE *************** \bigskip {\bf (b)} Prove that if $P: \{0, 1\}^* \rightarrow \{0, 1\}^*$ is an efficiently computable permutation, and $b$ is a hard-core predicate of $P$, then $P$ is a OWP. Also, prove that if $F: \{0, 1\}^* \rightarrow \{0, 1\}^*$ and $b(x)$ is hard to compute given only access to $F(x)$, then $F$ is not necessarily a OWF. \bigskip{\bf Solution:} ***************** INSERT PROBLEM 2b SOLUTION HERE *************** \section*{Problem 3: OWP $\Rightarrow$ PRG} (20 points) Let $P: \{0, 1\}^* \rightarrow \{0, 1\}^*$ be a OWP, and let $b$ be a hard-core predicate of $P$. Consider the following construction, due to Blum, Blum and Shubb, of a PRG. To generate a key $PK$, just use the key generation algorithm for $P$. Let $s$ denote the input, where $|s| = k$. Let $x_0 = s$, and let $x_i = P(x_{i-1})$ for all $i > 0$. Define $$G(s) = b(s = x_0) \circ b(x_1) \circ \ldots \circ b(x_k)$$ \bigskip {\bf (a)} Prove that $G$ is {\em next-bit unpredictable}. That is, that \begin{tabbing} $\forall \{A\}_n~\exists f \in neg~\forall k$\\ $\Pr[$\=$(\alpha, j) \leftarrow A_k; s \leftarrow \{0, 1\}^k;$\\ \>$r \leftarrow G(s); b \leftarrow A_k(\alpha, r_1 \circ \ldots \circ r_j): b = r_{j+1}] \leq 1/2 + f(k)$ \end{tabbing} where $r = r_1 \circ r_2 \circ \ldots \circ r_{k+1}$. \bigskip{\bf Solution:} ***************** INSERT PROBLEM 3a SOLUTION HERE *************** \bigskip {\bf (b)} Prove that for {\em any} $G$, if $G$ is next-bit unpredictable then it is a pseudorandom generator (in the sense we used in lecture - that $G$ is indistinguishable from random). \bigskip{\bf Solution:} ***************** INSERT PROBLEM 3b SOLUTION HERE *************** \section*{Problem 4: Pumping Pseudorandomness} (20 points) Suppose you are given a PRG $G$ such that $|G(x)| = |x| + 1$, and a polynomial $p$. Construct a $G'$ such that $|G'(x)| = p(|x|)$, and prove that $G'$ is also a PRG. Your proof should be along similar lines to the proof you use in problem 3. \bigskip{\bf Solution:} ***************** INSERT PROBLEM 4 SOLUTION HERE *************** \section*{Problem 5: Predictable OWFs} (5 points)\footnote{Consider this a challenge problem. At only 5 points it should not seriously impact your grade whether you do this problem or not.} Let $F: \{0, 1\}^* \rightarrow \{0, 1\}^*$ be a length-preserving one-way function. Let $b_i(x)$ be a predicate which returns the $i^{th}$ bit of $x$. Construct a length preserving function $F'$, and prove that $F'$ is (1) a one-way function, and (2) for any $1 \leq i \leq k$, it is easy for random $x$ of length $k$ to compute $b_i(x)$, given only $F'(x)$ as input. \bigskip{\bf Solution:} ***************** INSERT PROBLEM 5 SOLUTION HERE *************** \end{document} handouts/ps6.tex0100664003325600237220000002116007366271754013376 0ustar mliskovta6875\iffalse NOTE: If you do not collaborate with anyone, simply remove the '**** INSERT COLLABORATORS HERE****' and replace it with 'nobody.' Don't erase the collaboration statement! Thanks. INSTRUCTIONS: (if this is not ps1, use the right file name) Clip out the ********* INSERT HERE ********* bits below and insert appropriate TeX code. Once you are done with your file, run ``latex ps6.tex'' from the Athena shell. If your TeX code is clean, the latex will exit back to a prompt. Once this is done, run ``dvips ps6.dvi'' which should print your file to the nearest printer. There will be residual files called ps1.log, ps1.aux, and ps1.dvi. All these can be deleted, but do not delete ps1.tex. \fi \documentclass[11pt]{article} \usepackage{amsfonts} \usepackage{latexsym} \setlength{\oddsidemargin}{.25in} \setlength{\evensidemargin}{.25in} \setlength{\textwidth}{6in} \setlength{\topmargin}{-0.4in} \setlength{\textheight}{8.5in} \newcommand{\handout}[5]{ \renewcommand{\thepage}{#1-\arabic{page}} \noindent \begin{center} \framebox{ \vbox{ \hbox to 5.78in { {\bf 6.875J/18.425J Cryptography and Cryptanalysis} \hfill #2 } \vspace{4mm} \hbox to 5.78in { {\Large \hfill #5 \hfill} } \vspace{2mm} \hbox to 5.78in { {\it #3 \hfill #4} } } } \end{center} \vspace*{4mm} } \newcommand{\al}{\alpha} \newcommand{\Z}{\mathbb Z} \newcommand{\N}{\mathbb N} \begin{document} \handout{P6}{DUE: November 7, 2001}{Instructor: Anna Lysyanskaya} {Teaching Assistant: Moses Liskov} {*********** INSERT YOUR NAME HERE ***********: PS 6} I collaborated with ******** INSERT COLLABORATORS OR `nobody' HERE **********. \bigskip \section*{Problem 1} (24 points) Recall that in lecture we mentioned that pseudorandom generators come in handy when we want to reduce the communication complexity (i.e., length of the data sent) of encryption. This problem asks you to give a proof for this. \bigskip {\bf (a)} Let $g$ be a pseudorandom number generator, from $k$ bits to $\ell$ bits. Suppose a semantically secure cryptosystem $(G,E,D)$ is given. In this cryptosystem, encrypting a message of size $n$ results in communication complexity $p(n)$. Suppose Alice's public key is $PK$, which was obtained with security parameter $k$. Bob wants to send to Alice an $\ell$-bit message $m$. To do so, he proceeds as follows: he creates a random $k$-bit seed $s$. From that, he sets his pseudo-one-time-pad $r = g(s)$. The ciphertext he then sends is $(m\oplus r, E(s))$. Is the resulting encryption scheme semantically secure? Prove your answer. \bigskip {\bf Solution:} *************** INSERT PROBLEM 1a SOLUTION HERE *********** \bigskip {\bf (b)} Suppose that, instead, Alice's $PK$ is a public key of a trapdoor permutation $P_{PK}$. In order to send $m$, Bob comes up with a random $k$-bit seed $s$, and computes $r = g(s)$. He then sends $(m\oplus r, P_{PK}(s))$. Is the resulting encryption scheme semantically secure? Prove your answer. \bigskip {\bf Solution:} *************** INSERT PROBLEM 1b SOLUTION HERE *********** \bigskip {\bf (c)} Consider the following cryptosystem: Key generation: generate $n = pq$ where $p$ and $q$ are primes of length $k$ such that $p\equiv q \equiv 7 \bmod 8$. Public key is $n$, secret key is $p,q$. Encryption: to encrypt an $\ell$-bit message $m$, choose a random $s \in \Z_n^*$, create a random pad $r$ as follows: $x_1 = s^2 \bmod n$; $x_i = x_{i-1}^2 \bmod n$ for $2 \leq i \leq \ell+1$; $r_i = LSB(x_i)$ for $1 \leq i \leq \ell$. Output $(m \oplus r, x_{\ell+1})$. First, describe how to decrypt in this scheme with two modular exponentiations and $O(\ell)$ modular multiplications. {\it Hint: The fact that $p \equiv 7 \bmod 8$ comes in handy.} Second, formulate a hardness assumption that corresponds to factoring $n$ of the form used in this system, and prove that this cryptosystem is GM-secure under your assumption. \bigskip {\bf Solution:} *************** INSERT PROBLEM 1c SOLUTION HERE *********** \section*{Problem 2} (21 points) A message authentication code (MAC) is a method for ensuring that the data received over the network came from the right person. More precisely, it is an object that satisfies the following properties: Suppose Alice and Bob share a secret $s$, and Bob wants to send message $m$ to Alice. Then $MAC_s(m)$ is an efficiently computable function such that even if an active attacker Eve queries it on a set of messages of its choice, it still cannot authenticate any message not explicitly queried. To give a more formal definition, let us first extend our notation. If $A^{?}$ is an oracle Turing machine, then by $Q\leftarrow A^{O}(x)$ we denote the contents of $A$'s query tape upon termination with oracle $O$ and input $x$.\footnote{By the query tape, we mean the list of all oracle queries and the answers that were returned in the course of $A$'s execution.} Now: A function family $\{M_s(\circ)\}$ is a MAC if for all PPTF $\{A_n\}$, there exists a negligible function $\nu(k)$ such that \[\Pr[s \leftarrow \{0,1\}^k; (m,x),Q \leftarrow A_k^{M_s}: x = M_s(m) \mbox{ and } (m,x)\notin Q] \leq \nu(k)\] That is to say the adversary cannot find a pair $(m, x)$ such that $x = M_s(m)$ even with oracle access to $M_s$, so long as $M_s$ does not ask the oracle for $M_s(m)$ and receive the answer $x$. \bigskip {\bf (a)} Let $\{F_s~:~\{0,1\}^{|s|}\mapsto\{0,1\}^{|s|}\}$ be a PRF family. Show that it is a MAC. \bigskip {\bf Solution:} *************** INSERT PROBLEM 2a SOLUTION HERE *********** \bigskip {\bf (b)} Is it the case that a MAC is a PRF? Prove your answer. \bigskip {\bf Solution:} *************** INSERT PROBLEM 2b SOLUTION HERE *********** \bigskip {\bf (c)} Is it true that if MACs exist then PRFs exist? Prove your answer. \bigskip {\bf Solution:} *************** INSERT PROBLEM 2c SOLUTION HERE *********** \section*{Problem 3} (20 points) In this problem, we will try to relate various kinds of assumptions about one-way functions to each other. \bigskip {\bf (a)} Prove that if one-way function families exist, then one-way functions exist. Can your proof be adjusted easily to go through for OWPs? Informally, argue why or why not. \bigskip {\bf Solution:} *************** INSERT PROBLEM 3a SOLUTION HERE *********** \bigskip {\bf (b)} A function is length-preserving if the length of its output is always equal to the length of its input. Prove that if one-way functions exist, then length-preserving one-way functions exist. \bigskip {\bf Solution:} *************** INSERT PROBLEM 3b SOLUTION HERE *********** \section*{Problem 4} (25 points) Suppose that $F_s(x)$ is a pseudorandom function from $k$ bit input to $k$ bit output, where $s$ is the seed and $x$ is the input, decide whether each of the following are always pseudorandom functions. If so, prove it. If not, provide a counterexample and explain how to distinguish the given function from random. It is okay to leave out the details of your proofs so long as they can be fleshed out correctly without too much trouble. \bigskip {\bf (a)} $F_s^1(x) = F_{0^k}(x) \circ F_s(0^k)$ \bigskip {\bf Solution:} *************** INSERT PROBLEM 4a SOLUTION HERE *********** \bigskip {\bf (b)} $F_s^2(x) = F_{0^k}(s) \circ F_s(x)$ \bigskip {\bf Solution:} *************** INSERT PROBLEM 4b SOLUTION HERE *********** \bigskip {\bf (c)} $F_s^3(x) = F_s(x) \circ F_x(s)$ \bigskip {\bf Solution:} *************** INSERT PROBLEM 4c SOLUTION HERE *********** \bigskip {\bf (d)} $F_s^4(x) = F_{s_1}(x) \circ F_{s_2}(x)$ where $s_1 = F_s(x)$ and $s_2 = F_{\overline{s}}(x)$ \bigskip {\bf Solution:} *************** INSERT PROBLEM 4d SOLUTION HERE *********** \bigskip {\bf (e)} $F_s^5(x) = F_s(F_x(x))$ \bigskip {\bf Solution:} *************** INSERT PROBLEM 4e SOLUTION HERE *********** \section*{Problem 5} (10 points) Summarize what we know so far. We have discussed several primitives in this class: secure public-key cryptosystems (PKC), one-way functions and one-way function families (OWF and OWFF), one-way permutations and one-way permutation families (OWP and OWPF), pseudorandom generators (PRG), pseudorandom functions (PRF), hardcore bits (HB), and trapdoor predicates (TP). For each pair of these, specify whether we know if the existence of one implies the existence of the other. Give as many such inferences as you can, and for each inference give either a reference to a paper, book, or problem set problem, or give a brief proof idea. {\it Since implications are transitive, you do not need to give every arrow you can deduce - just give as many independent results as you can.} \bigskip {\bf Solution:} *************** INSERT PROBLEM 5 SOLUTION HERE *********** \end{document} handouts/ho11.ps0100664003325600237220000054461207366272100013256 0ustar mliskovta6875%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: ho11.dvi %%Pages: 3 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips ho11 -o ho11.ps %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2001.10.26:1033 %%BeginProcSet: tex.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul 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y Fn(\(20)24 b(p)s(oin)m(ts\))f(In)f(this)g(problem,)i(w)m (e)f(will)e(try)i(to)h(relate)f(v)-5 b(arious)22 b(kinds)g(of)h (assumptions)e(ab)s(out)i(one-w)m(a)m(y)150 2403 y(functions)29 b(to)i(eac)m(h)h(other.)291 2584 y Fo(\(a\))38 b Fn(Pro)m(v)m(e)i(that) f(if)f(one-w)m(a)m(y)i(function)d(families)g(exist,)k(then)d(one-w)m(a) m(y)j(functions)c(exist.)65 b(Can)150 2697 y(y)m(our)30 b(pro)s(of)g(b)s(e)f(adjusted)g(easily)h(to)h(go)g(through)e(for)h(O)m (WPs?)41 b(Informally)-8 b(,)28 b(argue)j(wh)m(y)f(or)g(wh)m(y)f(not.) 291 2878 y Fo(\(b\))38 b Fn(A)h(function)e(is)g(length-preserving)g(if) h(the)g(length)g(of)h(its)f(output)g(is)f(alw)m(a)m(ys)i(equal)f(to)i (the)150 2991 y(length)33 b(of)h(its)g(input.)49 b(Pro)m(v)m(e)35 b(that)g(if)e(one-w)m(a)m(y)i(functions)e(exist,)h(then)g (length-preserving)e(one-w)m(a)m(y)150 3104 y(functions)d(exist.)150 3383 y Fk(Problem)45 b(4)150 3586 y Fn(\(25)25 b(p)s(oin)m(ts\))e(Supp) s(ose)g(that)h Fj(F)1201 3600 y Fh(s)1238 3586 y Fn(\()p Fj(x)p 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known{end-hook}if %%EOF handouts/psheader.tex0100664003325600237220000000323507366271427014461 0ustar mliskovta6875\iffalse NOTE: If you do not collaborate with anyone, simply remove the '**** INSERT COLLABORATORS HERE****' and replace it with 'nobody.' Don't erase the collaboration statement! Thanks. INSTRUCTIONS: (if this is not ps1, use the right file name) Clip out the ********* INSERT HERE ********* bits below and insert appropriate TeX code. Once you are done with your file, run ``latex ps6.tex'' from the Athena shell. If your TeX code is clean, the latex will exit back to a prompt. Once this is done, run ``dvips ps6.dvi'' which should print your file to the nearest printer. There will be residual files called ps1.log, ps1.aux, and ps1.dvi. All these can be deleted, but do not delete ps1.tex. \fi \documentclass[11pt]{article} \usepackage{amsfonts} \usepackage{latexsym} \setlength{\oddsidemargin}{.25in} \setlength{\evensidemargin}{.25in} \setlength{\textwidth}{6in} \setlength{\topmargin}{-0.4in} \setlength{\textheight}{8.5in} \newcommand{\handout}[5]{ \renewcommand{\thepage}{#1-\arabic{page}} \noindent \begin{center} \framebox{ \vbox{ \hbox to 5.78in { {\bf 6.875J/18.425J Cryptography and Cryptanalysis} \hfill #2 } \vspace{4mm} \hbox to 5.78in { {\Large \hfill #5 \hfill} } \vspace{2mm} \hbox to 5.78in { {\it #3 \hfill #4} } } } \end{center} \vspace*{4mm} } \newcommand{\al}{\alpha} \newcommand{\Z}{\mathbb Z} \newcommand{\N}{\mathbb N} \begin{document} \handout{P5}{DUE: October 24, 2001}{Instructor: Anna Lysyanskaya} {Teaching Assistant: Moses Liskov} {*********** INSERT YOUR NAME HERE ***********: PS 5} I collaborated with ******** INSERT COLLABORATORS OR `nobody' HERE **********. 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Fl(1)p Fh(g)3194 1595 y Fe(n)3243 1628 y Fl(,)27 b(if)e(it)h(has)g(not)150 1741 y(y)m(et)35 b(giv)m(en)f(an)g(answ)m(er)g(for)g(that)g(input,)g (it)f(returns)g(a)h(random)f(v)-5 b(alue)34 b(in)f Fh(f)p Fl(0)p Ff(;)15 b Fl(1)p Fh(g)2994 1708 y Fe(n)3077 1741 y Fl(that)34 b(it)g(has)g(nev)m(er)150 1854 y(returned)29 b(b)s(efore.)291 1967 y(Pro)m(v)m(e)j(that)f Fh(O)817 1981 y Fg(0)856 1967 y Fl(\()p Ff(n)p Fl(\))g(is)e(indistinguishable)c (from)30 b Fh(O)2088 1981 y Fg(1)2128 1967 y Fl(\()p Ff(n)p Fl(\),)h(in)e(the)i(strongest)g(sense)f(y)m(ou)h(can.)150 2253 y Fi(Problem)45 b(2:)61 b(A)l(ttac)l(king)46 b(Lub)l(y-Rac)l(k)l (o\013)150 2456 y Fl(Recall)32 b(the)h(Lub)m(y-Rac)m(k)m(o\013)h (construction)f(for)f(a)h(pseudorandom)e(p)s(erm)m(utation)h(from)g(a)h (pseudoran-)150 2569 y(dom)d(function.)39 b(Let)31 b Ff(F)964 2583 y Fe(i)1023 2569 y Fl(b)s(e)f(a)h(set)g(of)f (pseudorandom)f(functions,)g(and)h(let)1398 2795 y Ff(LR)1529 2809 y Fe(i)1557 2795 y Fl(\()p Ff(L;)15 b(R)q Fl(\))26 b(=)f(\()p Ff(R)q(;)15 b(F)2124 2809 y Fe(i)2153 2795 y Fl(\()p Ff(R)q Fl(\))21 b Fh(\010)f Ff(L)p Fl(\))291 2962 y(de\014ne)36 b(the)h Ff(i)754 2929 y Fe(th)861 2962 y Fl(round)e(of)i(op)s(eration)f(of)h(the)g(Lub)m(y-Rac)m(k)m (o\013)h(construction.)60 b(F)-8 b(or)37 b(an)g(in)m(teger)g Ff(k)s Fl(,)150 3075 y(de\014ne)1107 3301 y Ff(LR)21 b Fh(\000)f Ff(k)s Fl(\()p Ff(L;)15 b(R)q Fl(\))26 b(=)f Ff(LR)1895 3316 y Fe(k)1938 3301 y Fl(\()p Ff(LR)2104 3316 y Fe(k)r Fd(\000)p Fg(1)2237 3301 y Fl(\()p Ff(:)15 b(:)g(:)h Fl(\()p Ff(L;)f(R)q Fl(\))g Ff(:)g(:)g(:)r Fl(\))291 3467 y(to)31 b(b)s(e)f(the)h(output)f(of)g Ff(k)s Fl(-round)f(Lub)m(y-Rac)m(k)m(o\013.)291 3580 y(W)-8 b(e)32 b(sa)m(y)f(a)g(p)s(erm)m(utation)e(family)g Ff(P)1537 3594 y Fe(S)1618 3580 y Fl(o)m(v)m(er)j Fh(f)p Fl(0)p Ff(;)15 b Fl(1)p Fh(g)2033 3547 y Fe(l)q Fg(\()p Fe(k)r Fg(\))2185 3580 y Fl(is)29 b(pseudorandom)g(if:)150 3793 y Fh(8f)p Ff(A)314 3760 y Fg(?)314 3821 y Fe(k)357 3793 y Fh(g)d(2)e Ff(P)13 b(P)g(T)43 b Fh(9)p Ff(\027)31 b Fh(2)25 b Ff(neg)s(l)32 b Fh(8)p Ff(k)150 3906 y Fl(Pr)o([)p Ff(S)f Fh( )25 b(f)p Fl(0)p Ff(;)15 b Fl(1)p Fh(g)695 3873 y Fe(k)740 3906 y Fl(;)g Fh(O)852 3920 y Fg(0)917 3906 y Fh( )26 b Ff(P)1092 3920 y Fe(S)1143 3906 y Fl(;)273 4019 y Fh(O)345 4033 y Fg(1)410 4019 y Fh( )f Fl(\005\()p Ff(l)r Fl(\()p Ff(k)s Fl(\)\);)273 4141 y Ff(b)312 4108 y Fd(0)360 4141 y Fh( )g Ff(A)544 4098 y Fd(O)600 4110 y Fc(b)544 4170 y Fe(k)661 4141 y Fl(:)h Ff(b)751 4108 y Fd(0)799 4141 y Fl(=)f Ff(b)p Fl(])h Ff(<)f Fl(1)p Ff(=)p Fl(2)c(+)f Ff(\027)6 b Fl(\()p Ff(k)s Fl(\))291 4354 y Fm(\(a\))30 b Fl(Pro)m(v)m(e)h(that)g Ff(LR)21 b Fh(\000)f Fl(2)31 b(is)e(not)i(a)f(pseudorandom)f(p)s(erm)m(utation)h (family)-8 b(.)291 4566 y(W)g(e)32 b(sa)m(y)f(a)g(p)s(erm)m(utation)e (family)g Ff(P)1537 4580 y Fe(S)1618 4566 y Fl(is)h(strongly)f (pseudorandom)g(if:)150 4779 y Fh(8f)p Ff(A)314 4734 y Fg(?)p Fe(;)p Fg(?)314 4790 y Fe(n)405 4779 y Fh(g)c(2)g Ff(P)13 b(P)g(T)43 b Fh(9)p Ff(\027)31 b Fh(2)24 b Ff(neg)s(l)33 b Fh(8)p Ff(k)150 4894 y Fl(Pr)o([)p Ff(S)e Fh( )25 b(f)p Fl(0)p Ff(;)15 b Fl(1)p Fh(g)695 4861 y Fe(k)740 4894 y Fl(;)g Fh(O)852 4908 y Fg(0)917 4894 y Fh( )26 b Ff(P)1092 4908 y Fe(S)1143 4894 y Fl(;)15 b Fh(O)1258 4861 y Fd(0)1281 4908 y Fg(0)1346 4894 y Fh( )25 b Ff(P)1533 4855 y Fd(\000)p Fg(1)1520 4922 y Fe(S)273 5006 y Fh(O)345 5020 y Fg(1)410 5006 y Fh( )g Fl(\005;)15 b Fh(O)709 4973 y Fd(0)733 5020 y Fg(1)797 5006 y Fh( )26 b Fl(\005)982 4973 y Fd(\000)p Fg(1)1076 5006 y Fl(;)273 5124 y Ff(b)312 5091 y Fd(0)360 5124 y Fh( )f Ff(A)544 5091 y Fd(O)600 5103 y Fc(b)632 5091 y Fe(;)p Fd(O)710 5068 y Fb(0)732 5103 y Fc(b)793 5124 y Fl(:)g Ff(b)882 5091 y Fd(0)931 5124 y Fl(=)g Ff(b)p Fl(])g Ff(<)g Fl(1)p Ff(=)p Fl(2)d(+)e Ff(\027)6 b Fl(\()p Ff(k)s Fl(\))1867 5416 y(12-1)p eop %%Page: 2 2 2 1 bop 291 159 a Fl(where)34 b(\005)626 126 y Fd(\000)p Fg(1)755 159 y Fl(in)m(v)m(erts)h(\005.)54 b(Think)32 b(of)j(\005)g(as)g(b)s(eing)e(a)i(\014xed)f(but)g(randomly)f(c)m(hosen) j(p)s(erum)m(tation,)150 271 y(and)30 b(\005)395 238 y Fd(\000)p Fg(1)520 271 y Fl(as)g(its)g(in)m(v)m(erse.)291 384 y Fm(\(b\))g Fl(Pro)m(v)m(e)h(that)g Ff(LR)21 b Fh(\000)f Fl(3)31 b(is)e(not)i(a)g(strong)f(pseudorandom)f(p)s(erm)m(utation)g (family)-8 b(.)150 671 y Fi(Problem)45 b(3:)61 b(F)-11 b(our-Round)43 b(Lub)l(y-Rac)l(k)l(o\013)150 874 y Fl(In)29 b(class)g(w)m(e)i(pro)m(v)m(ed)f(that)g Ff(LR)19 b Fh(\000)g Fl(3)30 b(is)f(a)h(pseudorandom)e(p)s(erm)m(utation)h(family)-8 b(.)39 b(Pro)m(v)m(e)31 b(that)f Ff(LR)20 b Fh(\000)f Fl(4)150 987 y(is)26 b(a)h(strong)f(pseudorandom)f(p)s(erm)m(utation)h (family)-8 b(.)38 b(This)25 b(is)h(rather)g(di\016cult)f(to)i(come)h (up)d(with)g(from)150 1100 y(scratc)m(h,)32 b(so)e(w)m(e)h(pro)m(vide)f (a)g(pro)s(of)g(sk)m(etc)m(h)i(that)f(y)m(ou)f(should)f(\014ll)f(in.) 291 1212 y Fm(Pro)s(of)i(sk)m(etc)m(h)p Fl(:)38 b(First)25 b(of)g(all,)g(assume)g(w)m(e)g(ha)m(v)m(e)h(an)f(adv)m(ersary)g Ff(A)2619 1179 y Fg(?)p Fe(;)p Fg(?)2735 1212 y Fl(whic)m(h)e(defeats)j (the)f(securit)m(y)150 1325 y(of)34 b Ff(LR)23 b Fh(\000)g Fl(4.)51 b(Pro)m(v)m(e)36 b(that)e(without)f(loss)g(of)h(generalit)m(y) g(w)m(e)h(can)f(assume)g(that)g(the)g(adv)m(ersary)h(nev)m(er)150 1438 y(asks)c Fh(O)j Fl(or)d Fh(O)639 1405 y Fd(0)693 1438 y Fl(the)g(same)g(question)f(t)m(wice,)i(that)g(the)f(adv)m (ersary)g(nev)m(er)g(queries)f Fh(O)k Fl(on)c(an)h(output)g(of)150 1551 y Fh(O)225 1518 y Fd(0)248 1551 y Fl(,)g(and)f(that)g(the)h(adv)m (ersary)g(nev)m(er)f(queries)g Fh(O)1871 1518 y Fd(0)1924 1551 y Fl(on)g(the)h(output)f(of)h Fh(O)s Fl(.)291 1664 y(Next,)f(form)m(ulate)e(a)h(construction)f(for)g Ff(LR)17 b Fh(\000)f Fl(4)p Ff(R)q Fl(,)29 b(whic)m(h)e(p)s(erforms)g(the)h(Lub) m(y-Rac)m(k)m(o\013)i(function)150 1777 y(with)23 b(4)i(random)f (functions)f(instead)h(of)g Ff(F)1598 1791 y Fg(1)1638 1777 y Ff(;)15 b(F)1736 1791 y Fg(2)1777 1777 y Ff(;)g(F)1875 1791 y Fg(3)1915 1777 y Ff(;)g(F)2013 1791 y Fg(4)2053 1777 y Fl(.)39 b(This)23 b(construction)h(is)f(just)h(lik)m(e)g(the)h (one)f(from)150 1890 y(class,)33 b(except)g(for)f Ff(LR)22 b Fh(\000)f Fl(4)32 b(instead)g(of)g Ff(LR)22 b Fh(\000)f Fl(3.)46 b(Pro)m(v)m(e)34 b(that)e Ff(LR)22 b Fh(\000)f Fl(4)p Ff(R)34 b Fl(is)d(indistinguishable)26 b(from)150 2003 y(a)31 b(random)e(function,)h(similarly)d(to)k(ho)m(w)f(w)m(e)h (did)e(in)g(class.)291 2116 y(Next,)c(form)m(ulate)e(a)g(construction)f (for)h(\()p Ff(LR)5 b Fh(\000)g Fl(4)p Ff(R)q Fl(\))2044 2083 y Fd(\000)p Fg(1)2139 2116 y Fl(,)24 b(whic)m(h)d(op)s(erates)j (with)d(the)i(oracle)g Ff(LR)5 b Fh(\000)g Fl(4)p Ff(R)q Fl(,)150 2229 y(and)22 b(pro)m(v)m(e)i(that)f(adv)m(ersaries)g(that)h (ha)m(v)m(e)g(oracle)f(access)h(to)g Ff(LR)6 b Fh(\000)f Fl(4)p Ff(R)23 b Fl(and)g(another)g(\(c)m(hallenge\))g(oracle)150 2342 y(cannot)36 b(tell)e(whether)g(the)i(c)m(hallenge)f(oracle)g(w)m (as)h(\()p Ff(LR)24 b Fh(\000)f Fl(4)p Ff(R)q Fl(\))2406 2309 y Fd(\000)p Fg(1)2536 2342 y Fl(or)35 b(a)h(random)e(function.)54 b(This)33 b(is)150 2454 y(the)e(meat)g(of)f(the)h(pro)s(of.)291 2567 y(Finally)-8 b(,)35 b(put)f(it)h(all)f(together)i(and)f(pro)m(v)m (e)h(that)f Ff(LR)24 b Fh(\000)f Fl(4)36 b(is)e(indistinguishable)c (from)k(a)i(random)150 2680 y(p)s(erm)m(utation)j(in)f(this)h(mo)s (del.)67 b(Y)-8 b(our)40 b(pro)s(of)f(should)f(assume)h(this)g(is)g (not)h(the)f(case)i(and)e(deriv)m(e)h(a)150 2793 y(con)m(tradiction)30 b(b)m(y)h(pro)m(ving)e(that)i(one)g(of)f(the)h(functions)e Ff(F)2232 2807 y Fe(i)2291 2793 y Fl(is)g(not)i(pseudorandom.)150 3080 y Fi(Problem)45 b(4:)61 b(Encrypting)45 b(the)g(Secret)g(Key)150 3283 y Fl(Let)39 b(\()p Ff(G;)15 b(E)5 b(;)15 b(D)s Fl(\))41 b(b)s(e)d(a)i(\(one-pass\))f(GM-secure)h(cryptosystem.)67 b(This)37 b(should)g(b)s(e)h(though)m(t)h(of)g(as)h(a)150 3395 y(\\secure)32 b(en)m(v)m(elop)s(e")h({)f(the)g(adv)m(ersary)h (cannot)f(see)h(its)e(con)m(ten)m(ts,)j(no)e(matter)h(what)f(the)g (message)h(is.)150 3508 y(This)38 b(problem)g(sho)m(ws)h(that)h(one)g (has)g(to)g(b)s(e)f(careful)g(with)f(this)h(in)m(tuition.)66 b(Namely)-8 b(,)43 b(w)m(e)d(ask)g(the)150 3621 y(question,)28 b(\\Can)g(I)g(trust)f(m)m(y)h(encryption)f(to)i(the)f(exten)m(t)h(that) g(I)e(can)i(publish)24 b(the)k(encryption)f(of)h(m)m(y)150 3734 y(secret)j(k)m(ey?")291 3947 y Fm(\(a\))26 b Fl(\(7)h(p)s(oin)m (ts\))f(Construct)g(a)h(cryptosystem)g(\()p Ff(G)2014 3914 y Fd(0)2038 3947 y Ff(;)15 b(E)2150 3914 y Fd(0)2174 3947 y Ff(;)g(D)2292 3914 y Fd(0)2316 3947 y Fl(\))26 b(based)h(on)f(\()p Ff(G;)15 b(E)5 b(;)15 b(D)s Fl(\))29 b(suc)m(h)d(that)h(giv)m(en)150 4060 y Ff(t)e Fl(=)g Ff(E)376 4027 y Fd(0)400 4060 y Fl(\()p Ff(P)13 b(K)590 4027 y Fd(0)613 4060 y Ff(;)i(S)5 b(K)798 4027 y Fd(0)821 4060 y Fl(\),)27 b(the)f(adv)m(ersary)f(can)g(easily)g(distinguish)c (the)k(encryptions)f(of)i Fj(any)f Fl(t)m(w)m(o)i(messages)150 4173 y Ff(m)230 4187 y Fg(0)302 4173 y Fl(and)32 b Ff(m)561 4187 y Fg(1)601 4173 y Fl(.)48 b(\(Hin)m(t:)e Ff(t)32 b Fl(should)f(someho)m(w)j(con)m(tain)f Ff(S)5 b(K)2171 4140 y Fd(0)2194 4173 y Fl(.)48 b(T)-8 b(ry)32 b(to)i(imagine)e(ho)m(w) h(this)f(can)h(b)s(e)f(done)150 4285 y(without)d(making)h(\()p Ff(G)915 4252 y Fd(0)939 4285 y Ff(;)15 b(E)1051 4252 y Fd(0)1075 4285 y Ff(;)g(D)1193 4252 y Fd(0)1217 4285 y Fl(\))30 b(insecure.\))291 4498 y Fm(\(b\))41 b Fl(\(7)i(p)s(oin)m (ts\))e(Construct)h(a)g(cryptosystem)g(\()p Ff(G)2113 4465 y Fd(00)2156 4498 y Ff(;)15 b(E)2268 4465 y Fd(00)2312 4498 y Ff(;)g(D)2430 4465 y Fd(00)2472 4498 y Fl(\))42 b(based)g(on)g(\()p Ff(G;)15 b(E)5 b(;)15 b(D)s Fl(\))44 b(suc)m(h)d(that)150 4611 y(ev)m(en)27 b(giving)e Ff(t)g Fl(=)g Ff(E)845 4578 y Fd(00)888 4611 y Fl(\()p Ff(P)13 b(K)1078 4578 y Fd(00)1121 4611 y Ff(;)i(S)5 b(K)1306 4578 y Fd(00)1348 4611 y Fl(\))27 b(to)g(the)f(adv)m(ersary)-8 b(,)28 b(the)f(sc)m(heme)g(is)e(still)g(GM-secure.)40 b(Y)-8 b(ou)27 b(should)150 4724 y(imagine)35 b(that)i(the)g(adv)m (ersary)f(learns)f Ff(t)h Fl(at)h(the)g(same)g(time)f(as)g Ff(P)13 b(K)2615 4691 y Fd(00)2657 4724 y Fl(.)59 b(\(Hin)m(t:)52 b Ff(t)36 b Fl(should)e(someho)m(w)150 4837 y(con)m(tain)d(random)e (garbage.)42 b(T)-8 b(ry)30 b(to)h(imagine)f(ho)m(w)g(w)m(e)h(can)g(do) f(this)f(but)h Fj(stil)5 b(l)33 b(b)-5 b(e)32 b(able)h(to)g(de)-5 b(crypt)34 b Ff(t)p Fl(\))291 5049 y Fm(\(c\))g Fl(\(6)i(p)s(oin)m (ts\))d(Supp)s(ose)g(Alice)h(and)g(Bob)h(eac)m(h)g(ha)m(v)m(e)h(k)m (eys,)g(\()p Ff(P)13 b(K)2687 5063 y Fe(A)2745 5049 y Ff(;)i(S)5 b(K)2923 5063 y Fe(A)2981 5049 y Fl(\))34 b(and)g(\()p Ff(P)13 b(K)3414 5063 y Fe(B)3475 5049 y Ff(;)i(S)5 b(K)3653 5063 y Fe(B)3715 5049 y Fl(\))150 5162 y(under)19 b(\()p Ff(G;)c(E)5 b(;)15 b(D)s Fl(\).)39 b(Pro)m(v)m(e)22 b(that)f(Alice)f(can)g(securely)g(encrypt)g Fj(her)h Fl(secret)g(k)m(ey)g(under)e Fj(Bob's)i Fl(public)c(k)m(ey)-8 b(.)1867 5416 y(12-2)p eop %%Page: 3 3 3 2 bop 150 159 a Fl(That)24 b(is,)g(pro)m(v)m(e)h(that)f(\()p Ff(G;)15 b(E)5 b(;)15 b(D)s Fl(\))26 b(is)d(still)f(GM-secure)j(ev)m (en)g(when)d(the)i(adv)m(ersary)h(kno)m(ws)e Ff(P)13 b(K)3418 173 y Fe(A)3475 159 y Ff(;)i(P)e(K)3663 173 y Fe(B)3725 159 y Fl(,)150 271 y(and)30 b Ff(E)5 b Fl(\()p Ff(P)13 b(K)582 285 y Fe(B)643 271 y Ff(;)i(S)5 b(K)821 285 y Fe(A)879 271 y Fl(\).)41 b(F)-8 b(ormally)g(,)30 b(pro)m(v)m(e)i(that)150 484 y Fh(8f)p Ff(A)314 499 y Fe(k)357 484 y Fh(g)26 b(2)e Ff(P)13 b(P)g(T)43 b Fh(9)p Ff(\027)31 b Fh(2)25 b Ff(neg)s(l)32 b Fh(8)p Ff(k)150 597 y Fl(Pr)o([\()p Ff(P)13 b(K)455 611 y Fe(A)513 597 y Ff(;)i(S)5 b(K)691 611 y Fe(A)749 597 y Fl(\))26 b Fh( )f Ff(G)p Fl(\(1)1077 564 y Fe(k)1121 597 y Fl(\);)15 b(\()p Ff(P)e(K)1379 611 y Fe(B)1441 597 y Ff(;)i(S)5 b(K)1619 611 y Fe(B)1680 597 y Fl(\))25 b Fh( )h Ff(G)p Fl(\(1)2008 564 y Fe(k)2052 597 y Fl(\);)15 b Ff(t)25 b Fh( )h Ff(E)5 b Fl(\()p Ff(P)13 b(K)2557 611 y Fe(B)2618 597 y Ff(;)i(S)5 b(K)2796 611 y Fe(A)2854 597 y Fl(\);)273 710 y(\()p Ff(\013;)15 b(m)486 724 y Fg(0)526 710 y Ff(;)g(m)646 724 y Fg(1)686 710 y Fl(\))25 b Fh( )h Ff(A)p Fl(\()p Ff(P)13 b(K)1114 724 y Fe(A)1171 710 y Ff(;)i(P)e(K)1359 724 y Fe(B)1421 710 y Ff(;)i(t)p Fl(\);)g Ff(b)26 b Fh( )f(f)p Fl(0)p Ff(;)15 b Fl(1)p Fh(g)p Fl(;)g Ff(b)2049 677 y Fd(0)2100 710 y Fh( )25 b Ff(A)p Fl(\()p Ff(\013;)15 b(E)5 b Fl(\()p Ff(P)13 b(K)2672 724 y Fe(A)2732 710 y Ff(;)i(m)2852 725 y Fe(b)2886 710 y Fl(\)\))26 b(:)g Ff(b)3072 677 y Fd(0)3120 710 y Fl(=)f Ff(b)p Fl(])h Ff(<)f Fl(1)p Ff(=)p Fl(2)c(+)f Ff(\027)6 b Fl(\()p Ff(k)s Fl(\))150 996 y Fi(Problem)45 b(5:)61 b(Insecure)45 b(Signatures)150 1199 y Fl(All)35 b(of)i(the)g(signature)f(sc)m(hemes)h(presen)m(ted)g (in)e(this)h(problem)f(are)i(insecure.)58 b(A)37 b(signature)f(sc)m (heme)150 1312 y(can)29 b(b)s(e)g(brok)m(en)g(in)e(a)j(n)m(um)m(b)s(er) d(of)j(w)m(a)m(ys:)40 b(it)29 b(can)g(b)s(e)g(sho)m(wn)f(to)i(b)s(e)e (existen)m(tially)g(forgeable,)i(it)e(can)i(b)s(e)150 1425 y(target-message)h(forgeable,)e(or)f(its)g(secret)h(k)m(ey)f(can)h (b)s(e)e(completely)h(reco)m(v)m(erable.)41 b(In)28 b(order)f(to)i (break)150 1538 y(a)34 b(signature)e(sc)m(heme,)j(di\013eren)m(t)d (\015a)m(v)m(ors)i(of)f(attac)m(ks)j(can)d(b)s(e)g(launc)m(hed:)45 b(a)33 b(public-k)m(ey)f(only)g(attac)m(k,)150 1651 y(or)e(an)h(in)m (teractiv)m(e)g(attac)m(k)i(where)d(the)g(adv)m(ersary)h(asks)f(for)g (signatures)g(on)g(messages)h(of)g(his)e(c)m(hoice.)291 1764 y(In)j(this)f(problem,)h(y)m(ou)h(are)g(ask)m(ed)g(to)h(break)e (eac)m(h)i(in)d(the)i(strongest)g(sense)g(p)s(ossible,)e(using)g(the) 150 1877 y(w)m(eak)m(est)h(attac)m(k)h(y)m(ou)e(can.)291 1990 y(On)h(the)g(other)h(hand,)f(eac)m(h)h(of)g(these)g(signature)e (sc)m(hemes)i(is)f(secure)g(in)f(a)i(w)m(eak)g(sense:)45 b(namely)-8 b(,)150 2103 y(on)33 b(input)d(a)j(public)d(k)m(ey)k(and)e (a)h(message,)i(it)d(is)g(infeasible)e(to)j(compute)g(the)g(signature.) 47 b(Pro)m(v)m(e)34 b(this)150 2215 y(w)m(eak)d(securit)m(y)f(prop)s (ert)m(y)g(for)g(eac)m(h)i(signature)e(sc)m(heme.)291 2428 y Fm(\(a\))h Fl(\(8)i(p)s(oin)m(ts\))f(Supp)s(ose)e(a)j(family)d (of)j(trap)s(do)s(or)e(p)s(erm)m(utations)g Fh(f)p Ff(p)2730 2442 y Fe(P)10 b(K)2853 2428 y Fh(g)33 b Fl(o)m(v)m(er)g Fh(f)p Fl(0)p Ff(;)15 b Fl(1)p Fh(g)3347 2395 y Fe(k)3424 2428 y Fl(is)32 b(giv)m(en.)150 2541 y(The)d(message)h(space)g(is)f Fh(f)p Fl(0)p Ff(;)15 b Fl(1)p Fh(g)1237 2508 y Fe(k)1282 2541 y Fl(.)40 b(A)30 b(public)c(k)m(ey)k(is)f(the)h(public)c(k)m(ey)k Ff(P)13 b(K)36 b Fl(of)30 b(a)g(mem)m(b)s(er)f(of)g(the)h(TDP;)150 2654 y(the)37 b(secret)h(k)m(ey)g(is)e(the)h(corresp)s(o)s(ding)e Ff(S)5 b(K)i Fl(.)60 b(The)37 b(signature)f(on)h(message)h Ff(m)f Fl(is)f(the)h(v)-5 b(alue)36 b Ff(x)h Fl(suc)m(h)150 2767 y(that)31 b Ff(p)393 2781 y Fe(P)10 b(K)516 2767 y Fl(\()p Ff(x)p Fl(\))26 b(=)e Ff(m)p Fl(.)291 2979 y Fm(\(b\))39 b Fl(\(6)h(p)s(oin)m(ts\))f(Assume)g(that)h(RSA)f(is)g(a) h(family)e(of)h(trap)s(do)s(or)g(p)s(erm)m(utations.)67 b(A)40 b(signature)150 3092 y(sc)m(heme)34 b(is)e(as)h(follo)m(ws:)45 b(the)33 b(public)e(k)m(ey)j(is)e(an)h(RSA)f(public)f(k)m(ey)i(\()p Ff(n;)15 b(e)p Fl(\),)35 b(and)e(the)g(secret)h(k)m(ey)g(is)e(the)150 3205 y(corresp)s(onding)c(RSA)h(secret)i(k)m(ey)f Ff(d)p Fl(.)41 b(The)29 b(message)i(space)g(is)d Fa(Z)2416 3172 y Fd(\003)2416 3228 y Fe(n)2459 3205 y Fl(.)40 b(The)30 b(signature)f(on)g(message)i Ff(m)f Fl(is)150 3318 y(the)h(v)-5 b(alue)29 b Ff(m)621 3285 y Fe(d)687 3318 y Fl(mo)s(d)24 b Ff(n)p Fl(.)291 3530 y Fm(\(c\))33 b Fl(\(6)h(p)s(oin)m(ts\))f (Assume)f(that)i(factoring)g(is)e(hard.)49 b(The)32 b(public)f(k)m(ey)j (of)f(this)g(signature)f(sc)m(heme)150 3643 y(consists)h(of)h(a)h(Blum) d(in)m(teger)j Ff(n)30 b Fl(=)h Ff(pq)s Fl(.)51 b(\(Recall)34 b(that)g(Blum)f(in)m(tegers)h(ha)m(v)m(e)h(the)f(prop)s(ert)m(y)f(that) i Fh(\000)p Fl(1)150 3756 y(is)c(a)h(quadratic)f(non-residue)f(with)h (Jacobi)g(sym)m(b)s(ol)g(1\).)45 b(The)31 b(secret)i(k)m(ey)g(is)e Ff(n)p Fl('s)g(factorization.)45 b(The)150 3869 y(message)25 b(space)f(is)f Ff(J)864 3883 y Fg(1)904 3869 y Fl(\()p Ff(n)p Fl(\),)i(i.e.)39 b(the)24 b(set)g(of)g(elemen)m(ts)g(of)g Fa(Z)2143 3836 y Fd(\003)2143 3892 y Fe(n)2209 3869 y Fl(that)h(ha)m(v)m(e)g(Jacobi)e(sym)m(b)s(ol)g(1.)39 b(A)24 b(signature)150 3982 y Ff(\033)34 b Fl(on)d(message)g Ff(m)26 b Fh(2)g Ff(J)955 3996 y Fg(1)995 3982 y Fl(\()p Ff(n)p Fl(\))31 b(is)e(computed)i(as)g(follo)m(ws:)41 b(if)29 b Ff(m)i Fl(is)f(a)h(qudratic)f(residue,)f(then)i Ff(\033)j Fl(is)c(some)150 4095 y(arbitrary)f(square)h(ro)s(ot)h(of)f Ff(m)p Fl(.)41 b(Otherwise,)29 b Ff(\033)34 b Fl(is)29 b(some)i(arbitrary)e(square)h(ro)s(ot)h(of)f Fh(\000)p Ff(m)p Fl(.)1867 5416 y(12-3)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF handouts/ps7.tex0100664003325600237220000002364107374544700013376 0ustar mliskovta6875\iffalse NOTE: If you do not collaborate with anyone, simply remove the '**** INSERT COLLABORATORS HERE****' and replace it with 'nobody.' Don't erase the collaboration statement! Thanks. INSTRUCTIONS: (if this is not ps1, use the right file name) Clip out the ********* INSERT HERE ********* bits below and insert appropriate TeX code. Once you are done with your file, run ``latex ps7.tex'' from the Athena shell. If your TeX code is clean, the latex will exit back to a prompt. Once this is done, run ``dvips ps7.dvi'' which should print your file to the nearest printer. There will be residual files called ps1.log, ps1.aux, and ps1.dvi. All these can be deleted, but do not delete ps1.tex. \fi \documentclass[11pt]{article} \usepackage{amsfonts} \usepackage{latexsym} \setlength{\oddsidemargin}{.25in} \setlength{\evensidemargin}{.25in} \setlength{\textwidth}{6in} \setlength{\topmargin}{-0.4in} \setlength{\textheight}{8.5in} \newcommand{\handout}[5]{ \renewcommand{\thepage}{#1-\arabic{page}} \noindent \begin{center} \framebox{ \vbox{ \hbox to 5.78in { {\bf 6.875J/18.425J Cryptography and Cryptanalysis} \hfill #2 } \vspace{4mm} \hbox to 5.78in { {\Large \hfill #5 \hfill} } \vspace{2mm} \hbox to 5.78in { {\it #3 \hfill #4} } } } \end{center} \vspace*{4mm} } \newcommand{\al}{\alpha} \newcommand{\Z}{\mathbb Z} \newcommand{\N}{\mathbb N} \newcommand{\calO}{{\cal O}} \begin{document} \handout{P7}{DUE: November 21, 2001}{Instructor: Anna Lysyanskaya} {Teaching Assistant: Moses Liskov} {*********** INSERT YOUR NAME HERE ***********: PS 7} I collaborated with ******** INSERT COLLABORATORS OR `nobody' HERE **********. \bigskip \section*{Problem 1: Random Functions and Random Permutations} Let $\calO_0 = R(n)$ be a random (length-preserving) function oracle: on any input from $\{0, 1\}^n$, if it has not yet given an answer for that input, it returns a random value from $\{0, 1\}^n$. Let $\calO_1 = \Pi(n)$ be a random permutation oracle: on any input from $\{0, 1\}^n$, if it has not yet given an answer for that input, it returns a random value in $\{0, 1\}^n$ that it has never returned before. Prove that $\calO_0(n)$ is indistinguishable from ${\mathcal O}_1(n)$, in the strongest sense you can. \bigskip {\bf Solution:} *************** INSERT PROBLEM 1 SOLUTION HERE *********** \section*{Problem 2: Attacking Luby-Rackoff} Recall the Luby-Rackoff construction for a pseudorandom permutation from a pseudorandom function. Let $F_i$ be a set of pseudorandom functions, and let $$LR_i(L, R) = (R, F_i(R) \oplus L )$$ define the $i^{th}$ round of operation of the Luby-Rackoff construction. For an integer $k$, define $$LR-k(L, R) = LR_k(LR_{k-1}( \ldots (L, R) \ldots )$$ to be the output of $k$-round Luby-Rackoff. We say a permutation family $P_S$ over $\{0, 1\}^{l(k)}$ is pseudorandom if: \begin{tabbing} $\forall \{A^{?}_k\} \in PPT~\exists \nu \in negl~\forall k$\\ $\Pr[$\=$S \leftarrow \{0, 1\}^k; \calO_0 \leftarrow P_S;$\\ \>${\mathcal O}_1 \leftarrow \Pi(l(k));$\\ \>$b' \leftarrow A^{\calO_b}_k: b' = b] < 1/2 + \nu(k)$ \end{tabbing} {\bf (a)} Prove that $LR-2$ is not a pseudorandom permutation family. \bigskip {\bf Solution:} *************** INSERT PROBLEM 2a SOLUTION HERE *********** \bigskip We say a permutation family $P_S$ is strongly pseudorandom if: \begin{tabbing} $\forall \{A^{?,?}_n\} \in PPT~\exists \nu \in negl~\forall k$\\ $\Pr[$\=$S \leftarrow \{0, 1\}^k; \calO_0 \leftarrow P_S; {\mathcal O'}_0 \leftarrow P^{-1}_S$\\ \>${\mathcal O}_1 \leftarrow \Pi; {\mathcal O'}_1 \leftarrow \Pi^{-1};$\\ \>$b' \leftarrow A^{\calO_b,{\mathcal O'}_b}: b' = b] < 1/2 + \nu(k)$ \end{tabbing} where $\Pi^{-1}$ inverts $\Pi$. Think of $\Pi$ as being a fixed but randomly chosen perumtation, and $\Pi^{-1}$ as its inverse. {\bf (b)} Prove that $LR-3$ is not a strong pseudorandom permutation family. \bigskip {\bf Solution:} *************** INSERT PROBLEM 2b SOLUTION HERE *********** \section*{Problem 3: Four-Round Luby-Rackoff} In class we proved that $LR-3$ is a pseudorandom permutation family. Prove that $LR-4$ is a strong pseudorandom permutation family. This is rather difficult to come up with from scratch, so we provide a proof sketch that you should fill in. {\bf Proof sketch}: First of all, assume we have an adversary $A^{?,?}$ which defeats the security of $LR-4$. Prove that without loss of generality we can assume that the adversary never asks $\calO$ or $\calO'$ the same question twice, that the adversary never queries $\calO$ on an output of $\calO'$, and that the adversary never queries $\calO'$ on the output of $\calO$. Next, formulate a construction for $LR-4R$, which performs the Luby-Rackoff function with 4 random functions instead of $F_1, F_2, F_3, F_4$. This construction is just like the one from class, except for $LR-4$ instead of $LR-3$. Prove that $LR-4R$ is indistinguishable from a random function, similarly to how we did in class. Next, formulate a construction for $(LR-4R)^{-1}$, which operates with the oracle $LR-4R$, and prove that adversaries that have oracle access to $LR-4R$ and another (challenge) oracle cannot tell whether the challenge oracle was $(LR-4R)^{-1}$ or a random function. This is the meat of the proof. Finally, put it all together and prove that $LR-4$ is indistinguishable from a random permutation in this model. Your proof should assume this is not the case and derive a contradiction by proving that one of the functions $F_i$ is not pseudorandom. \bigskip {\bf Solution:} *************** INSERT PROBLEM 3 SOLUTION HERE *********** \section*{Problem 4: Encrypting the Secret Key} Let $(G, E, D)$ be a (one-pass) GM-secure cryptosystem. This should be thought of as a ``secure envelope'' -- the adversary cannot see its contents, no matter what the message is. This problem shows that one has to be careful with this intuition. Namely, we ask the question, ``Can I trust my encryption to the extent that I can publish the encryption of my secret key?'' \bigskip {\bf (a)} (7 points) Construct a cryptosystem $(G', E', D')$ based on $(G, E, D)$ such that given $t = E'(PK', SK')$, the adversary can easily distinguish the encryptions of {\em any} two messages $m_0$ and $m_1$. (Hint: $t$ should somehow contain $SK'$. Try to imagine how this can be done without making $(G', E', D')$ insecure.) \bigskip {\bf Solution:} *************** INSERT PROBLEM 4a SOLUTION HERE *********** \bigskip {\bf (b)} (7 points) Construct a cryptosystem $(G'', E'', D'')$ based on $(G, E, D)$ such that even giving $t = E''(PK'', SK'')$ to the adversary, the scheme is still GM-secure. You should imagine that the adversary learns $t$ at the same time as $PK''$. (Hint: $t$ should somehow contain random garbage. Try to imagine how we can do this but {\em still be able to decrypt $t$}) \bigskip {\bf Solution:} *************** INSERT PROBLEM 4b SOLUTION HERE *********** \bigskip {\bf (c)} (6 points) Suppose Alice and Bob each have keys, $(PK_A, SK_A)$ and $(PK_B, SK_B)$ under $(G, E, D)$. Prove that Alice can securely encrypt {\em her} secret key under {\em Bob's} public key. That is, prove that $(G, E, D)$ is still GM-secure even when the adversary knows $PK_A, PK_B$, and $E(PK_B, SK_A)$. Formally, prove that \begin{tabbing} $\forall \{A_k\} \in PPT~\exists \nu \in negl~\forall k$\\ $\Pr[$\=$(PK_A, SK_A) \leftarrow G(1^k); (PK_B, SK_B) \leftarrow G(1^k); t \leftarrow E(PK_B, SK_A)$;\\ \>$(\alpha, m_0, m_1) \leftarrow A(PK_A, PK_B, t); b \leftarrow \{0, 1\}; b' \leftarrow A(\alpha, E(PK_A, m_b)): b'=b] < 1/2 + \nu(k)$ \end{tabbing} \bigskip {\bf Solution:} *************** INSERT PROBLEM 4c SOLUTION HERE *********** \section*{Problem 5: Insecure Signatures} All of the signature schemes presented in this problem are insecure. A signature scheme can be broken in a number of ways: it can be shown to be existentially forgeable, it can be target-message forgeable, or its secret key can be completely recoverable. In order to break a signature scheme, different flavors of attacks can be launched: a public-key only attack, or an interactive attack where the adversary asks for signatures on messages of his choice. In this problem, you are asked to break each in the strongest sense possible, using the weakest attack you can. On the other hand, each of these signature schemes is secure in a weak sense: namely, on input a public key and a message, it is infeasible to compute the signature. Prove this weak security property for each signature scheme. \bigskip {\bf (a)} (8 points) Suppose a family of trapdoor permutations $\{p_{PK}\}$ over $\{0,1\}^k$ is given. The message space is $\{0,1\}^k$. A public key is the public key $PK$ of a member of the TDP; the secret key is the correspoding $SK$. The signature on message $m$ is the value $x$ such that $p_{PK}(x) = m$. \bigskip {\bf Solution:} *************** INSERT PROBLEM 5a SOLUTION HERE *********** \bigskip {\bf (b)} (6 points) Assume that RSA is a family of trapdoor permutations. A signature scheme is as follows: the public key is an RSA public key $(n,e)$, and the secret key is the corresponding RSA secret key $d$. The message space is $\Z_n^*$. The signature on message $m$ is the value $m^d \bmod n$. \bigskip {\bf Solution:} *************** INSERT PROBLEM 5b SOLUTION HERE *********** \bigskip {\bf (c)} (6 points) Assume that factoring is hard. The public key of this signature scheme consists of a Blum integer $n = pq$. (Recall that Blum integers have the property that $-1$ is a quadratic non-residue with Jacobi symbol $1$). The secret key is $n$'s factorization. The message space is $J_1(n)$, i.e. the set of elements of $\Z_n^*$ that have Jacobi symbol $1$. A signature $\sigma$ on message $m \in J_1(n)$ is computed as follows: if $m$ is a qudratic residue, then $\sigma$ is some arbitrary square root of $m$. Otherwise, $\sigma$ is some arbitrary square root of $-m$. \bigskip {\bf Solution:} *************** INSERT PROBLEM 5c SOLUTION HERE *********** \end{document} handouts/ps8.tex0100664003325600237220000001622407400767556013404 0ustar mliskovta6875\iffalse NOTE: If you do not collaborate with anyone, simply remove the '**** INSERT COLLABORATORS HERE****' and replace it with 'nobody.' Don't erase the collaboration statement! Thanks. INSTRUCTIONS: (if this is not ps1, use the right file name) Clip out the ********* INSERT HERE ********* bits below and insert appropriate TeX code. Once you are done with your file, run ``latex ps7.tex'' from the Athena shell. If your TeX code is clean, the latex will exit back to a prompt. Once this is done, run ``dvips ps7.dvi'' which should print your file to the nearest printer. There will be residual files called ps1.log, ps1.aux, and ps1.dvi. All these can be deleted, but do not delete ps1.tex. \fi \documentclass[11pt]{article} \usepackage{amsfonts} \usepackage{latexsym} \setlength{\oddsidemargin}{.25in} \setlength{\evensidemargin}{.25in} \setlength{\textwidth}{6in} \setlength{\topmargin}{-0.4in} \setlength{\textheight}{8.5in} \newcommand{\handout}[5]{ \renewcommand{\thepage}{#1-\arabic{page}} \noindent \begin{center} \framebox{ \vbox{ \hbox to 5.78in { {\bf 6.875J/18.425J Cryptography and Cryptanalysis} \hfill #2 } \vspace{4mm} \hbox to 5.78in { {\Large \hfill #5 \hfill} } \vspace{2mm} \hbox to 5.78in { {\it #3 \hfill #4} } } } \end{center} \vspace*{4mm} } \newcommand{\al}{\alpha} \newcommand{\Z}{\mathbb Z} \newcommand{\N}{\mathbb N} \newcommand{\calO}{{\cal O}} \begin{document} \handout{P8}{DUE: December 5, 2001}{Instructor: Anna Lysyanskaya} {Teaching Assistant: Moses Liskov} {*********** INSERT YOUR NAME HERE ***********: PS 8} I collaborated with ******** INSERT COLLABORATORS OR `nobody' HERE **********. \bigskip \section*{Problem 1: Adaptive Attacks} (40 points) When we considered the security of public-key cryptosystems, we basically only considered allowing the adversary access to the public key. However, a stronger attack scenario may be considered: where the adversary has access to a decryption oracle. Informally, we want that the adversary cannot succeed in a one-pass GM security scenario even when the adversary has access to a decryption oracle under the relevant secret key, though of course querying the oracle on the actual challenge should be disallowed. \bigskip {\bf (a)} (15 points) Give a formal definition of a public-key cryptosystem which is one-pass GM secure against adaptive chosen ciphertext attack, as described above. \bigskip {\bf Solution:} ****************** INSERT PROBLEM 1a SOLUTION HERE ***************** \bigskip {\bf (b)} (15 points) Recall the ElGamal-A cryptosystem. Alice selects at random (1) a $k$-bit prime $p = 2q+1$ where $q$ is also prime, (2) an element $g$ in $Z_p^*$ of order $q$, and (3) an $x$ in $\Z_q^*$. Then, she sets her public key to be $(p,g,h)$, where $h=g^x \bmod p$, and her secret key to be $x$. Alice's public key restricts the message space to consist of all elements in $Z_p^*$ having order $q$. To send such a message $m$ to Alice, one picks an $r \leftarrow \Z_q$ and computes his ciphertext $c = (g^r \bmod p, h^rm \bmod p)$. To decrypt $c = (a,b)$, Alice computes $b/(a^x) \bmod p$. In this problem we go beyond proving that ElGamal-A is not secure against adaptive chosen ciphertext attack. Suppose the adversary has access to a challenge ciphertext $c$, and may ask only {\em one} query of the decryption oracle, which much be different from $c$. Show how to design a $c'$ from $c$ such that given the decryption of $c'$ it is easy to recover the decryption of $c$. Can you strengthen this result so that if $c = (a, b)$ and $c' = (a', b')$ then $a \neq a'$ and $b \neq b'$? If so, show how. \bigskip {\bf Solution:} ****************** INSERT PROBLEM 1b SOLUTION HERE ***************** \bigskip {\bf (c)} (10 points) Suppose the adversary has access to a ciphertext $c$ of a message under the (1-bit) Goldwasser-Micali cryptosystem, and may ask one query of a decryption oracle, other than $c$. Show how to design a $c'$ from $c$ such that given the decryption of $c'$ it is easy to recover the decryption of $c$. Also, generalize your attack on the 1-bit scheme to an attack on the $\ell$-bit scheme. \bigskip {\bf Solution:} ****************** INSERT PROBLEM 1c SOLUTION HERE ***************** \section*{Problem 2: Collision-resistant hashing} (30 points) Define a family of collision-resistant hash functions ${\mathcal H}$ as follows: \begin{enumerate} \item On input $1^k$, ${\mathcal H}$ outputs the description of a key $PK$. \item For any $PK$, it is efficient to compute $H_{PK}$. \item $H_{PK}$ is length decreasing: $\forall k, \forall PK \in {\mathcal H}(1^k) \forall x \in \{0, 1\}^*, |H_{PK}(x)| \leq $ min$(k, |x|)$. \item It is hard to find collisions: \begin{tabbing} $\forall \{A_k\} PPTF~\exists \nu \in negl~\forall k$\\ $\Pr[PK \leftarrow {\mathcal H}(1^k); (x, y) \leftarrow A_k(PK): H_{PK}(x) = H_{PK}(y) \wedge x \neq y] < \nu(k)$ \end{tabbing} \end{enumerate} \bigskip {\bf (a)} Prove that the existence of a family of collision-resistant hash functions implies the existence of a one-way function. \bigskip {\bf Solution:} ****************** INSERT PROBLEM 2a SOLUTION HERE ***************** \bigskip {\bf (b)} Suppose a family of collision-resistant hash functions ${\mathcal H}$ is given, and let ${\mathcal H'}$ be a family of hash functions that takes two security parameters, $l$ and $n$, so that it has two keys $PK_l \leftarrow {\mathcal H}(1^l)$ and $PK_n \leftarrow {\mathcal H}(1^n)$. Define $H'_{PK_l,PK_n}(x) = H_{PK_l}(H_{PK_n}(x))$. Prove or disprove: there is an infinite set of security parameter pairs for which ${\mathcal H'}$ is not collision-resistant. Prove or disprove: for all pairs $(k, k)$, ${\mathcal H'}$ is a family of collision-resistant hash functions. \bigskip {\bf Solution:} ****************** INSERT PROBLEM 2b SOLUTION HERE ***************** \bigskip {\bf (c)} Suppose $G$ is a PRG. Now suppose ${\mathcal H}^G$ is a family of hash functions defined by ${\mathcal H}^G(1^k) = {\mathcal H}(1^k)$ and $H^G_{PK}(x) = G(H_{PK}(x))$. Prove or disprove: ${\mathcal H}^G$ is a family of collision-resistant hash functions. Prove or disprove: ${\mathcal H}^G$ is a family of functions indistinguishable from random. \bigskip {\bf Solution:} ****************** INSERT PROBLEM 2c SOLUTION HERE ***************** \section*{Problem 3: Signature Schemes in the Random Oracle Model} (30 points) In the ``random oracle model,'' we make the assumption that some function behaves as if it were a random oracle ${\mathcal O}$. We assume the adversary, signer, and verifier have access to the random oracle. Furthermore, we make the assumption only that for every $x \in \{0, 1\}^*, {\mathcal O}(x)$ is a truly random string of the same length as $x$. Assume that a TDP family $p_{PK}$ exists, and design a simple signature scheme using $p$ and a random oracle. Prove that the scheme is non-existentially forgeable against an adaptive chosen message attack. (Hint: in order to do this, you will have to describe exactly how the oracle ${\mathcal O}$ works, but ${\mathcal O}$ must still be truly random.) \bigskip {\bf Solution:} ****************** INSERT PROBLEM 3 SOLUTION HERE ***************** \end{document} handouts/ho16.ps0100664003325600237220000044362207403467602013267 0ustar mliskovta6875%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: ho16.dvi %%Pages: 2 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips ho16.dvi -o ho16.ps %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2001.12.05:1403 %%BeginProcSet: tex.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ 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3465 y(c)m(hosen)27 b(ciphertext)g(attac)m(k.)42 b(Supp)s(ose)24 b(the)j(adv)m(ersary)g(has)g(access)h(to)f(a)g(c)m(hallenge)g (ciphertext)g Ff(c)p Fj(,)h(and)150 3578 y(ma)m(y)d(ask)h(only)e Fh(one)h Fj(query)f(of)h(the)g(decryption)f(oracle,)j(whic)m(h)c(m)m (uc)m(h)i(b)s(e)g(di\013eren)m(t)f(from)g Ff(c)p Fj(.)39 b(Sho)m(w)25 b(ho)m(w)150 3690 y(to)i(design)f(a)h Ff(c)643 3657 y Fe(0)693 3690 y Fj(from)f Ff(c)h Fj(suc)m(h)g(that)g(giv)m(en)g (the)g(decryption)e(of)i Ff(c)2339 3657 y Fe(0)2389 3690 y Fj(it)f(is)g(easy)h(to)h(reco)m(v)m(er)g(the)f(decryption)150 3803 y(of)k Ff(c)p Fj(.)291 3916 y(Can)38 b(y)m(ou)g(strengthen)g(this) e(result)h(so)h(that)h(if)e Ff(c)h Fj(=)f(\()p Ff(a;)15 b(b)p Fj(\))39 b(and)f Ff(c)2633 3883 y Fe(0)2694 3916 y Fj(=)g(\()p Ff(a)2886 3883 y Fe(0)2909 3916 y Ff(;)15 b(b)2988 3883 y Fe(0)3012 3916 y Fj(\))38 b(then)g Ff(a)g Fb(6)p Fj(=)f Ff(a)3542 3883 y Fe(0)3604 3916 y Fj(and)150 4029 y Ff(b)25 b Fb(6)p Fj(=)g Ff(b)349 3996 y Fe(0)373 4029 y Fj(?)40 b(If)30 b(so,)h(sho)m(w)f(ho)m(w.)291 4229 y Fk(\(c\))i Fj(\(10)i(p)s(oin)m(ts\))e(Supp)s(ose)f(the)i(adv)m (ersary)g(has)f(access)i(to)g(a)f(ciphertext)f Ff(c)h Fj(of)g(a)g(message)g(under)150 4342 y(the)c(\(1-bit\))g(Goldw)m (asser-Micali)e(cryptosystem,)i(and)f(ma)m(y)h(ask)g(one)g(query)e(of)i (a)g(decryption)e(oracle,)150 4455 y(other)i(than)g Ff(c)p Fj(.)41 b(Sho)m(w)29 b(ho)m(w)g(to)h(design)e(a)h Ff(c)1629 4422 y Fe(0)1682 4455 y Fj(from)g Ff(c)g Fj(suc)m(h)g(that)h(giv)m(en)f (the)g(decryption)f(of)i Ff(c)3348 4422 y Fe(0)3400 4455 y Fj(it)f(is)f(easy)150 4568 y(to)e(reco)m(v)m(er)i(the)e(decryption)e (of)i Ff(c)p Fj(.)40 b(Also,)26 b(generalize)g(y)m(our)g(attac)m(k)i (on)e(the)f(1-bit)h(sc)m(heme)g(to)h(an)f(attac)m(k)150 4681 y(on)k(the)h Ff(`)p Fj(-bit)e(sc)m(heme.)150 4964 y Fg(Problem)45 b(2:)61 b(Collision-resistan)l(t)48 b(hashing)150 5167 y Fj(\(30)32 b(p)s(oin)m(ts\))d(De\014ne)i(a)f(family)f(of)i (collision-resistan)m(t)d(hash)i(functions)f Fb(H)i Fj(as)f(follo)m (ws:)1867 5416 y(14-1)p eop %%Page: 2 2 2 1 bop 261 159 a Fj(1.)46 b(On)30 b(input)e(1)816 126 y Fd(k)859 159 y Fj(,)j Fb(H)g Fj(outputs)f(the)g(description)f(of)h(a) h(k)m(ey)g Ff(P)13 b(K)7 b Fj(.)261 346 y(2.)46 b(F)-8 b(or)31 b(an)m(y)g Ff(P)13 b(K)7 b Fj(,)30 b(it)g(is)g(e\016cien)m(t)h (to)g(compute)f Ff(H)1996 360 y Fd(P)10 b(K)2119 346 y Fj(.)261 534 y(3.)46 b Ff(H)453 548 y Fd(P)10 b(K)606 534 y Fj(is)29 b(length)h(decreasing:)41 b Fb(8)p Ff(k)s(;)15 b Fb(8)p Ff(P)e(K)31 b Fb(2)25 b(H)q Fj(\(1)2066 501 y Fd(k)2109 534 y Fj(\))p Fb(8)p Ff(x)g Fb(2)g(f)p Fj(0)p Ff(;)15 b Fj(1)p Fb(g)2578 501 y Fe(\003)2620 534 y Ff(;)g Fb(j)p Ff(H)2761 548 y Fd(P)10 b(K)2884 534 y Fj(\()p Ff(x)p Fj(\))p Fb(j)26 b(\024)k Fj(min\()p Ff(k)s(;)15 b Fb(j)p Ff(x)p Fb(j)p Fj(\).)261 721 y(4.)46 b(It)31 b(is)e(hard)h(to)h(\014nd)d(collisions:)377 971 y Fb(8f)p Ff(A)541 986 y Fd(k)584 971 y Fb(g)p Ff(P)13 b(P)g(T)g(F)44 b Fb(9)p Ff(\027)30 b Fb(2)25 b Ff(neg)s(l)33 b Fb(8)p Ff(k)377 1084 y Fj(Pr[)p Ff(P)13 b(K)32 b Fb( )25 b(H)q Fj(\(1)954 1051 y Fd(k)997 1084 y Fj(\);)15 b(\()p Ff(x;)g(y)s Fj(\))27 b Fb( )e Ff(A)1493 1099 y Fd(k)1536 1084 y Fj(\()p Ff(P)13 b(K)7 b Fj(\))26 b(:)f Ff(H)1913 1098 y Fd(P)10 b(K)2036 1084 y Fj(\()p Ff(x)p Fj(\))26 b(=)f Ff(H)2356 1098 y Fd(P)10 b(K)2478 1084 y Fj(\()p Ff(y)s Fj(\))21 b Fb(^)f Ff(x)25 b Fb(6)p Fj(=)g Ff(y)s Fj(])g Ff(<)g(\027)6 b Fj(\()p Ff(k)s Fj(\))291 1434 y Fk(\(a\))28 b Fj(Pro)m(v)m(e)j(that)e (the)g(existence)h(of)f(a)h(family)d(of)i(collision-resistan)m(t)f (hash)g(functions)f(implies)g(the)150 1547 y(existence)k(of)f(a)h (one-w)m(a)m(y)h(function.)291 1759 y Fk(\(b\))j Fj(Supp)s(ose)f(a)j (family)d(of)i(collision-resistan)m(t)e(hash)h(functions)g Fb(H)h Fj(is)f(giv)m(en,)i(and)f(let)g Fb(H)3517 1726 y Fe(0)3575 1759 y Fj(b)s(e)g(a)150 1872 y(family)h(of)h(hash)f (functions)g(that)i(tak)m(es)g(t)m(w)m(o)h(securit)m(y)e(parameters,)i Ff(l)g Fj(and)e Ff(n)p Fj(,)i(so)e(that)h(it)f(has)g(t)m(w)m(o)150 1985 y(k)m(eys)33 b Ff(P)13 b(K)500 2000 y Fd(l)555 1985 y Fb( )29 b(H)q Fj(\(1)833 1952 y Fd(l)859 1985 y Fj(\))k(and)f Ff(P)13 b(K)1254 1999 y Fd(n)1330 1985 y Fb( )28 b(H)q Fj(\(1)1607 1952 y Fd(n)1655 1985 y Fj(\).)47 b(De\014ne)33 b Ff(H)2129 1952 y Fe(0)2122 2012 y Fd(P)10 b(K)2237 2024 y Fa(l)2260 2012 y Fd(;P)g(K)2395 2020 y Fa(n)2440 1985 y Fj(\()p Ff(x)p Fj(\))29 b(=)g Ff(H)2767 1999 y Fd(P)10 b(K)2882 2011 y Fa(l)2909 1985 y Fj(\()p Ff(H)3020 1999 y Fd(P)g(K)3135 2007 y Fa(n)3180 1985 y Fj(\()p Ff(x)p Fj(\)\).)48 b(Pro)m(v)m(e)34 b(or)150 2098 y(dispro)m(v)m(e:)49 b(there)36 b(is)e(an)h(in\014nite)d(set)k(of)f(securit)m(y)g(parameter) h(pairs)d(for)i(whic)m(h)f Fb(H)3081 2065 y Fe(0)3139 2098 y Fj(is)g(not)h(collision-)150 2211 y(resistan)m(t.)59 b(Pro)m(v)m(e)37 b(or)g(dispro)m(v)m(e:)52 b(for)36 b(all)f(pairs)g(\() p Ff(k)s(;)15 b(k)s Fj(\),)40 b Fb(H)2214 2178 y Fe(0)2273 2211 y Fj(is)35 b(a)i(family)e(of)h(collision-resistan)m(t)f(hash)150 2324 y(functions.)291 2536 y Fk(\(c\))k Fj(Supp)s(ose)f Ff(G)i Fj(is)e(a)i(PR)m(G.)g(No)m(w)g(supp)s(ose)e Fb(H)2027 2503 y Fd(G)2125 2536 y Fj(is)g(a)i(family)e(of)h(hash)g(functions)f (de\014ned)g(b)m(y)150 2649 y Fb(H)228 2616 y Fd(G)287 2649 y Fj(\(1)367 2616 y Fd(k)410 2649 y Fj(\))26 b(=)f Fb(H)q Fj(\(1)725 2616 y Fd(k)768 2649 y Fj(\))g(and)f Ff(H)1082 2616 y Fd(G)1075 2676 y(P)10 b(K)1197 2649 y Fj(\()p Ff(x)p Fj(\))26 b(=)f Ff(G)p Fj(\()p Ff(H)1623 2663 y Fd(P)10 b(K)1747 2649 y Fj(\()p Ff(x)p Fj(\)\).)39 b(Pro)m(v)m(e)26 b(or)f(dispro)m(v)m(e:)37 b Fb(H)2792 2616 y Fd(G)2875 2649 y Fj(is)24 b(a)h(family)e(of)i(collision-)150 2762 y(resistan)m(t)34 b(hash)g(functions.)50 b(Pro)m(v)m(e)36 b(or)e(dispro)m(v)m(e:)48 b Fb(H)2030 2729 y Fd(G)2123 2762 y Fj(is)33 b(a)i(family)d(of)j(functions)d(indistinguishable)150 2875 y(from)e(random.)150 3162 y Fg(Problem)45 b(3:)61 b(Signature)45 b(Sc)l(hemes)h(in)e(the)i(Random)f(Oracle)h(Mo)t(del)150 3364 y Fj(\(30)41 b(p)s(oin)m(ts\))e(In)g(the)g(\\random)h(oracle)g(mo) s(del,")h(w)m(e)f(mak)m(e)h(the)f(assumption)e(that)i(some)g(function) 150 3477 y(b)s(eha)m(v)m(es)27 b(as)g(if)f(it)h(w)m(ere)g(a)g(random)f (oracle)h Fb(O)s Fj(.)40 b(W)-8 b(e)28 b(assume)e(the)h(adv)m(ersary)-8 b(,)29 b(signer,)e(and)f(v)m(eri\014er)g(ha)m(v)m(e)150 3590 y(access)40 b(to)f(the)g(random)f(oracle.)66 b(F)-8 b(urthermore,)40 b(w)m(e)g(mak)m(e)f(the)g(assumption)e(only)h(that)h (for)f(ev)m(ery)150 3703 y Ff(x)25 b Fb(2)g(f)p Fj(0)p Ff(;)15 b Fj(1)p Fb(g)533 3670 y Fe(\003)575 3703 y Ff(;)g Fb(O)s Fj(\()p Ff(x)p Fj(\))31 b(is)e(a)i(truly)e(random)h(string)f(of) i(the)f(same)h(length)f(as)g Ff(x)p Fj(.)291 3816 y(Assume)22 b(that)i(a)f(TDP)g(family)e Ff(p)1414 3830 y Fd(P)10 b(K)1560 3816 y Fj(exists,)24 b(and)e(design)g(a)h(simple)e(signature)h (sc)m(heme)i(using)d Ff(p)i Fj(and)150 3929 y(a)33 b(random)f(oracle.) 47 b(Pro)m(v)m(e)34 b(that)f(the)g(sc)m(heme)g(is)f(non-existen)m (tially)f(forgeable)h(against)h(an)g(adaptiv)m(e)150 4042 y(c)m(hosen)j(message)h(attac)m(k.)59 b(\(Hin)m(t:)51 b(in)35 b(order)g(to)h(do)g(this,)g(y)m(ou)g(will)d(ha)m(v)m(e)k(to)f (describ)s(e)e(exactly)j(ho)m(w)150 4155 y(the)31 b(oracle)f Fb(O)k Fj(w)m(orks,)c(but)g Fb(O)j Fj(m)m(ust)d(still)f(b)s(e)g(truly)g (random.\))1867 5416 y(14-2)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF handouts/ps9.tex0100664003325600237220000001411507403467674013403 0ustar mliskovta6875\iffalse NOTE: If you do not collaborate with anyone, simply remove the '**** INSERT COLLABORATORS HERE****' and replace it with 'nobody.' Don't erase the collaboration statement! Thanks. INSTRUCTIONS: (if this is not ps1, use the right file name) Clip out the ********* INSERT HERE ********* bits below and insert appropriate TeX code. Once you are done with your file, run ``latex ps7.tex'' from the Athena shell. If your TeX code is clean, the latex will exit back to a prompt. Once this is done, run ``dvips ps7.dvi'' which should print your file to the nearest printer. There will be residual files called ps1.log, ps1.aux, and ps1.dvi. All these can be deleted, but do not delete ps1.tex. \fi \documentclass[11pt]{article} \usepackage{amsfonts} \usepackage{latexsym} \setlength{\oddsidemargin}{.25in} \setlength{\evensidemargin}{.25in} \setlength{\textwidth}{6in} \setlength{\topmargin}{-0.4in} \setlength{\textheight}{8.5in} \newcommand{\handout}[5]{ \renewcommand{\thepage}{#1-\arabic{page}} \noindent \begin{center} \framebox{ \vbox{ \hbox to 5.78in { {\bf 6.875J/18.425J Cryptography and Cryptanalysis} \hfill #2 } \vspace{4mm} \hbox to 5.78in { {\Large \hfill #5 \hfill} } \vspace{2mm} \hbox to 5.78in { {\it #3 \hfill #4} } } } \end{center} \vspace*{4mm} } \newcommand{\al}{\alpha} \newcommand{\Z}{\mathbb Z} \newcommand{\N}{\mathbb N} \newcommand{\calO}{{\cal O}} \begin{document} \handout{P9}{DUE: December 12, 2001}{Instructor: Anna Lysyanskaya} {Teaching Assistant: Moses Liskov} {*********** INSERT YOUR NAME HERE ***********: PS 9} I collaborated with ******** INSERT COLLABORATORS OR `nobody' HERE **********. \bigskip \section*{Problem 1} (33 points) Recall that in a commitment scheme there are two ITMs, $C$ and $R$ (for committer and receiver), which operate in two stages (they are given an auxillary input, either $`c'$ or $`r'$, to indicate which stage they are in at the time). Recall the definition: \begin{tabbing} Correctness:\\ $\forall b \in \{0, 1\}~\forall k$\\ $\Pr[$\=$u \leftarrow C(b, 1^k, `c') \leftrightarrow R(1^k, `c') \rightarrow v;$\\ \>$C(u, `r') \leftrightarrow R(v, `r') \rightarrow b': b' = b] = 1$\\ \\ Hiding:\\ $\forall \{R'_k\} PPTF~\exists \nu \in negl~\forall k$\\ $\Pr[b \leftarrow \{0, 1\}; u \leftarrow C(b, 1^k, `c') \leftrightarrow R'_k(1^k, `c') \rightarrow b':$\\ \>$b = b'] < 1/2 + \nu(k)$\\ \\ Binding:\\ $\forall \{C'_k\} PPTF~\exists \nu \in negl~\forall k$\\ $\Pr[u \leftarrow C'_k(1^k, `c') \leftrightarrow R(1^k, `c') \rightarrow v; C'_k(u, 0, `r') \leftrightarrow R(v, `r') \rightarrow b_0;$\\ \>$C'_k(u, 1, `r') \leftrightarrow R(v, `r') \rightarrow b_1: b_0 \neq b_1] < \nu(k)$\\ \end{tabbing} Prove that if a commitment scheme exists, then a one-way function exists. Hint: Be careful to remember that in a commitment scheme, the commit stage may take multiple messages. \bigskip {\bf Solution:} ******************* INSERT PROBLEM 1 SOLUTION HERE *************** \section*{Problem 2} Consider the following commitment scheme. Suppose there are public parameters $p$ (a prime), and $g$ and $h$, two generators modulo $p$, and that these are chosen at random at the start and made available to both parties. To commit to a bit $b$, the committer chooses a random $r$ and computes $g^rh^b$, which is sent to the verifier. To open the commitment, the committer reveals $r$ and $b$. \bigskip {\bf (a)} (11 points) Prove that this is a secure commitment scheme. More precisely, prove that this scheme is correct, that it is hiding unconditionally, and that it is binding if the discrete log problem is hard. \bigskip {\bf Solution:} ******************* INSERT PROBLEM 2a SOLUTION HERE *************** \bigskip {\bf (b)} (11 points) If instead the committer generates $p$, $g$, and $h$, is the scheme secure? Prove your answer. If no, is there a way to fix it easily? \bigskip {\bf Solution:} ******************* INSERT PROBLEM 2b SOLUTION HERE *************** \bigskip {\bf (c)} (11 points) If instead the receiver generates $p$, $g$, and $h$, is the scheme secure? Prove your answer. If no, is there a way to fix it easily? \bigskip {\bf Solution:} ******************* INSERT PROBLEM 2c SOLUTION HERE *************** \section*{Problem 3} Consider the following zero-knowledge protocol for proving that a value $a$ is a quadratic residue modulo $n$, in which Input: $n$ and $a$. Let $\alpha^2 \equiv a \bmod n$ be the square root of $a$. \begin{enumerate} \item $P$ chooses a random prime $p = 2q+1$ where $q$ is a random $k$-bit prime, and a random $g$ or order $q$, and randomly chooses $\beta$ and sets $h = g^{\beta} \bmod p$. $P$ sends $p, q, g, h$ to $V$. \item $V$ computes a random string $c$ of length $k$ and a random $r$, and sends $x = g^rh^c$ to $P$. \item $P$ chooses $k$ random values $r_1, \ldots, r_k$ and computes $s_i = r_i^2$ and sends $s_1, \ldots, s_k$ to $V$. \item $V$ sends $r$ and $c$ to $P$.\\ \item $P$ checks that $x = g^rh^c$. If so, $P$ sends to $V$, for each $i$, the value $t_i = r_i \alpha^{c_i}$, where $c = c_1 \circ \ldots \circ c_k$. $P$ also sends $\beta$ to $V$ \item $V$ checks that $t_i^2 \equiv s_i a^{c_i}$ for each $i$, and that $h = g^{\beta}$. If so, $V$ accepts, otherwise $V$ rejects. \end{enumerate} This defines a $P, V$ which you will prove is a zero-knowledge proof system. It should be clear that $P, V$ has completeness 1. The idea of this construction is to simply parallelize the ZKP we presented in class for quadratic residuosity. In order to prove zero-knowledgeness we need to add the commitment $x$ into the protocol, as you will see when proving this. \bigskip {\bf (a)} (17 points) Prove that $P, V$ has negligible soundness (ie, $V$ can only be convinced with negligible probability if $a$ is a quadratic non-residue modulo $n$). \bigskip {\bf Solution:} ******************* INSERT PROBLEM 3a SOLUTION HERE *************** \bigskip {\bf (b)} (17 points) Prove that if the discrete logarithm problem is hard, then $P, V$ is zero-knowledge. \bigskip {\bf Solution:} ******************* INSERT PROBLEM 3b SOLUTION HERE *************** \end{document} handouts/ps9.tex~0100664003325600237220000001622407403467432013574 0ustar mliskovta6875\iffalse NOTE: If you do not collaborate with anyone, simply remove the '**** INSERT COLLABORATORS HERE****' and replace it with 'nobody.' Don't erase the collaboration statement! Thanks. INSTRUCTIONS: (if this is not ps1, use the right file name) Clip out the ********* INSERT HERE ********* bits below and insert appropriate TeX code. Once you are done with your file, run ``latex ps7.tex'' from the Athena shell. If your TeX code is clean, the latex will exit back to a prompt. Once this is done, run ``dvips ps7.dvi'' which should print your file to the nearest printer. There will be residual files called ps1.log, ps1.aux, and ps1.dvi. All these can be deleted, but do not delete ps1.tex. \fi \documentclass[11pt]{article} \usepackage{amsfonts} \usepackage{latexsym} \setlength{\oddsidemargin}{.25in} \setlength{\evensidemargin}{.25in} \setlength{\textwidth}{6in} \setlength{\topmargin}{-0.4in} \setlength{\textheight}{8.5in} \newcommand{\handout}[5]{ \renewcommand{\thepage}{#1-\arabic{page}} \noindent \begin{center} \framebox{ \vbox{ \hbox to 5.78in { {\bf 6.875J/18.425J Cryptography and Cryptanalysis} \hfill #2 } \vspace{4mm} \hbox to 5.78in { {\Large \hfill #5 \hfill} } \vspace{2mm} \hbox to 5.78in { {\it #3 \hfill #4} } } } \end{center} \vspace*{4mm} } \newcommand{\al}{\alpha} \newcommand{\Z}{\mathbb Z} \newcommand{\N}{\mathbb N} \newcommand{\calO}{{\cal O}} \begin{document} \handout{P8}{DUE: December 5, 2001}{Instructor: Anna Lysyanskaya} {Teaching Assistant: Moses Liskov} {*********** INSERT YOUR NAME HERE ***********: PS 8} I collaborated with ******** INSERT COLLABORATORS OR `nobody' HERE **********. \bigskip \section*{Problem 1: Adaptive Attacks} (40 points) When we considered the security of public-key cryptosystems, we basically only considered allowing the adversary access to the public key. However, a stronger attack scenario may be considered: where the adversary has access to a decryption oracle. Informally, we want that the adversary cannot succeed in a one-pass GM security scenario even when the adversary has access to a decryption oracle under the relevant secret key, though of course querying the oracle on the actual challenge should be disallowed. \bigskip {\bf (a)} (15 points) Give a formal definition of a public-key cryptosystem which is one-pass GM secure against adaptive chosen ciphertext attack, as described above. \bigskip {\bf Solution:} ****************** INSERT PROBLEM 1a SOLUTION HERE ***************** \bigskip {\bf (b)} (15 points) Recall the ElGamal-A cryptosystem. Alice selects at random (1) a $k$-bit prime $p = 2q+1$ where $q$ is also prime, (2) an element $g$ in $Z_p^*$ of order $q$, and (3) an $x$ in $\Z_q^*$. Then, she sets her public key to be $(p,g,h)$, where $h=g^x \bmod p$, and her secret key to be $x$. Alice's public key restricts the message space to consist of all elements in $Z_p^*$ having order $q$. To send such a message $m$ to Alice, one picks an $r \leftarrow \Z_q$ and computes his ciphertext $c = (g^r \bmod p, h^rm \bmod p)$. To decrypt $c = (a,b)$, Alice computes $b/(a^x) \bmod p$. In this problem we go beyond proving that ElGamal-A is not secure against adaptive chosen ciphertext attack. Suppose the adversary has access to a challenge ciphertext $c$, and may ask only {\em one} query of the decryption oracle, which much be different from $c$. Show how to design a $c'$ from $c$ such that given the decryption of $c'$ it is easy to recover the decryption of $c$. Can you strengthen this result so that if $c = (a, b)$ and $c' = (a', b')$ then $a \neq a'$ and $b \neq b'$? If so, show how. \bigskip {\bf Solution:} ****************** INSERT PROBLEM 1b SOLUTION HERE ***************** \bigskip {\bf (c)} (10 points) Suppose the adversary has access to a ciphertext $c$ of a message under the (1-bit) Goldwasser-Micali cryptosystem, and may ask one query of a decryption oracle, other than $c$. Show how to design a $c'$ from $c$ such that given the decryption of $c'$ it is easy to recover the decryption of $c$. Also, generalize your attack on the 1-bit scheme to an attack on the $\ell$-bit scheme. \bigskip {\bf Solution:} ****************** INSERT PROBLEM 1c SOLUTION HERE ***************** \section*{Problem 2: Collision-resistant hashing} (30 points) Define a family of collision-resistant hash functions ${\mathcal H}$ as follows: \begin{enumerate} \item On input $1^k$, ${\mathcal H}$ outputs the description of a key $PK$. \item For any $PK$, it is efficient to compute $H_{PK}$. \item $H_{PK}$ is length decreasing: $\forall k, \forall PK \in {\mathcal H}(1^k) \forall x \in \{0, 1\}^*, |H_{PK}(x)| \leq $ min$(k, |x|)$. \item It is hard to find collisions: \begin{tabbing} $\forall \{A_k\} PPTF~\exists \nu \in negl~\forall k$\\ $\Pr[PK \leftarrow {\mathcal H}(1^k); (x, y) \leftarrow A_k(PK): H_{PK}(x) = H_{PK}(y) \wedge x \neq y] < \nu(k)$ \end{tabbing} \end{enumerate} \bigskip {\bf (a)} Prove that the existence of a family of collision-resistant hash functions implies the existence of a one-way function. \bigskip {\bf Solution:} ****************** INSERT PROBLEM 2a SOLUTION HERE ***************** \bigskip {\bf (b)} Suppose a family of collision-resistant hash functions ${\mathcal H}$ is given, and let ${\mathcal H'}$ be a family of hash functions that takes two security parameters, $l$ and $n$, so that it has two keys $PK_l \leftarrow {\mathcal H}(1^l)$ and $PK_n \leftarrow {\mathcal H}(1^n)$. Define $H'_{PK_l,PK_n}(x) = H_{PK_l}(H_{PK_n}(x))$. Prove or disprove: there is an infinite set of security parameter pairs for which ${\mathcal H'}$ is not collision-resistant. Prove or disprove: for all pairs $(k, k)$, ${\mathcal H'}$ is a family of collision-resistant hash functions. \bigskip {\bf Solution:} ****************** INSERT PROBLEM 2b SOLUTION HERE ***************** \bigskip {\bf (c)} Suppose $G$ is a PRG. Now suppose ${\mathcal H}^G$ is a family of hash functions defined by ${\mathcal H}^G(1^k) = {\mathcal H}(1^k)$ and $H^G_{PK}(x) = G(H_{PK}(x))$. Prove or disprove: ${\mathcal H}^G$ is a family of collision-resistant hash functions. Prove or disprove: ${\mathcal H}^G$ is a family of functions indistinguishable from random. \bigskip {\bf Solution:} ****************** INSERT PROBLEM 2c SOLUTION HERE ***************** \section*{Problem 3: Signature Schemes in the Random Oracle Model} (30 points) In the ``random oracle model,'' we make the assumption that some function behaves as if it were a random oracle ${\mathcal O}$. We assume the adversary, signer, and verifier have access to the random oracle. Furthermore, we make the assumption only that for every $x \in \{0, 1\}^*, {\mathcal O}(x)$ is a truly random string of the same length as $x$. Assume that a TDP family $p_{PK}$ exists, and design a simple signature scheme using $p$ and a random oracle. Prove that the scheme is non-existentially forgeable against an adaptive chosen message attack. (Hint: in order to do this, you will have to describe exactly how the oracle ${\mathcal O}$ works, but ${\mathcal O}$ must still be truly random.) \bigskip {\bf Solution:} ****************** INSERT PROBLEM 3 SOLUTION HERE ***************** \end{document} handouts/solutions/0042750003326500237220000000000007534742234013423 5ustar annata6875handouts/solutions/sol1.ps0100664003325600237220000067570307352123353015431 0ustar mliskovta6875%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: ho3.dvi %%Pages: 6 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips ho3.dvi -o ho3.ps %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2001.09.19:1008 %%BeginProcSet: tex.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 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Fp(\()p Fl(S;)15 b(')p Fp(\()p Fl(n)p Fp(\)\))32 b(=)d(1)k(then)f(there)h(is)f(only)g (one)h(v)-5 b(alid)150 3521 y(preimage)26 b(of)h Fl(z)681 3488 y Fk(S)732 3521 y Fp(,)g(but)f(if)f Fl(g)s(cd)p Fp(\()p Fl(S;)15 b(')p Fp(\()p Fl(n)p Fp(\)\))29 b Fl(>)c Fp(1)i(then)f(there)g(is)g(more)g(than)h(one,)g(so)g(the)g Fl(R)q(S)5 b(A)3304 3488 y Fi(\000)p Fh(1)3424 3521 y Fp(function)150 3634 y(w)m(ould)29 b(only)h(return)f(the)h(righ)m(t)g (one)h(with)e(probabilit)m(y)f(at)j(most)g(1)p Fl(=)p Fp(2.)150 3846 y Fq(Theorem)j(1)46 b Fn(F)-7 b(or)40 b(any)g Fl(n)p Fn(,)g Fl(S)5 b Fn(,)41 b(for)f(any)g Fl(x)f Fn(ther)-5 b(e)41 b(is)e(one)h(and)g(only)g(one)g(value)f Fl(y)j Fn(such)e(that)g Fl(y)3591 3813 y Fk(S)3679 3846 y Fj(\021)150 3959 y Fl(x)25 b Fp(mo)s(d)g Fl(n)k Fn(if)g(and)h(only)h (if)e Fl(S)34 b Fn(is)c(r)-5 b(elatively)30 b(prime)h(to)f Fl(')p Fp(\()p Fl(n)p Fp(\))p Fn(.)41 b(That)31 b(is,)e(the)h(function) g Fl(f)10 b Fp(\()p Fl(y)s Fp(\))25 b(=)g Fl(y)3420 3926 y Fk(S)3496 3959 y Fp(mo)s(d)f Fl(n)150 4072 y Fn(is)33 b(a)h(bije)-5 b(ction)33 b(fr)-5 b(om)35 b Fg(Z)974 4039 y Fi(\003)974 4095 y Fk(n)1049 4072 y Fn(to)f Fg(Z)1224 4039 y Fi(\003)1224 4095 y Fk(n)1300 4072 y Fn(if)f(and)h(only)g(if)f Fl(g)s(cd)p Fp(\()p Fl(S;)15 b(')p Fp(\()p Fl(n)p Fp(\)\))30 b(=)c(1)p Fn(.)44 b(Mor)-5 b(e)g(over,)34 b(if)f Fl(g)s(cd)p Fp(\()p Fl(S;)15 b(')p Fp(\()p Fl(n)p Fp(\)\))29 b Fl(>)e Fp(1)150 4185 y Fn(then)33 b Fp(an)m(y)g Fl(y)28 b Fj(2)d Fg(Z)750 4152 y Fi(\003)750 4208 y Fk(n)825 4185 y Fn(in)33 b(the)g(image)f(of)h Fl(f)42 b Fn(has)34 b(at)f(le)-5 b(ast)34 b(2)f(pr)-5 b(eimages.)291 4398 y(Pr)g(o)g(of:)529 4365 y Fh(1)616 4398 y Fp(First)33 b(of)g(all,)h(if)e Fl(S)38 b Fp(and)33 b Fl(')p Fp(\()p Fl(n)p Fp(\))h(are)f(relativ)m (ely)g(prime,)g(the)g(function)f Fl(f)43 b Fp(is)32 b(just)h(the)g(RSA) 150 4511 y(function)c(and)h(is)f(in)m(v)m(ertible.)39 b(No)m(w)31 b(supp)s(ose)e Fl(g)s(cd)p Fp(\()p Fl(S;)15 b(')p Fp(\()p Fl(n)p Fp(\)\))29 b(=)c Fl(d)g(>)g Fp(1.)291 4624 y(W)-8 b(e)32 b(use)e(the)g(follo)m(wing)f(theorem)i(due)f(to)h (Cauc)m(h)m(y:)150 4836 y Fq(Theorem)j(2)h(\(Cauc)m(h)m(y\))45 b Fn(L)-5 b(et)33 b Fl(G)f Fn(b)-5 b(e)32 b(a)h(gr)-5 b(oup.)42 b(Then,)33 b(for)g(any)f(prime)h Fl(p)f Fn(dividing)g Fj(j)p Fl(G)p Fj(j)p Fn(,)h(ther)-5 b(e)33 b(is)f(an)150 4949 y(element)h Fl(g)c Fj(2)c Fl(G)33 b Fn(of)f(or)-5 b(der)35 b Fl(p)p Fn(.)p 150 5010 1440 4 v 254 5064 a Ff(1)288 5096 y Fe(due)25 b(in)h(part)f(to)h(Y)-6 b(an)25 b(Zhong)h(Ding)1889 5416 y Fp(3-1)p eop %%Page: 2 2 2 1 bop 291 159 a Fp(W)-8 b(e)37 b(consider)d Fl(G)g Fp(=)f Fg(Z)1086 126 y Fi(\003)1086 181 y Fk(n)1128 159 y Fp(.)56 b(The)35 b(size)g(of)g Fl(G)h Fp(is)e Fl(')p Fp(\()p Fl(n)p Fp(\).)56 b(Let)36 b Fl(p)d(>)g Fp(1)j(b)s(e)f(an)m(y)g (prime)f(dividing)e Fl(d)p Fp(.)56 b(By)150 271 y(Cauc)m(h)m(y's)32 b(theorem,)h(there)e(is)g(an)g(elemen)m(t)h Fl(g)f Fj(2)c Fg(Z)1936 238 y Fi(\003)1936 294 y Fk(n)2010 271 y Fp(of)32 b(order)f Fl(p)p Fp(.)43 b(Since)31 b Fl(p)c(>)g Fp(1,)32 b Fl(g)f Fj(6\021)c Fp(1)f(mo)s(d)e Fl(n)p Fp(,)32 b(so)f(w)m(e)150 384 y(kno)m(w)f(that)h(for)f(an)m(y)h Fl(y)s Fp(,)1679 610 y Fl(g)s(y)d Fj(6\021)d Fl(y)j Fp(mo)s(d)d Fl(n)291 777 y Fp(but)1221 1003 y(\()p Fl(g)s(y)s Fp(\))1385 965 y Fk(S)1462 1003 y Fj(\021)g Fl(y)1606 965 y Fk(S)1656 1003 y Fl(g)1702 965 y Fk(S)1779 1003 y Fj(\021)g Fl(x)p Fp(\()p Fl(g)2008 965 y Fk(p)2049 1003 y Fp(\))2084 965 y Fk(S=p)2227 1003 y Fj(\021)g Fl(x)g Fp(mo)s(d)g Fl(n;)291 1170 y Fp(so)36 b Fl(x)f Fp(has)g(at)i(least)e(2)h(\(in)f(fact,)j(it)d (has)g(at)h(least)g Fl(p)p Fp(\))f(preimages,)i(namely)e Fl(y)j Fp(and)d Fl(g)s(y)s Fp(.)56 b(Since)35 b(an)m(y)150 1283 y(elemen)m(t)29 b(in)f(the)h(range)g(of)g Fl(f)38 b Fp(can)29 b(b)s(e)f(expressed)h(as)g Fl(y)2027 1250 y Fk(S)2106 1283 y Fp(for)g(some)g Fl(y)s Fp(,)g Fn(every)g Fp(elemen)m(t)g(in)e Fl(f)10 b Fp(\()p Fg(Z)3402 1250 y Fi(\003)3402 1305 y Fk(n)3445 1283 y Fp(\))29 b(has)f(at)150 1396 y(least)j(2)f(preimages.)2824 b Fd(2)291 1608 y Fp(No)m(w,)31 b(w)m(e)g(will)d(argue)j(the)f(correctness)h(of)g(the)g (algorithm.)39 b(If)30 b Fl(g)s(cd)p Fp(\()p Fl(S;)15 b(')p Fp(\()p Fl(n)p Fp(\)\))29 b(=)c(1)30 b(then)1388 1812 y Fl(x)25 b Fj( )g Fl(R)q(S)5 b(A)1780 1775 y Fi(\000)p Fh(1)1874 1812 y Fp(\()p Fl(n;)15 b(S;)g(z)2146 1775 y Fk(S)2223 1812 y Fp(mo)s(d)25 b Fl(n)p Fp(\))150 2017 y(will)f(alw)m(a)m(ys)j(b)s(e)f(correct,)j(so)e(if)f(the)h(righ)m(t)f (answ)m(er)h(is)f(\\y)m(es")i(w)m(e)f(will)d(alw)m(a)m(ys)k(output)e (\\y)m(es.")41 b(W)-8 b(e)28 b(kno)m(w)150 2129 y(that)33 b(if)e Fl(g)s(cd)p Fp(\()p Fl(S;)15 b(')p Fp(\()p Fl(n)p Fp(\)\))32 b Fl(>)d Fp(1,)k(then)f(for)h(an)m(y)g Fl(z)t Fp(,)g(there)g(are)f(at)i(least)e Fn(two)i Fl(y)h Fp(suc)m(h)d(that)h Fl(y)3200 2096 y Fk(S)3280 2129 y Fp(=)28 b Fl(z)3425 2096 y Fk(S)3476 2129 y Fp(.)47 b(No)m(w,)150 2242 y(w)m(e)29 b(don't)g(kno)m(w)f(ho)m(w)h Fl(R)q(S)5 b(A)1140 2209 y Fi(\000)p Fh(1)1263 2242 y Fp(w)m(orks)29 b(when)e Fl(g)s(cd)p Fp(\()p Fl(S;)15 b(')p Fp(\()p Fl(n)p Fp(\)\))29 b Fj(6)p Fp(=)c(1,)k(but)f(w)m(e)h(kno)m(w)g(what)f(it)h Fn(c)-5 b(an)c('t)29 b Fp(do.)40 b(It)150 2355 y(can't)29 b(guess)e(whic)m(h)g(answ)m(er)g(w)m(e)h(w)m(an)m(t)h(when)d(there)i (is)f(more)g(than)h(one)g(correct)h(answ)m(er.)39 b(Th)m(us,)28 b(since)150 2468 y(there)h(are)g(at)h(least)f(t)m(w)m(o)h(correct)g (answ)m(ers,)f(the)g(probabilit)m(y)d(that)j(w)m(e)g(output)g(\\y)m (es")h(is)e(at)h(most)g(1/2.)150 2581 y(\(It)34 b(migh)m(t)f(b)s(e)g (less,)h(but)e(w)m(e)i(don't)g(kno)m(w)f(ho)m(w)h(the)f Fl(R)q(S)5 b(A)2212 2548 y Fi(\000)p Fh(1)2340 2581 y Fp(function)32 b(op)s(erates)i(and)f(w)m(e)h(shouldn't)150 2694 y(put)c(an)m(y)g(assumptions)f(on)h(it.\))291 2807 y(This)22 b(giv)m(es)h(an)h(algorithm)e(whic)m(h)g(giv)m(es)i(us)f(the) g(correct)i(answ)m(er)e(all)f(the)i(time)f(if)f Fl(g)s(cd)p Fp(\()p Fl(S;)15 b(')p Fp(\()p Fl(n)p Fp(\)\))29 b(=)c(1)150 2920 y(and)h(at)h(least)g(half)e(the)i(time)f(if)f Fl(g)s(cd)p Fp(\()p Fl(S;)15 b(')p Fp(\()p Fl(n)p Fp(\)\))29 b Fl(>)c Fp(1.)39 b(W)-8 b(e)28 b(can)f(simply)d(run)h(the)h(algorithm)g(man)m (y)g(times)150 3033 y(in)c(succession)g(and)h(output)f(\\y)m(es")j (only)d(if)g(ev)m(ery)i(execution)g(of)f(the)g(algorithm)f(outputs)h (\\y)m(es,")j(giving)150 3146 y(\\no")36 b(as)f(our)g(answ)m(er)g (otherwise.)55 b(This)33 b(will)g(reduce)i(the)g(probabilit)m(y)e(of)j (erroneously)e(outputting)150 3259 y(\\y)m(es")f(to)f(an)f(arbitrarily) d(small)i(amoun)m(t.)44 b(\(This)29 b(is)i(the)g(standard)f(pumping)f (of)i(probabilistic)d(algo-)150 3371 y(rithms.\))291 3584 y Fq(Common)36 b(Errors:)46 b Fp(A)33 b(common)g(mistak)m(e)g(p)s (eople)f(made)h(w)m(as)g(misunderstanding)c(the)k(di\013er-)150 3697 y(ence)h(b)s(et)m(w)m(een)g(in)m(v)m(erting)e(RSA)h(\(giv)m(en)g Fl(n;)15 b(e;)g(u)p Fp(,)35 b(\014nd)d(a)h Fl(v)k Fp(suc)m(h)c(that)g Fl(v)2674 3664 y Fk(e)2741 3697 y Fj(\021)d Fl(u)25 b Fp(mo)s(d)g Fl(n)p Fp(\))33 b(and)f(reco)m(ving)150 3810 y(the)j(RSA)f(secret)h(k)m(ey)h(from)e(the)h(public)d(k)m(ey)j(\(giv)m (en)g Fl(n;)15 b(e)p Fp(,)36 b(\014nd)d Fl(d)p Fp(\).)54 b(If)34 b(one)h(can)g(reco)m(v)m(er)h(the)f(secret)150 3923 y(k)m(ey)-8 b(,)38 b(one)d(can)g(certainly)f(in)m(v)m(ert)h(RSA,)g (but)f(it)h(migh)m(t)g(b)s(e)f(that)i(in)m(v)m(erting)e(RSA)g(is)g (easy)i(but)e(reco)m(v-)150 4036 y(ering)d(the)h(secret)g(k)m(ey)h(is)d (hard.)44 b(Another)32 b(frequen)m(t)f(mistak)m(e)h(w)m(as)g(asserting) g(that)g(if)f(w)m(e)h(w)m(ere)g(giv)m(en)150 4149 y Fl(m)230 4116 y Fk(S)306 4149 y Fp(mo)s(d)24 b Fl(n)h Fp(where)f Fl(g)s(cd)p Fp(\()p Fl(S;)15 b(')p Fp(\()p Fl(n)p Fp(\)\))29 b Fl(>)c Fp(1,)h(then)f Fl(m)g Fp(cannot)h(b)s(e)e(reco)m(v)m(ered)j (since)d(there)h(is)f(no)h Fl(d)h Fp(suc)m(h)e(that)150 4261 y Fl(S)5 b(d)31 b Fj(\021)f Fp(1)c(mo)s(d)e Fl(')p Fp(\()p Fl(n)p Fp(\).)51 b(It)34 b(is)e(true)i(there)f(is)g(no)h Fl(d)p Fp(,)g(but)f(w)m(e)h(cannot)g(assume)g(the)f(RSA)g(in)m(v)m (erter)h(w)m(orks)150 4374 y(this)c(w)m(a)m(y)-8 b(.)45 b(As)32 b(an)f(example,)g(supp)s(ose)f Fl(g)s(cd)p Fp(\()p Fl(S;)15 b(')p Fp(\()p Fl(n)p Fp(\)\))30 b Fl(>)d Fp(1)k(but)g Fl(m)c Fp(=)f(1.)44 b(Then,)31 b Fl(m)3047 4341 y Fk(S)3124 4374 y Fj(\021)c Fp(1)f(mo)s(d)e Fl(n)p Fp(,)32 b(and)150 4487 y(w)m(e)f(can)g(see)f(that)h(1)g(is)f(a)g(correct)i(answ)m(er)e (ev)m(en)h(though)f(there)h(is)e(no)i(decrypting)e(k)m(ey)-8 b(.)291 4600 y(Another)40 b(common)g(error)g(w)m(as)g(in)f(the)h(pro)s (of)f(that)i(w)m(e)f(needed,)j(asserting)c(that)i(w)m(e)g(c)m(ho)s(ose) g(a)150 4713 y(generator)g Fl(g)i Fp(for)c Fg(Z)863 4680 y Fi(\003)863 4736 y Fk(n)906 4713 y Fp(.)67 b(If)39 b Fl(n)g Fp(is)g(an)g(RSA)g(mo)s(dulus,)h Fg(Z)2122 4680 y Fi(\003)2122 4736 y Fk(n)2204 4713 y Fp(will)c(not)k(b)s(e)f(a)h (cyclic)f(group,)i(so)f(there)g(is)150 4826 y(no)32 b(generator)h Fl(g)s Fp(.)46 b(The)31 b(source)h(of)g(this)f(confusion)g(w)m(as)h (probably)e(that)j Fg(Z)2780 4793 y Fh(+)2780 4848 y Fk(n)2834 4826 y Fp(,)g(the)f(additiv)m(e)f(group,)h Fn(is)150 4939 y Fp(cyclic,)e(and)g(1)h(is)e(a)i(generator.)1889 5416 y(3-2)p eop %%Page: 3 3 3 2 bop 150 159 a Fm(Problem)45 b(2)150 361 y Fp(Assume)36 b(one)h(is)e(using)g(RSA)h(with)g(the)g(same)h(mo)s(dulus)d Fl(n)i Fp(but)g(di\013eren)m(t)g(relativ)m(ely)g(prime)f(public)150 474 y(exp)s(onen)m(ts)c Fl(e)620 488 y Fh(1)691 474 y Fp(and)g Fl(e)911 488 y Fh(2)981 474 y Fp(to)h(encrypt)f(a)h(message)g Fl(m)p Fp(.)43 b(Pro)m(v)m(e)32 b(that)g(the)g(adv)m(ersary)f(can)h (reconstruct)f Fl(m)150 587 y Fp(from)f(the)g(t)m(w)m(o)i(encryptions)d (\(i.e.)41 b(from)30 b Fl(c)1627 601 y Fh(1)1692 587 y Fp(=)25 b Fl(m)1868 554 y Fk(e)1901 563 y Ff(1)1965 587 y Fp(mo)s(d)49 b Fl(n)30 b Fp(and)g Fl(c)2490 601 y Fh(2)2555 587 y Fp(=)25 b Fl(m)2731 554 y Fk(e)2764 563 y Ff(2)2828 587 y Fp(mo)s(d)49 b Fl(n)p Fp(\).)291 800 y Fq(Solution:)60 b Fp(Since)39 b Fl(e)1046 814 y Fh(1)1126 800 y Fp(and)g Fl(e)1354 814 y Fh(2)1434 800 y Fp(are)h(relativ)m(ely)f(prime,)j(the)e(adv)m(ersary)g(can)g(use)g (the)g(extended)150 913 y(GCD)32 b(algorithm)g(to)g(\014nd)f(in)m (tegers)h Fl(u)g Fp(and)g Fl(v)j Fp(suc)m(h)d(that)h Fl(ue)2281 927 y Fh(1)2342 913 y Fp(+)21 b Fl(v)s(e)2523 927 y Fh(2)2591 913 y Fp(=)28 b(1.)46 b(No)m(w,)34 b(the)e(adv)m (ersary)h(can)150 1026 y(simply)28 b(compute)j Fl(c)849 993 y Fk(u)849 1050 y Fh(1)894 1026 y Fl(c)933 993 y Fk(v)933 1050 y Fh(2)1004 1026 y Fp(and)f(since)1073 1251 y Fl(c)1112 1214 y Fk(u)1112 1274 y Fh(1)1157 1251 y Fl(c)1196 1214 y Fk(v)1196 1274 y Fh(2)1262 1251 y Fp(=)25 b(\()p Fl(m)1473 1214 y Fk(e)1506 1223 y Ff(1)1545 1251 y Fp(\))1580 1214 y Fk(u)1625 1251 y Fp(\()p Fl(m)1740 1214 y Fk(e)1773 1223 y Ff(2)1812 1251 y Fp(\))1847 1214 y Fk(v)1914 1251 y Fp(=)g Fl(m)2090 1214 y Fk(ue)2164 1223 y Ff(1)2198 1214 y Fh(+)p Fk(v)r(e)2322 1223 y Ff(2)2386 1251 y Fp(=)g Fl(m)2562 1214 y Fh(1)2626 1251 y Fp(=)g Fl(m;)291 1418 y Fp(this)k(reconstructs)i(the)g(message.)291 1631 y Fq(Common)g(Errors:)39 b Fp(The)28 b(only)f(common)h(error)g(in) e(this)h(problem)g(w)m(as)h(that)g(p)s(eople)f(men)m(tioned)150 1744 y(that)e Fl(u)g Fp(and)f Fl(v)k Fp(exist,)e(and)f(then)f(use)h (them,)h(without)e(remarking)f(that)j(they)f(can)g(b)s(e)f(found)g (e\016cien)m(tly)-8 b(.)150 2030 y Fm(Problem)45 b(3)150 2233 y Fp(Assume)e(Alice,)k(Bob)d(and)f(Carol)g(ha)m(v)m(e)i(3)g (randomly)d(c)m(hosen)i(mo)s(duli)d Fl(n)2838 2247 y Fk(A)2895 2233 y Fp(,)47 b Fl(n)3022 2247 y Fk(B)3126 2233 y Fp(and)c Fl(n)3371 2247 y Fk(C)3430 2233 y Fp(,)k(s.t.)82 b(3)150 2346 y(is)40 b(a)h(common)g(RSA)f(exp)s(onen)m(t)h(for)f(all)g (of)h(them)g(\(i.e.)72 b Fl(g)s(cd)p Fp(\(3)p Fl(;)15 b(')p Fp(\()p Fl(n)2607 2360 y Fk(X)2677 2346 y Fp(\)\))43 b(=)g(1,)g Fl(X)50 b Fp(=)43 b Fl(A;)15 b(B)5 b(;)15 b(C)7 b Fp(\).)72 b(A)150 2459 y(user)31 b Fl(U)41 b Fp(encrypts)31 b(the)h(same)g(message)g Fl(m)g Fp(using)d(their)i (public)e(k)m(eys.)44 b(Sho)m(w)31 b(that)i(an)e(adv)m(ersary)h(can)150 2572 y(reconstruct)21 b Fl(m)f Fp(from)g(the)h(3)g(encryptions)e (\(i.e.)38 b(from)20 b Fl(m)2086 2539 y Fh(3)2151 2572 y Fp(mo)s(d)49 b Fl(n)2430 2586 y Fk(A)2487 2572 y Fp(,)23 b Fl(m)2615 2539 y Fh(3)2679 2572 y Fp(mo)s(d)50 b Fl(n)2959 2586 y Fk(B)3039 2572 y Fp(and)20 b Fl(m)3286 2539 y Fh(3)3351 2572 y Fp(mo)s(d)50 b Fl(n)3631 2586 y Fk(C)3689 2572 y Fp(\).)291 2784 y Fq(Solution:)40 b Fp(W)-8 b(e)29 b(assume)f(the)g(mo)s(duli)e Fl(n)1716 2798 y Fk(A)1772 2784 y Fp(,)j Fl(n)1881 2798 y Fk(B)1941 2784 y Fp(,)g(and)f Fl(n)2225 2798 y Fk(C)2311 2784 y Fp(are)h(distinct.)2786 2751 y Fh(2)2864 2784 y Fp(There)e(are)i(t)m(w)m(o)g(cases)g(to)150 2897 y(consider.)39 b(First)28 b(of)g(all,)g(the)g(adv)m(ersary)h(c)m (hec)m(ks)g(to)g(mak)m(e)h(sure)d(that)i Fl(n)2638 2911 y Fk(A)2695 2897 y Fp(,)f Fl(n)2803 2911 y Fk(B)2864 2897 y Fp(,)g(and)g Fl(n)3147 2911 y Fk(C)3234 2897 y Fp(are)h(relativ)m(ely)150 3010 y(prime.)51 b(Supp)s(ose)33 b(this)g(c)m(hec)m(k)j(fails,)e(and)g(\(for)g(instance\))g(w)m(e)h (learn)f(that)h Fl(g)s(cd)p Fp(\()p Fl(n)3011 3024 y Fk(A)3069 3010 y Fl(;)15 b(n)3164 3024 y Fk(B)3225 3010 y Fp(\))32 b(=)f Fl(p)p Fp(.)53 b(Then,)150 3123 y(since)28 b Fl(n)426 3137 y Fk(A)511 3123 y Fp(and)g Fl(n)741 3137 y Fk(B)829 3123 y Fp(are)h(the)g(pro)s(ducts)e(of)h(t)m(w)m(o)i (primes,)e(w)m(e)h(ha)m(v)m(e)g(factored)h(them.)40 b(Once)28 b(w)m(e)h(ha)m(v)m(e)h(the)150 3236 y(factorization)h(of)g Fl(n)837 3250 y Fk(A)893 3236 y Fp(,)g(w)m(e)g(can)f(simply)e (calculate)j(the)f(decryption)g(k)m(ey)h(and)f(decrypt)g Fl(m)3280 3203 y Fh(3)3344 3236 y Fp(mo)s(d)25 b Fl(n)3599 3250 y Fk(A)3655 3236 y Fp(.)291 3349 y(Supp)s(ose)38 b(instead)h(that)h Fl(n)1241 3363 y Fk(A)1298 3349 y Fl(;)15 b(n)1393 3363 y Fk(B)1454 3349 y Fp(,)42 b(and)d Fl(n)1762 3363 y Fk(C)1861 3349 y Fp(are)h(relativ)m(ely)f(prime.)67 b(By)40 b(the)g(Chinese)e(Remain-)150 3462 y(der)h(Theorem)h(\(CR)-8 b(T\),)40 b(w)m(e)g(can)h(\014nd)d Fl(y)44 b Fj(2)c Fg(Z)1802 3429 y Fi(\003)1802 3484 y Fk(n)1845 3495 y Fc(A)1891 3484 y Fk(n)1934 3495 y Fc(B)1986 3484 y Fk(n)2029 3495 y Fc(C)2125 3462 y Fp(suc)m(h)f(that)i Fl(y)j Fj(\021)c Fl(m)2826 3429 y Fh(3)2891 3462 y Fp(mo)s(d)24 b Fl(n)3145 3476 y Fk(A)3202 3462 y Fl(;)15 b(n)3297 3476 y Fk(B)3357 3462 y Fp(,)43 b(and)c Fl(n)3666 3476 y Fk(C)3725 3462 y Fp(.)150 3575 y(By)46 b(the)g(uniqueness)e(of)h(the)h(CR)-8 b(T,)46 b(w)m(e)g(m)m(ust)g(ha)m(v)m(e)h Fl(y)54 b Fj(\021)c Fl(m)2421 3542 y Fh(3)2486 3575 y Fp(mo)s(d)24 b Fl(n)2740 3589 y Fk(A)2797 3575 y Fl(n)2852 3589 y Fk(B)2912 3575 y Fl(n)2967 3589 y Fk(C)3026 3575 y Fp(.)87 b(Ho)m(w)m(ev)m(er,)52 b Fl(m)e(<)150 3688 y(min)p Fp(\()p Fl(n)406 3702 y Fk(A)463 3688 y Fl(;)15 b(n)558 3702 y Fk(B)618 3688 y Fl(;)g(n)713 3702 y Fk(C)772 3688 y Fp(\),)41 b(and)d(th)m(us)h Fl(m)1347 3655 y Fh(3)1425 3688 y Fl(<)f(min)p Fp(\()p Fl(n)1790 3702 y Fk(A)1847 3688 y Fl(;)15 b(n)1942 3702 y Fk(B)2002 3688 y Fl(;)g(n)2097 3702 y Fk(C)2156 3688 y Fp(\))2191 3655 y Fh(3)2270 3688 y Fl(<)39 b(n)2435 3702 y Fk(A)2491 3688 y Fl(n)2546 3702 y Fk(B)2606 3688 y Fl(n)2661 3702 y Fk(C)2720 3688 y Fp(,)i(so)e Fl(y)i Fp(=)e Fl(m)3182 3655 y Fh(3)3221 3688 y Fp(,)i(and)d(w)m(e)h(can)150 3801 y(simply)28 b(tak)m(e)k(the)e(cub)s(e)g(ro)s(ot)h(of)f Fl(y)j Fp(o)m(v)m(er)f(the)f(real)f(n)m(um)m(b)s(ers)f(to)i(reco)m(v)m (er)h Fl(m)p Fp(.)150 4087 y Fm(Problem)45 b(4)150 4290 y Fp(Supp)s(ose)25 b Fl(S)5 b(QR)q(T)13 b Fp(\()p Fl(a;)i(p)p Fp(\))28 b(is)e(a)h(p)s(olynomial-time)e(algorithm)h(that,)j(if)d Fl(p)g Fp(is)h(prime)e(and)i Fl(a)g Fp(is)f(a)h(quadratic)150 4403 y(residue)i Fl(a)c Fj(2)g Fl(Z)687 4370 y Fi(\003)680 4425 y Fk(p)726 4403 y Fp(,)31 b(outputs)e(\(w.h.p.\))41 b(some)31 b(square)f(ro)s(ot)h(of)f Fl(a)g Fp(mo)s(d)g Fl(p)p Fp(.)291 4516 y Fq(\(a\))35 b Fp(Pro)m(v)m(e)i(that)f(the)g (follo)m(wing)e(algorithm)g(is)h(a)h(primalit)m(y)d(test)k(\(i.e.)56 b(that)36 b(on)g(a)g(prime)e(input)150 4629 y(outputs)k(PRIME)g(with)g (high)f(probabilit)m(y)-8 b(,)39 b(and)e(on)i(a)g(comp)s(osite)f(input) f(outputs)h(COMPOSITE)150 4742 y(with)29 b(probabilit)m(y)f(at)j(least) g(1)p Fl(=)p Fp(2.\))p 150 4803 1440 4 v 254 4857 a Ff(2)288 4889 y Fe(If)26 b(t)n(w)n(o)g(or)g(more)f(of)i(the)e(mo)r(duli)g(are)h (the)g(same,)f(w)n(e)h(ma)n(y)f(not)g(b)r(e)h(able)g(to)g(reco)n(v)n (er)g(the)f(message)h(e\016cien)n(tly)-6 b(.)1889 5416 y Fp(3-3)p eop %%Page: 4 4 4 3 bop 150 159 a Fp(1.)41 b(If)30 b Fl(n)25 b Fp(=)g(2,)31 b(output)f(PRIME.)150 271 y(2.)41 b(If)30 b Fl(n)g Fp(is)f(ev)m(en)i (or)g(of)f(the)h(form)f Fl(n)25 b Fp(=)g Fl(p)1544 239 y Fk(b)1608 271 y Fp(for)30 b(some)h Fl(p;)15 b(b)25 b(>)g Fp(1,)31 b(output)f(COMPOSITE.)150 384 y(3.)41 b(Cho)s(ose)30 b(a)h(random)f Fl(r)d Fj(2)e(f)p Fp(1)p Fl(;)15 b(:)g(:)g(:)j(;)d(n)20 b Fj(\000)g Fp(1)p Fj(g)p Fp(.)41 b(If)30 b Fl(g)s(cd)p Fp(\()p Fl(r)m(;)15 b(n)p Fp(\))28 b Fl(>)d Fp(1,)31 b(output)f(COMPOSITE.)150 497 y(4.)41 b(Compute)30 b Fl(x)25 b Fj( )g Fl(S)5 b(QR)q(T)13 b Fp(\()p Fl(r)1196 464 y Fh(2)1260 497 y Fp(mo)s(d)25 b Fl(n;)15 b(n)p Fp(\).)150 610 y(5.)41 b(If)30 b Fl(x)25 b Fp(=)g Fj(\006)p Fl(r)j Fp(mo)s(d)c Fl(n)p Fp(,)30 b(output)g(PRIME.)h(Otherwise,)e(output)h(COMPOSITE.)291 922 y Fq(Solution:)41 b Fp(W)-8 b(e)32 b(m)m(ust)e(\014rst)g(pro)m(v)m (e)h(the)f(follo)m(wing)f(theorem:)150 1135 y Fq(Theorem)34 b(3)46 b Fn(The)29 b(fol)5 b(lowing)31 b(holds)g(for)f(any)g(o)-5 b(dd)31 b Fl(n)p Fn(:)40 b(If)29 b Fl(n)g Fn(is)g(prime,)i(then)f (every)f(quadr)-5 b(atic)31 b(r)-5 b(esidue)150 1248 y(has)30 b(exactly)g(two)h(squar)-5 b(e)30 b(r)-5 b(o)g(ots.)42 b(If)30 b Fl(n)e Fn(is)i(the)f(pr)-5 b(o)g(duct)32 b(of)d(at)h(le)-5 b(ast)31 b(two)f(primes,)h(then)e(every)g(quadr)-5 b(atic)150 1361 y(r)g(esidue)33 b(has)h(at)f(le)-5 b(ast)34 b(four)f(squar)-5 b(e)33 b(r)-5 b(o)g(ots.)291 1573 y(Pr)g(o)g(of:)51 b Fp(Supp)s(ose)33 b Fl(b)i Fp(is)f(a)h(quadratic)g(residue)e(mo)s(dulo)g Fl(n)p Fp(,)j(and)e Fl(n)f Fp(=)f Fl(p)2780 1533 y Fk(\013)2825 1542 y Ff(1)2780 1599 y Fh(1)2864 1573 y Fl(p)2910 1533 y Fk(\013)2955 1542 y Ff(2)2910 1599 y Fh(2)3009 1573 y Fl(:)15 b(:)g(:)h(p)3176 1530 y Fk(\013)3221 1542 y Fc(k)3176 1603 y Fk(k)3299 1573 y Fp(for)34 b(some)i Fl(k)s Fp(,)150 1686 y(where)30 b Fl(\013)471 1700 y Fk(i)524 1686 y Fl(>)25 b Fp(0)31 b(and)f Fl(p)919 1700 y Fk(i)972 1686 y Fj(6)p Fp(=)25 b Fl(p)1114 1700 y Fk(j)1181 1686 y Fp(for)30 b Fl(i)25 b Fj(6)p Fp(=)g Fl(j)5 b Fp(.)42 b(W)-8 b(e)31 b(are)g(in)m(terested)f(in)f(the)i(solutions)e(to)i(the)f (equation)1685 1912 y Fl(x)1737 1874 y Fh(2)1801 1912 y Fj(\021)25 b Fl(b)g Fp(mo)s(d)g Fl(n)1418 b Fp(\(1\))291 2079 y(Supp)s(ose)27 b Fl(a)i Fp(is)f(one)h(solution)e(to)j(\(1\).)41 b(By)29 b(the)g(CR)-8 b(T,)28 b(w)m(e)i(can)f(represen)m(t)g Fl(a)f Fp(uniquely)e(as)j(a)g Fl(k)s Fp(-tuple)150 2192 y(\()p Fl(a)233 2206 y Fh(1)273 2192 y Fl(;)15 b(:)g(:)g(:)h(;)f(a)522 2207 y Fk(k)566 2192 y Fp(\))30 b(where)g Fl(a)942 2206 y Fk(i)996 2192 y Fj(\021)25 b Fl(a)g Fp(mo)s(d)f Fl(p)1410 2150 y Fk(\013)1455 2160 y Fc(i)1410 2219 y Fk(i)1486 2192 y Fp(.)41 b(Since)29 b(w)m(e)i(kno)m(w)f(that)1674 2418 y Fl(a)1722 2380 y Fh(2)1787 2418 y Fj(\021)25 b Fl(b)g Fp(mo)s(d)f Fl(n;)291 2584 y Fp(w)m(e)31 b(kno)m(w)1536 2810 y Fl(a)1584 2773 y Fh(2)1649 2810 y Fj(\021)25 b Fl(a)1793 2773 y Fh(2)1793 2833 y Fk(i)1857 2810 y Fj(\021)g Fl(b)h Fp(mo)s(d)e Fl(p)2263 2769 y Fk(\013)2308 2779 y Fc(i)2263 2837 y Fk(i)2339 2810 y Fl(:)291 2977 y Fp(No)m(w,)38 b(w)m(e)e(can)f(\014nd)f(2)1072 2944 y Fk(k)1151 2977 y Fp(total)i(solutions)e(to)i(\(1\))h(b)m(y)e(using)f(the)h(CR)-8 b(T)35 b(to)i(reconstruct)e(the)h(2)3591 2944 y Fk(k)3670 2977 y Fl(k)s Fp(-)150 3090 y(tuples)i(\()p Fj(\006)p Fl(a)581 3104 y Fh(1)621 3090 y Fl(;)15 b(:)g(:)g(:)h(;)f Fj(\006)p Fl(a)941 3105 y Fk(k)984 3090 y Fp(\).)67 b(Since)38 b Fl(n)h Fp(is)f(kno)m(wn)h(to)h(b)s(e)e(o)s(dd,)j(these)e Fl(k)s Fp(-tuples)g(are)g(all)f(distinct)g(\(that)150 3203 y(is,)i Fl(a)325 3217 y Fk(i)392 3203 y Fj(6\021)e(\000)p Fl(a)620 3217 y Fk(i)674 3203 y Fp(mo)s(d)24 b Fl(p)919 3162 y Fk(\013)964 3172 y Fc(i)919 3230 y Fk(i)995 3203 y Fp(\).)65 b(Note)40 b(that)f(it)f(is)g(p)s(ossible)e(to)j(mak)m(e)g (this)f(exact)i(\(ie,)h(there)d(are)h(exactly)150 3316 y(2)195 3283 y Fk(k)278 3316 y Fp(solutions\),)i(but)e(it)h(is)f (unnecessary)g(for)h(us:)59 b(kno)m(wing)39 b(there)h(are)g(at)h(least) f(2)3100 3283 y Fk(k)3183 3316 y Fp(su\016ces)g(if)e Fl(n)i Fp(is)150 3429 y(comp)s(osite.)55 b(Ho)m(w)m(ev)m(er,)38 b(if)d Fl(n)f Fp(is)g(prime,)h(there)h(m)m(ust)f(b)s(e)f(exactly)i(t)m (w)m(o)g(square)f(ro)s(ots.)55 b(Supp)s(ose)34 b Fl(x;)15 b(y)150 3542 y Fp(are)31 b(square)f(ro)s(ots)g(of)h Fl(a)f Fp(mo)s(dulo)f Fl(n)h Fp(where)g Fl(n)g Fp(is)f(prime.)39 b(Then,)1648 3768 y Fl(x)1700 3730 y Fh(2)1764 3768 y Fj(\021)25 b Fl(y)1908 3730 y Fh(2)1973 3768 y Fp(mo)s(d)f Fl(n;)291 3934 y Fp(so)1692 4160 y Fl(x)1744 4123 y Fh(2)1804 4160 y Fj(\000)c Fl(y)1943 4123 y Fh(2)2007 4160 y Fp(=)25 b Fl(k)s(n)150 4327 y Fp(for)34 b(some)h(in)m(teger)g Fl(k)s Fp(.)53 b(Th)m(us,)34 b Fl(n)p Fj(j)p Fp(\()p Fl(x)23 b Fj(\000)g Fl(y)s Fp(\)\()p Fl(x)g Fp(+)g Fl(y)s Fp(\))34 b(so)h(since)f Fl(n)f Fp(is)h(prime,)g(either)g Fl(n)p Fj(j)p Fp(\()p Fl(x)23 b Fj(\000)f Fl(y)s Fp(\),)36 b(in)d(whic)m(h)150 4440 y(case)f Fl(x)26 b Fj(\021)g Fl(y)i Fp(mo)s(d)d Fl(n)p Fp(,)30 b(or)h Fl(n)p Fj(j)p Fp(\()p Fl(x)21 b Fp(+)f Fl(y)s Fp(\),)32 b(in)d(whic)m(h)h(case)i Fl(x)26 b Fj(\021)g(\000)p Fl(y)i Fp(mo)s(d)c Fl(n)p Fp(.)42 b(Th)m(us,)30 b(there)h(are)h(at)f(most)h(t)m(w)m(o)150 4553 y(square)25 b(ro)s(ots)g(of)g(an)m(y)g(quadratic)g(residue)e(mo)s (dulo)h Fl(n)p Fp(.)38 b(There)25 b(are)g(at)h(least)f(t)m(w)m(o,)j (since)c Fl(x)h Fj(6)p Fp(=)g Fj(\000)p Fl(x)g Fp(unless)150 4666 y Fl(x)g Fp(=)g(0)31 b(\(since)f Fl(n)25 b(>)g Fp(2\),)31 b(and)f Fl(a)25 b Fp(=)g(0)31 b(is)e(not)i(an)f(elemen)m(t)h(w)m(e)g (consider.)1092 b Fd(2)291 4779 y Fp(Supp)s(ose)20 b(that)j Fl(p)p Fp(\()p Fj(j)p Fl(n)p Fj(j)p Fp(\))f(is)g(the)g(probabilit)m(y)d (that)k Fl(S)5 b(QR)q(T)13 b Fp(\()p Fl(a;)i(n)p Fp(\))22 b(returns)f(a)h(square)g(ro)s(ot)g(of)h Fl(a)f Fp(mo)s(dulo)150 4892 y Fl(n)30 b Fp(when)f Fl(n)h Fp(is)g(prime.)291 5005 y(If)f Fl(n)g Fp(is)f(a)i(prime)e(n)m(um)m(b)s(er,)h(then)g (either)g Fl(n)g Fp(is)f(o)s(dd)g(or)i Fl(n)25 b Fp(=)g(2.)40 b(If)29 b Fl(n)c Fp(=)g(2,)30 b(the)g(algorithm)e(succeeds.)150 5117 y(If)35 b Fl(n)f Fp(is)g(o)s(dd,)i(then)f(with)e(probabilit)m(y)g Fl(p)p Fp(\()p Fj(j)p Fl(n)p Fj(j)p Fp(\))j(the)f(v)-5 b(alue)35 b Fl(x)g Fp(is)f(a)h(square)g(ro)s(ot)g(of)h Fl(r)3099 5085 y Fh(2)3173 5117 y Fp(mo)s(dulo)d Fl(n)p Fp(.)55 b(By)1889 5416 y(3-4)p eop %%Page: 5 5 5 4 bop 150 159 a Fp(theorem)26 b(3,)g(the)g(only)e(t)m(w)m(o)j(square) e(ro)s(ots)g(are)h Fl(r)h Fp(and)e Fj(\000)p Fl(r)s Fp(,)h(so)f Fl(x)g Fp(=)g Fj(\006)p Fl(r)j Fp(with)c(probabilit)m(y)e Fl(p)p Fp(\()p Fj(j)p Fl(n)p Fj(j)p Fp(\),)27 b(whic)m(h)150 271 y(is)i(high.)291 384 y(If)35 b Fl(n)g Fp(is)g(comp)s(osite,)i(then) e(if)g Fl(n)g Fp(is)g(ev)m(en)h(or)f(of)h(the)g(form)f Fl(p)2386 351 y Fk(b)2456 384 y Fp(for)g(some)h Fl(p;)15 b(b)34 b(>)g Fp(1,)j(the)f(algorithm)150 497 y(succeeds)31 b(with)e(probabilit)m(y)f(1.)1257 464 y Fh(3)1337 497 y Fp(Otherwise,)h(w)m(e)i(get)h(to)f(step)f(5,)h(and)f Fl(n)g Fp(is)f(o)s(dd)g(and)h(is)g(the)g(pro)s(duct)150 610 y(of)j(at)h(least)g(t)m(w)m(o)h(distinct)c(o)s(dd)i(primes.)47 b(Th)m(us,)34 b(b)m(y)f(theorem)h(3,)g(there)g(are)f(at)h(least)g(4)g (square)f(ro)s(ots)150 723 y(of)j Fl(r)303 690 y Fh(2)367 723 y Fp(mo)s(d)25 b Fl(n)p Fp(.)56 b(Call)34 b(these)i(square)g(ro)s (ots)f Fl(r)m(;)15 b Fj(\000)p Fl(r)m(;)g(r)1936 690 y Fi(0)1961 723 y Fl(;)36 b Fp(and)f Fj(\000)p Fl(r)2319 690 y Fi(0)2341 723 y Fp(.)57 b(Whatev)m(er)37 b Fl(S)5 b(QR)q(T)48 b Fp(do)s(es,)37 b(it)e(cannot)150 836 y(return)h Fj(\006)p Fl(r)j Fp(when)d(w)m(e)i(c)m(hose)g Fl(r)i Fp(and)c(return)g Fj(\006)p Fl(r)1882 803 y Fi(0)1942 836 y Fp(when)g(w)m(e)h(c)m(hose)h Fl(r)2617 803 y Fi(0)2640 836 y Fp(,)h(since)d(its)h(input)e(is)h(iden)m(tical)150 949 y(in)30 b(those)i(t)m(w)m(o)h(cases.)46 b(Th)m(us,)31 b(the)h(algorithm)e(will)f(output)i(PRIME)h(with)e(probabilit)m(y)f(at) j(most)g(1/2.)150 1062 y(Therefore,)e(the)h(algorithm)e(outputs)h (COMPOSITE)f(with)g(probabilit)m(y)f(at)j(least)f(1/2.)291 1274 y Fq(Common)25 b(Errors:)37 b Fp(The)22 b(most)h(common)h(error)e (w)m(as)h(not)h(explaining)c(wh)m(y)i(step)h(2)h(is)e(imp)s(ortan)m(t.) 150 1387 y(If)28 b Fl(n)h Fp(is)f(a)h(prime)e(p)s(o)m(w)m(er,)j(then)e (quadratic)h(residues)e(will)f(ha)m(v)m(e)k(only)e(t)m(w)m(o)i(square)f (ro)s(ots,)h(so)f(w)m(e)g(w)m(ould)150 1500 y(return)i(PRIME)i(with)e (high)g(probabilit)m(y)f(if)i(w)m(e)h(reac)m(h)g(step)f(5.)48 b(Similarly)-8 b(,)30 b(for)i(some)h(ev)m(en)g(n)m(um)m(b)s(ers)150 1613 y(\(lik)m(e)g(6,)i(for)f(instance\),)h(quadratic)e(residues)f(ha)m (v)m(e)j(only)d(t)m(w)m(o)j(square)f(ro)s(ots.)51 b(Ho)m(w)m(ev)m(er,) 36 b(if)d Fl(n)g Fp(is)g(o)s(dd)150 1726 y(and)d(is)f(not)i(a)g(prime)d (p)s(o)m(w)m(er,)j(then)f(quadratic)g(residues)f(m)m(ust)h(ha)m(v)m(e)h (4)g(square)f(ro)s(ots,)h(as)g(w)m(e)g(pro)m(v)m(e.)291 1939 y Fq(Remarks:)40 b Fp(W)-8 b(e)32 b(attribute)e(this)f(primalit)m (y)f(test)k(to)f(Silvio)d(Micali.)291 2151 y Fq(\(b\))j Fp(Supp)s(ose)f Fl(p)d Fp(=)h(4)p Fl(m)21 b Fp(+)g(3)32 b(is)f(a)h(prime)e(and)i(that)g Fl(a)g Fp(is)e(a)j(quadratic)e(residue) f(mo)s(dulo)g Fl(p)p Fp(.)45 b(Pro)m(v)m(e)150 2264 y(that)31 b Fl(a)395 2231 y Fk(m)p Fh(+1)582 2264 y Fp(is)e(a)i(square)f(ro)s(ot) h(of)g Fl(a)f Fp(mo)s(dulo)f Fl(p)p Fp(.)291 2476 y Fq(Solution:)45 b Fp(Let)33 b Fl(a)g Fp(b)s(e)e(a)i(quadratic)f(residue)f(mo)s(d)g Fl(p)e Fp(=)f(4)p Fl(m)22 b Fp(+)f(3,)34 b(and)e(let)g Fl(b)2985 2443 y Fh(2)3053 2476 y Fj(\021)d Fl(a)c Fp(mo)s(d)f Fl(p)p Fp(.)47 b(Then)150 2589 y(w)m(e)31 b(kno)m(w)f(that)1462 2907 y(\()p Fl(a)1545 2869 y Fk(m)p Fh(+1)1702 2907 y Fp(\))1737 2869 y Fh(2)1860 2907 y Fj(\021)82 b Fp(\()p Fl(b)2087 2869 y Fh(2)2127 2907 y Fl(m)20 b Fp(+)g(2\))2398 2869 y Fh(2)1860 3044 y Fj(\021)82 b Fl(b)2052 3007 y Fh(4)p Fk(m)p Fh(+4)1860 3182 y Fj(\021)g Fl(b)2052 3145 y Fh(4)p Fk(m)p Fh(+2)2244 3182 y Fl(b)2283 3145 y Fh(2)1860 3320 y Fj(\021)g Fl(b)2052 3282 y Fk(p)p Fi(\000)p Fh(1)2182 3320 y Fl(a)1860 3458 y Fj(\021)g Fp(1)p Fl(a)26 b Fp(mo)s(d)f Fl(p;)291 3662 y Fp(b)s(ecause)30 b Fl(b)666 3629 y Fk(p)p Fi(\000)p Fh(1)821 3662 y Fj(\021)25 b Fp(1)g(mo)s(d)g Fl(p)k Fp(for)g(prime)f Fl(p)p Fp(,)h(b)m(y)h(F)-8 b(ermat's)31 b(theorem.)40 b(Th)m(us,)29 b Fl(a)2949 3629 y Fk(m)p Fh(+1)3135 3662 y Fp(is)g(a)g(square)h(ro)s(ot)150 3775 y(of)h Fl(a)f Fp(mo)s(dulo)f Fl(p)p Fp(.)291 3987 y Fq(\(c\))j Fp(Use)g(\(a\))h(and)e(\(b\))h(to)h(devise)e(a)h(primalit)m(y)e (testing)i(algorithm)f(for)h(in)m(tegers)g(equiv)-5 b(alen)m(t)31 b(to)i(3)150 4100 y(mo)s(dulo)c(4.)291 4313 y Fq(Solution:)39 b Fp(Implemen)m(t)25 b(the)i(function)e Fl(S)5 b(QR)q(T)13 b Fp(\()p Fl(a;)i(n)p Fp(\))26 b(b)m(y)g(computing)f Fl(a)2814 4280 y Fh(\()p Fk(n)p Fh(+1\))p Fk(=)p Fh(4)3102 4313 y Fp(mo)s(d)g Fl(n)p Fp(.)38 b(No)m(w)27 b(use)150 4426 y(the)34 b(algorithm)f(from)g(part)h(\(a\).)52 b(If)33 b Fl(n)h Fp(is)f(a)h(prime,)f(then)h Fl(n)c Fp(=)h Fl(p)g Fp(=)f(4)p Fl(m)23 b Fp(+)f(3)35 b(for)e(some)h Fl(m)p Fp(,)h(and)e(then)150 4539 y(\()p Fl(n)20 b Fp(+)g(1\))p Fl(=)p Fp(4)27 b(=)e(\(4)p Fl(m)c Fp(+)f(4\))p Fl(=)p Fp(4)27 b(=)e Fl(m)20 b Fp(+)g(1,)31 b(so)f(b)m(y)g(part)h(\(b\))f(the) h Fl(S)5 b(QR)q(T)42 b Fp(function)29 b(giv)m(es)i(correct)h(output.)p 150 4600 1440 4 v 254 4656 a Ff(3)288 4688 y Fe(Finding)27 b(whether)f Fb(n)d Fe(=)f Fb(p)1059 4656 y Fc(b)1116 4688 y Fe(is)27 b(easy;)g(w)n(e)g(need)f(only)g(c)n(hec)n(k)g(v)l (alues)g(of)i Fb(b)e Fe(up)f(to)i(log)2738 4707 y Ff(2)2785 4688 y Fb(n)p Fe(,)h(and)e(eac)n(h)g(one)h(is)g(simply)e(an)150 4779 y Fb(n)p Fe(th)g(ro)r(ot)i(computation)e(o)n(v)n(er)g(the)h(real)g (n)n(um)n(b)r(ers.)1889 5416 y Fp(3-5)p eop %%Page: 6 6 6 5 bop 150 159 a Fm(Problem)45 b(5)150 361 y Fp(Supp)s(ose)29 b Fl(p)h Fp(is)f(a)i(prime)e(and)g Fl(g)34 b Fp(is)c(a)g(generator)i (mo)s(dulo)d Fl(p)p Fp(.)291 474 y(Exp)s(erimen)m(t)g(1:)41 b(Pic)m(k)30 b Fl(x)h Fp(at)g(random)e(in)g Fj(f)p Fp(1)p Fl(;)15 b(:)g(:)g(:)j(;)d(p)20 b Fj(\000)g Fp(1)p Fj(g)p Fp(.)42 b(Output)29 b Fl(g)2708 441 y Fk(x)2753 474 y Fp(.)291 587 y(Exp)s(erimen)m(t)g(2:)41 b(Pic)m(k)30 b Fl(x;)15 b(y)34 b Fp(at)d(random)f(in)f Fj(f)p Fp(1)p Fl(;)15 b(:)g(:)g(:)i(;)e(p)21 b Fj(\000)f Fp(1)p Fj(g)p Fp(.)41 b(Output)29 b Fl(g)2796 554 y Fk(xy)2878 587 y Fp(.)291 700 y(Pro)m(v)m(e)46 b(or)g(dispro)m(v)m(e:)69 b(Exp)s(erimen)m(t)44 b(1)h(and)g(Exp)s(erimen)m(t)e(2)j(pro)s(duce)e (iden)m(tically)f(distributed)150 813 y(outputs.)291 1026 y Fq(Solution:)58 b Fp(If)38 b Fl(p)h Fp(=)g(2,)i(it)d(is)g (trivial)f(to)i(sho)m(w)g(the)f(t)m(w)m(o)i(exp)s(erimen)m(ts)e(are)h (iden)m(tical;)j(the)d(only)150 1139 y(output)25 b(is)f Fl(g)575 1106 y Fh(1)615 1139 y Fp(.)39 b(Ho)m(w)m(ev)m(er,)29 b(if)24 b Fl(p)h(>)g Fp(2,)i(The)e(t)m(w)m(o)i(exp)s(erimen)m(ts)d(do)i (not)f(pro)s(duce)f(iden)m(tically)g(distributed)150 1251 y(outputs.)52 b(In)34 b(exp)s(erimen)m(t)g(1,)i Fl(g)1278 1218 y Fk(x)1357 1251 y Fp(is)d(a)i(quadratic)f(residue)f(if) g(and)h(only)g(if)f Fl(x)i Fp(is)e(ev)m(en;)38 b(a)c(probabilit)m(y)150 1364 y(of)f(1)p Fl(=)p Fp(2.)49 b(In)32 b(exp)s(erimen)m(t)g(2,)i Fl(g)1206 1331 y Fk(xy)1320 1364 y Fp(is)e(a)h(quadratic)f(residue)f (if)h(and)g(only)g(if)f Fl(x)i Fp(is)f(ev)m(en)h(or)g Fl(y)i Fp(is)d(ev)m(en.)48 b(If)150 1477 y(either)27 b(one)g(is)f(ev)m(en,)j(then)e Fl(xy)s(=)p Fp(2)h(is)e(an)h(in)m (teger,)i(and)d Fl(g)2062 1444 y Fk(xy)r(=)p Fh(2)2242 1477 y Fp(will)e(b)s(e)j(a)h(square)e(ro)s(ot)i(of)f Fl(g)3224 1444 y Fk(xy)3306 1477 y Fp(.)40 b(Th)m(us,)27 b(the)150 1590 y(probabilit)m(y)h(that)j(the)f(output)g(of)h(exp)s (erimen)m(t)e(2)i(will)d(b)s(e)i(a)g(quadratic)g(residue)f(is)g(3)p Fl(=)p Fp(4.)291 1803 y(Another)22 b(solution:)35 b(In)21 b(exp)s(erimen)m(t)f(1,)k Fl(g)1722 1770 y Fk(x)1792 1803 y Fj(\021)h Fp(1)h(mo)s(d)e Fl(p)e Fp(if)e(and)h(only)g(if)g Fl(x)k Fp(=)g Fl(p)s Fj(\000)s Fp(1.)37 b(The)21 b(probabilit)m(y)150 1916 y(of)33 b(this)e(is)h(1)p Fl(=)p Fp(\()p Fl(p)23 b Fj(\000)e Fp(1\).)49 b(On)32 b(the)h(other)g(hand,)f(in)f(exp)s (erimen)m(t)h(2,)i(if)e(either)g Fl(x)d Fp(=)g Fl(p)21 b Fj(\000)h Fp(1)33 b(or)g Fl(y)f Fp(=)c Fl(p)22 b Fj(\000)f Fp(1,)150 2029 y(then)30 b Fl(g)403 1996 y Fk(xy)510 2029 y Fj(\021)25 b Fp(1)h(mo)s(d)e Fl(p)p Fp(.)41 b(The)30 b(probabilit)m(y)d(of)k(this)e(is)h(at)h(least)1415 2208 y(2)p Fl(p)20 b Fj(\000)g Fp(1)p 1383 2249 313 4 v 1383 2332 a(\()p Fl(p)g Fj(\000)g Fp(1\))1655 2306 y Fh(2)1730 2270 y Fl(>)1891 2208 y(p)g Fj(\000)g Fp(1)p 1836 2249 V 1836 2332 a(\()p Fl(p)h Fj(\000)e Fp(1\))2108 2306 y Fh(2)2184 2270 y Fp(=)2368 2208 y(1)p 2290 2249 203 4 v 2290 2332 a Fl(p)h Fj(\000)g Fp(1)2502 2270 y Fl(:)291 2480 y Fp(Th)m(us,)30 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FF80007FF80003FF80007FF80003FF8000FFF80001FF8001FFF80001FF8003FFFC0000FF C007DFFFE000FFF01F9FFFE0007FFFFF1FFFE0003FFFFE1FFFE0000FFFF81FFFE00001FF E01FE000332A7CA83A>III121 D E %EndDVIPSBitmapFont end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: Letter %%EndSetup %%Page: 1 1 1 0 bop 157 71 3586 4 v 157 574 4 504 v 246 164 a Fr(6.875J/18.425J)38 b(Cryptograph)m(y)d(and)g(Cryptanalysis)546 b Fq(Octob)s(er)30 b(1,)h(2001)1256 371 y Fp(Handout)37 b(6:)51 b(Solution)37 b(Set)i(2)246 531 y Fo(Instructor:)k(A)n(nna)33 b(Lysyanskaya)853 b(T)-7 b(e)i(aching)33 b(Assistant:)43 b(Moses)33 b(Liskov)p 3739 574 V 157 577 3586 4 v 150 987 a Fn(Problem)45 b(1)150 1190 y Fq(F)-8 b(or)32 b(eac)m(h)h(of)f(the)f(follo)m(wing)f (de\014nitions)f(of)j(securit)m(y)f(for)h(a)g(cryptosystem,)g(compare)g (them)g(to)g(GM-)150 1303 y(securit)m(y)-8 b(.)42 b(Some)31 b(of)g(the)g(de\014nitions)d(ma)m(y)j(b)s(e)f(to)s(o)i(strong,)f(that)g (is,)f(unattainable.)41 b(If)30 b(the)h(de\014nition)150 1416 y(is)e(unattainable,)h(pro)m(v)m(e)h(it.)40 b(Otherwise,)30 b(for)g(eac)m(h)h(de\014nition)d(y)m(ou)j(should)e(pro)m(v)m(e)i(or)f (dispro)m(v)m(e)g(that)261 1628 y(1.)46 b(If)30 b(\()p Fm(G;)15 b(E)5 b(;)15 b(D)s Fq(\))33 b(is)c(secure)i(under)d(that)j (de\014nition)d(then)i(it)g(is)g(GM-secure,)h(and)261 1816 y(2.)46 b(If)30 b(\()p Fm(G;)15 b(E)5 b(;)15 b(D)s Fq(\))33 b(is)c(GM-secure)i(then)g(it)e(is)h(secure)g(under)f(the)i (giv)m(en)f(de\014nition.)291 2128 y Fr(\(a\))g Fq(2)499 2095 y Fl(\000)p Fk(k)629 2128 y Fo(se)-5 b(curity.)41 b Fq(Do)31 b(w)m(e)g(need)f(to)h(b)s(e)f(so)h(general)f(ab)s(out)g(the) h(negligible)d(function?)150 2340 y Fj(8)d(f)p Fm(A)339 2354 y Fk(n)386 2340 y Fj(g)h(2)f Fm(P)13 b(P)g(T)38 b Fj(9)p Fm(K)31 b Fj(2)25 b Fi(N)43 b Fq(suc)m(h)30 b(that)h Fj(8)p Fm(k)d(>)d(K)q(;)355 2453 y Fq(Pr)o([\()p Fm(P)13 b(K)q(;)i(S)5 b(K)i Fq(\))27 b Fj( )e Fm(G)p Fq(\(1)1175 2420 y Fk(k)1219 2453 y Fq(\);)15 b Fm(b)26 b Fj( )f(f)p Fq(0)p Fm(;)15 b Fq(1)p Fj(g)p Fq(;)355 2566 y Fm(c)25 b Fj( )g Fm(E)5 b Fq(\()p Fm(P)13 b(K)q(;)i(b)p Fq(\);)g Fm(b)984 2533 y Fl(0)1035 2566 y Fj( )26 b Fm(A)1220 2581 y Fk(k)1262 2566 y Fq(\()p Fm(P)13 b(K)q(;)i(c)p Fq(\))28 b(:)d Fm(b)1677 2533 y Fl(0)1726 2566 y Fq(=)g Fm(b)p Fq(])g Fj(\024)g Fq(1)p Fm(=)p Fq(2)d(+)e(2)2300 2533 y Fl(\000)p Fk(k)291 2879 y Fr(Solution:)44 b Fq(This)30 b(de\014nition)f(is)i Fr(unattainable.)43 b Fq(Supp)s(ose)30 b(that)i(\()p Fm(G;)15 b(E)5 b(;)15 b(D)s Fq(\))34 b(is)c(a)i (cryptosystem)150 2991 y(suc)m(h)38 b(that)h Fm(G)p Fq(\(1)719 2958 y Fk(k)763 2991 y Fq(\))g(uses)f(exactly)h Fm(k)i Fq(bits)c(of)i(randomness.)63 b(Then)38 b(an)g(adv)m(ersary)h Fm(A)f Fq(whic)m(h)f(defeats)150 3104 y(\()p Fm(G;)15 b(E)5 b(;)15 b(D)s Fq(\))45 b(is)d(one)h(that,)k(on)c(input)e(\()p Fm(P)13 b(K)q(;)i(c)p Fq(\))45 b(runs)c Fm(G)p Fq(\(1)2190 3071 y Fk(k)2234 3104 y Fq(\))i(four)g(times)f(with)g(four)g(indep)s (enden)m(tly)150 3217 y(generated)k(random)d(tap)s(es.)84 b(If)44 b(on)h(either)f(attempt,)49 b(\()p Fm(P)13 b(K)q(;)i(s)p Fq(\))46 b(is)e(the)h(output)f(from)g Fm(G)p Fq(\(1)3471 3184 y Fk(k)3515 3217 y Fq(\),)49 b(the)150 3330 y(adv)m(ersary)31 b(uses)f Fm(s)g Fq(as)h(the)f(secret)i(k)m(ey)f(to)g(decrypt)g Fm(c)f Fq(and)g(returns)g(the)g(decryption.)40 b(Otherwise,)30 b(the)150 3443 y(adv)m(ersary)h(simply)d(outputs)i(0)h(or)g(1)g(with)e (equal)i(probabilit)m(y)-8 b(.)39 b(Since)30 b(there)h(is)e(at)j(least) f(one)g(random)150 3556 y(tap)s(e)i(whic)m(h)f(pro)s(duces)g Fm(P)13 b(K)7 b Fq(,)34 b(the)f(probabilit)m(y)e(that)j(w)m(e)f(pic)m (k)g(a)h(\\go)s(o)s(d")g(random)e(tap)s(e)h(\(one)h(whic)m(h)150 3669 y(giv)m(es)d(us)e(a)i(usable)e(secret)j(k)m(ey\))f(is)f(at)h (least)f(2)1761 3636 y Fl(\000)p Fk(k)r Fh(+2)1950 3669 y Fq(.)40 b(No)m(w,)32 b(let)e Fm(S)2435 3683 y Fk(GM)2569 3669 y Fq(\()p Fm(k)s(;)15 b(A)p Fq(\))31 b(b)s(e)f(the)h(ev)m(en)m(t)h (that)567 3895 y(\()p Fm(P)13 b(K)q(;)i(S)5 b(K)i Fq(\))26 b Fj( )f Fm(G)p Fq(\(1)1264 3857 y Fk(k)1308 3895 y Fq(\);)15 b Fm(b)26 b Fj( )g(f)p Fq(0)p Fm(;)15 b Fq(1)p Fj(g)p Fq(;)g Fm(c)28 b Fj( )d Fm(E)5 b Fq(\()p Fm(P)13 b(K)q(;)i(b)p Fq(\);)g Fm(b)2457 3857 y Fl(0)2508 3895 y Fj( )25 b Fm(A)2692 3910 y Fk(k)2735 3895 y Fq(\()p Fm(P)13 b(K)q(;)i(c)p Fq(\))27 b(:)f Fm(b)3150 3857 y Fl(0)3198 3895 y Fq(=)f Fm(b)291 4062 y Fq(and)30 b(let)g Fm(G)670 4077 y Fk(k)744 4062 y Fq(b)s(e)f(the)i(ev)m(en)m(t)h(that)150 4379 y Fm(r)c Fj( )d(f)p Fq(0)p Fm(;)15 b Fq(1)p Fj(g)555 4341 y Fk(k)600 4379 y Fq(;)g(\()p Fm(P)e(K)q(;)i(S)5 b(K)i Fq(\))27 b Fj( )e Fm(G)p Fq(\(1)1338 4341 y Fk(k)1382 4379 y Fm(;)15 b(r)s Fq(\);)g Fm(r)1582 4393 y Fh(0)1622 4379 y Fm(;)g(r)1703 4393 y Fh(1)1743 4379 y Fm(;)g(r)1824 4393 y Fh(2)1864 4379 y Fm(;)g(r)1945 4393 y Fh(3)2010 4379 y Fj( )25 b(f)p Fq(0)p Fm(;)15 b Fq(1)p Fj(g)2346 4341 y Fk(k)2416 4379 y Fq(:)26 b Fj(9)p Fm(i)f Fj(2)g Fq([0)p Fm(;)15 b Fq(3])22 b(\()p Fm(P)13 b(K)q(;)i(S)5 b(K)i Fq(\))27 b(=)d Fm(G)p Fq(\(1)3539 4341 y Fk(k)3583 4379 y Fm(;)15 b(r)3664 4393 y Fk(i)3693 4379 y Fq(\))p Fm(;)291 4583 y Fq(that)37 b(is,)h(that)f(w)m(e)g(guess)g(an)g Fm(r)1381 4597 y Fk(i)1445 4583 y Fq(whic)m(h)f(giv)m(es)h(us)f(the)g (righ)m(t)h(public)d(k)m(ey)j(and)f(secret)i(k)m(ey)-8 b(.)61 b(Note)150 4696 y(that)30 b(the)f(probabilit)m(y)e(that)i(w)m(e) h(get)g(the)f(righ)m(t)g(public)d(k)m(ey)k(but)f(a)g(di\013eren)m(t)g (secret)h(k)m(ey)f(that)h(do)s(esn't)150 4809 y(decrypt)g(m)m(ust)g(b)s (e)g(0,)h(b)m(y)f(the)h(cryptosystem's)g(faithfulness.)38 b(Th)m(us,)1889 5416 y(6-1)p eop %%Page: 2 2 2 1 bop 481 363 a Fm(P)13 b(r)s Fq([)p Fm(S)677 377 y Fk(GM)811 363 y Fq(\()p Fm(k)s(;)i(A)p Fq(\)])84 b Fj(\025)f Fm(P)13 b(r)s Fq([)p Fm(S)1498 377 y Fk(GM)1632 363 y Fq(\()p Fm(k)s(;)i(A)p Fq(\))p Fj(j)p Fm(G)1956 378 y Fk(k)2001 363 y Fq(])p Fm(P)e(r)s Fq([)p Fm(G)2237 378 y Fk(k)2280 363 y Fq(])20 b(+)g Fm(P)13 b(r)s Fq([)p Fm(S)2612 377 y Fk(GM)2746 363 y Fq(\()p Fm(k)s(;)i(A)p Fq(\))p Fj(j)p 2999 290 115 4 v Fm(G)3070 378 y Fk(k)3114 363 y Fq(])p Fm(P)e(r)s Fq([)p 3279 290 V Fm(G)3350 378 y Fk(k)3394 363 y Fq(])1148 501 y(=)83 b(2)1347 463 y Fl(\000)p Fk(k)r Fh(+2)1556 501 y Fq(+)19 b(\(1)p Fm(=)p Fq(2\)\(1)k Fj(\000)d Fq(2)2090 463 y Fl(\000)p Fk(k)r Fh(+2)2279 501 y Fq(\))1148 638 y(=)83 b(2)1347 601 y Fl(\000)p Fk(k)r Fh(+2)1556 638 y Fq(+)19 b(1)p Fm(=)p Fq(2)j Fj(\000)e Fq(2)1939 601 y Fl(\000)p Fk(k)r Fh(+1)1148 776 y Fq(=)83 b(1)p Fm(=)p Fq(2)22 b(+)e(2)1595 739 y Fl(\000)p Fk(k)r Fh(+1)1148 914 y Fj(\025)83 b Fq(1)p Fm(=)p Fq(2)22 b(+)e(2)1595 876 y Fl(\000)p Fk(k)291 1118 y Fq(Th)m(us,)28 b(this)e(adv)m(ersary)i(defeats)g(the)g(2)1631 1085 y Fl(\000)p Fk(k)1757 1118 y Fq(securit)m(y)f(of)h(this)f (cryptosystem,)i(since)e(this)f(attac)m(k)k(has)150 1231 y(to)s(o)h(high)e(a)i(probabilit)m(y)d(of)i(success)h(at)g Fo(every)f Fm(k)s Fq(.)291 1344 y(No)m(w)i(it)e(is)g(imp)s(ortan)m(t)f (to)j(note)f(that)h(w)m(e)f(had)f(to)h(assume)g(that)g Fm(G)p Fq(\(1)2708 1311 y Fk(k)2752 1344 y Fq(\))g(used)f(only)g Fm(k)k Fq(bits)29 b(of)i(ran-)150 1457 y(domness.)39 b(Actually)-8 b(,)29 b(a)g(similar)d(argumen)m(t)j(w)m(orks)f(if)f Fm(G)p Fq(\(1)2198 1424 y Fk(k)2242 1457 y Fq(\))i(uses)f Fm(k)k Fq(bits)27 b(plus)f(or)j(min)m(us)e(a)h(constan)m(t,)150 1570 y(but)j(if)g(it)h(uses)g(more)g(bits,)f(this)g(argumen)m(t)i(only) e(pro)m(v)m(es)i(that)f(no)g(2)2567 1537 y Fl(\000)p Fk(k)2665 1570 y Fq(-secure)h(cryptosystems)f(exist)150 1683 y(where)26 b Fm(G)p Fq(\(1)560 1650 y Fk(k)604 1683 y Fq(\))g(uses)g Fm(k)15 b Fq(+)d Fm(O)s Fq(\(1\))27 b(bits)e(of)h(randomness.)38 b(As)26 b(it)g(turns)f(out,)j(w)m(e)e(can) h(pro)m(v)m(e)g(that)f(if)g(no)g(2)3622 1650 y Fl(\000)p Fk(k)3720 1683 y Fq(-)150 1796 y(secure)34 b(cryptosystems)g(exist)f (where)g Fm(G)p Fq(\(1)1658 1763 y Fk(k)1702 1796 y Fq(\))h(uses)f Fm(k)26 b Fq(+)c Fm(O)s Fq(\(1\))34 b(bits)f(of)h(randomness,)f(y)m(et) i(GM-secure)150 1909 y(cryptosystems)i(exist,)h(then)e(no)g(2)1387 1876 y Fl(\000)p Fk(k)1485 1909 y Fq(-secure)h(cryptosystems)f(exist,)i (regardless)e(of)g(the)h(n)m(um)m(b)s(er)e(of)150 2022 y(bits)27 b(of)i(randomness)e(used)g(in)g Fm(G)p Fq(.)40 b(W)-8 b(e)30 b(ma)m(y)f(revisit)e(this)g(problem)f(later)j(in)d(the)j (semester,)g(to)g(handle)150 2134 y(this)g(issue.)291 2347 y Fr(Common)39 b(errors:)51 b Fq(It)35 b(is)f(not)h(a)h (su\016cien)m(t)f(argumen)m(t)g(to)h(sa)m(y)g(something)f(along)g(the)g (lines)f(of)150 2460 y(\\w)m(ell,)26 b Fm(f)10 b Fq(\()p Fm(k)s Fq(\))25 b(ma)m(y)h(b)s(e)e(negligible)f(but)h(greater)j(than)d (2)2026 2427 y Fl(\000)p Fk(k)2124 2460 y Fq(,)j(for)e(example)f Fm(f)10 b Fq(\()p Fm(k)s Fq(\))26 b(=)f(1)p Fm(:)p Fq(5)3072 2427 y Fk(k)3116 2460 y Fq(,)h(so)f(2)3318 2427 y Fl(\000)p Fk(k)3441 2460 y Fq(securit)m(y)150 2573 y(do)s(es)e(not)h(imply)e (GM-securit)m(y)-8 b(.")40 b(The)23 b(problem)f(with)g(this)h(is)f (that)j(the)f(de\014nition)d(sa)m(ys)j(that)g(for)g(eac)m(h)150 2686 y(adv)m(ersary)29 b Fm(A)g Fq(there)h(m)m(ust)e(b)s(e)h(a)g (function)f Fm(f)38 b Fq(whic)m(h)28 b(is)g(negligible)e(suc)m(h)j (that)h(the)f(condition)f(applies.)150 2799 y(W)-8 b(ell,)41 b(this)e(argumen)m(t)g(w)m(ould)f(b)s(e)h(su\016cien)m(t)g(if)f(w)m(e)i (could)e(argue)i(that)f(there)h(is,)h(no)e(matter)h(what)150 2912 y(\()p Fm(G;)15 b(E)5 b(;)15 b(D)s Fq(\))36 b(w)m(e)f(c)m(ho)s (ose,)i(some)e(adv)m(ersary)f Fm(A)h Fq(where)e(ALL)i(functions)d Fm(f)44 b Fq(that)35 b(satisfy)f(the)g(condition)150 3024 y(with)e(that)h Fm(A)g Fq(are)h(suc)m(h)f(that)g Fm(f)10 b Fq(\()p Fm(k)s Fq(\))30 b Fm(>)f Fq(2)1572 2991 y Fl(\000)p Fk(k)1703 3024 y Fq(for)k(su\016cien)m(tly)f(large)h Fm(k)s Fq(.)49 b(This)31 b(is)h(essen)m(tially)g(what)h(the)150 3137 y(pro)s(of)d(ab)s(o)m(v)m(e)h(do)s(es.)291 3350 y Fr(\(b\))f Fo(A)n(lmost-al)5 b(l-p)-5 b(ar)g(ameter)36 b(se)-5 b(curity.)40 b Fq(Do)32 b Fo(al)5 b(l)30 b Fq(su\016cien)m(tly) g(large)g Fm(k)j Fq(ha)m(v)m(e)f(to)f(w)m(ork?)150 3562 y Fj(8)25 b(f)p Fm(A)339 3576 y Fk(n)386 3562 y Fj(g)h(2)f Fm(P)13 b(P)g(T)38 b Fj(9)p Fm(f)c Fj(2)25 b Fm(neg)s(;)15 b(K)33 b Fj(2)25 b Fi(N)43 b Fq(suc)m(h)30 b(that)h Fj(8)p Fm(k)c(>)e(K)37 b Fq(suc)m(h)31 b(that)f Fm(k)f Fj(6)p Fq(=)c(2)2823 3529 y Fk(i)2882 3562 y Fq(for)30 b(an)m(y)h Fm(i;)355 3675 y Fq(Pr)o([\()p Fm(P)13 b(K)q(;)i(S)5 b(K)i Fq(\))27 b Fj( )e Fm(G)p Fq(\(1)1175 3642 y Fk(k)1219 3675 y Fq(\);)15 b Fm(b)26 b Fj( )f(f)p Fq(0)p Fm(;)15 b Fq(1)p Fj(g)p Fq(;)355 3788 y Fm(c)25 b Fj( )g Fm(E)5 b Fq(\()p Fm(P)13 b(K)q(;)i(b)p Fq(\);)g Fm(b)984 3755 y Fl(0)1035 3788 y Fj( )26 b Fm(A)1220 3803 y Fk(k)1262 3788 y Fq(\()p Fm(P)13 b(K)q(;)i(c)p Fq(\))28 b(:)d Fm(b)1677 3755 y Fl(0)1726 3788 y Fq(=)g Fm(b)p Fq(])g Fj(\024)g Fq(1)p Fm(=)p Fq(2)d(+)e Fm(f)10 b Fq(\()p Fm(k)s Fq(\))291 4100 y Fr(Solution:)47 b Fq(This)31 b(de\014nition)g(is)h Fr(to)s(o)38 b(w)m(eak.)48 b Fq(Supp)s(ose)32 b(\()p Fm(G;)15 b(E)5 b(;)15 b(D)s Fq(\))35 b(is)d(a)h(GM-secure)i(cryptosys-) 150 4213 y(tem,)c(and)f(de\014ne)f(\()p Fm(G)902 4180 y Fl(0)927 4213 y Fm(;)15 b(E)1039 4180 y Fl(0)1063 4213 y Fm(;)g(D)1181 4180 y Fl(0)1204 4213 y Fq(\))31 b(as)f(follo)m(ws:)286 4426 y Fj(\017)46 b Fm(G)448 4393 y Fl(0)472 4426 y Fq(\(1)552 4393 y Fk(k)596 4426 y Fq(\))25 b(=)g Fm(G)p Fq(\(1)903 4393 y Fk(k)947 4426 y Fq(\))31 b(if)e Fm(k)34 b Fq(is)29 b(not)i(a)f(p)s(o)m(w)m(er)h(of)f(2.)286 4614 y Fj(\017)46 b Fm(G)448 4581 y Fl(0)472 4614 y Fq(\(1)552 4581 y Fk(k)596 4614 y Fq(\))25 b(=)g(\(0)p Fm(;)15 b Fq(0\))32 b(if)e Fm(k)j Fq(is)d(a)g(p)s(o)m(w)m(er)h(of)f(2.)286 4801 y Fj(\017)46 b Fm(E)449 4768 y Fl(0)473 4801 y Fq(\(0)p Fm(;)15 b(m)p Fq(\))27 b(=)e(0)20 b Fj(\016)h Fm(m)p Fq(,)30 b(and)g(for)g Fm(P)13 b(K)32 b Fj(6)p Fq(=)25 b(0,)31 b Fm(E)1862 4768 y Fl(0)1886 4801 y Fq(\()p Fm(P)13 b(K)q(;)i(m)p Fq(\))27 b(=)d(1)d Fj(\016)g Fm(E)5 b Fq(\()p Fm(P)13 b(K)q(;)i(m)p Fq(\).)286 4989 y Fj(\017)46 b Fm(D)455 4956 y Fl(0)478 4989 y Fq(\()p Fm(S)5 b(K)q(;)15 b Fq(0)22 b Fj(\016)f Fm(c)p Fq(\))26 b(=)f Fm(c)p Fq(,)31 b(and)f Fm(D)1371 4956 y Fl(0)1394 4989 y Fq(\()p Fm(S)5 b(K)q(;)15 b Fq(1)22 b Fj(\016)e Fm(c)p Fq(\))26 b(=)f Fm(D)s Fq(\()p Fm(S)5 b(K)q(;)15 b(c)p Fq(\))1889 5416 y(6-2)p eop %%Page: 3 3 3 2 bop 291 159 a Fq(It)29 b(is)f(clear)g(that)i(this)d(system)i(is)f (faithful.)38 b(It)28 b(is)g(also)h(clear)f(that)i(this)d(cryptosystem) i(is)f(not)h(GM-)150 271 y(secure.)41 b(The)29 b(adv)m(ersary)h(that)g (defeats)h(it)e(is)g(one)h(that,)h(for)e Fm(k)k Fq(not)d(a)g(p)s(o)m(w) m(er)g(of)g(2)g(just)g(guesses)g(0)g(or)g(1)150 384 y(with)c(equal)h (lik)m(eliho)s(o)s(d,)e(and)i(for)g Fm(k)k Fq(a)c(p)s(o)m(w)m(er)h(of)f (2)h(just)f(outputs)g Fm(c)h Fq(whenev)m(er)f(0)14 b Fj(\016)g Fm(c)29 b Fq(is)d(the)i(ciphertext.)150 497 y(Ho)m(w)m(ev)m(er,)46 b(\()p Fm(G)661 464 y Fl(0)685 497 y Fm(;)15 b(E)797 464 y Fl(0)821 497 y Fm(;)g(D)939 464 y Fl(0)963 497 y Fq(\))41 b(is)g(almost-all-parameter-secure.)74 b(If)40 b(there)i(is)e(a)i(non)m(uniform)d(adv)m(ersary)150 610 y Fj(f)p Fm(A)263 625 y Fk(k)306 610 y Fj(g)32 b Fq(where)e(for)h(in\014nitely)d(man)m(y)j Fm(k)k Fq(other)c(than)g(p)s (o)m(w)m(ers)g(of)g(2,)h Fj(f)p Fm(A)2570 625 y Fk(k)2613 610 y Fj(g)g Fq(is)e(able)h(to)g(distinguish,)d(then)150 723 y Fj(f)p Fm(A)263 738 y Fk(k)306 723 y Fj(g)36 b Fq(also)e(con)m(tradicts)i(the)f(GM-securit)m(y)g(of)g(\()p Fm(G;)15 b(E)5 b(;)15 b(D)s Fq(\),)38 b(since)d(for)f(in\014nitely)e (man)m(y)j Fm(k)s Fq(,)h(it)f(is)e(able)150 836 y(to)f(distinguish.)40 b Fj(f)p Fm(A)874 851 y Fk(k)918 836 y Fj(g)31 b Fq(ma)m(y)h(fail)e (utterly)h(when)g Fm(k)j Fq(is)d(a)g(p)s(o)m(w)m(er)h(of)f(2,)i(but)d (that)i(isn't)f(necessary)h(-)f(w)m(e)150 949 y(only)j(need)h(in\014n)m (tely)e(man)m(y)i Fm(k)j Fq(for)d(whic)m(h)f Fj(f)p Fm(A)1788 964 y Fk(k)1831 949 y Fj(g)h Fq(succeeds.)55 b(Th)m(us,)36 b(almost-all-parameter-securit)m(y)150 1062 y(do)s(es)30 b(not)h(imply)d(GM-securit)m(y)-8 b(.)291 1257 y(On)30 b(the)i(other)f(hand,)f(\()p Fm(G;)15 b(E)5 b(;)15 b(D)s Fq(\))33 b(m)m(ust)e(b)s(e)g(almost-all-parameter-secure,)h(since)e (the)h(condition)150 1370 y(required)38 b(for)i (almost-all-parameter-securit)m(y)g(is)g(a)g(sub)s(condition)d(of)j (GM-securit)m(y)-8 b(.)71 b(Th)m(us,)42 b(GM-)150 1483 y(securit)m(y)30 b(implies)e(almost-all-parameter-securit)m(y)-8 b(.)291 1679 y Fr(Common)33 b(errors:)41 b Fq(Note)32 b(that)f(just)f(b)s(ecause)g(AAP)h(securit)m(y)f(do)s(esn't)g(sp)s (ecify)f(what)i(happ)s(ens)150 1792 y(when)42 b Fm(k)49 b Fq(=)d(2)658 1759 y Fk(i)730 1792 y Fq(this)c(do)s(esn't)h(mean)g (that)g Fo(any)h Fq(sc)m(hemes)f(are)h(necessarily)e(not)h(GM-secure.) 79 b(The)150 1904 y(solution)36 b(here)i(giv)m(es)g(an)g(example)f (that)i(pro)m(v)m(es)f(there)g(is)f(a)h(signi\014can)m(t)f(problem.)62 b(Ho)m(w)m(ev)m(er,)42 b(just)150 2017 y(noticing)29 b(that)i(there)g(is)e(that)i(hole)f(left)g(b)m(y)h(the)f(de\014nition)e (do)s(esn't)i(su\016ce.)291 2213 y Fr(\(c\))d Fo(A)n(l)5 b(l-p)-5 b(ar)g(ameter)32 b(se)-5 b(curity.)39 b Fq(Do)28 b(w)m(e)g(need)f(the)g(assertion)g(that)h Fm(k)i Fq(has)d(to)h(b)s(e)f (su\016cien)m(tly)f(large?)150 2400 y Fj(8)f(f)p Fm(A)339 2414 y Fk(n)386 2400 y Fj(g)h(2)f Fm(P)13 b(P)g(T)38 b Fj(9)p Fm(f)c Fj(2)25 b Fm(neg)34 b Fq(suc)m(h)c(that)h Fj(8)p Fm(k)d Fj(2)c Fi(N)7 b Fm(;)355 2513 y Fq(Pr)o([\()p Fm(P)13 b(K)q(;)i(S)5 b(K)i Fq(\))27 b Fj( )e Fm(G)p Fq(\(1)1175 2480 y Fk(k)1219 2513 y Fq(\);)15 b Fm(b)26 b Fj( )f(f)p Fq(0)p Fm(;)15 b Fq(1)p Fj(g)p Fq(;)355 2626 y Fm(c)25 b Fj( )g Fm(E)5 b Fq(\()p Fm(P)13 b(K)q(;)i(b)p Fq(\);)g Fm(b)984 2593 y Fl(0)1035 2626 y Fj( )26 b Fm(A)1220 2641 y Fk(k)1262 2626 y Fq(\()p Fm(P)13 b(K)q(;)i(c)p Fq(\))28 b(:)d Fm(b)1677 2593 y Fl(0)1726 2626 y Fq(=)g Fm(b)p Fq(])g Fj(\024)g Fq(1)p Fm(=)p Fq(2)d(+)e Fm(f)10 b Fq(\()p Fm(k)s Fq(\))291 2895 y Fr(Solution:)47 b Fq(This)31 b(de\014nition)g(is)h Fr(equiv)-6 b(alen)m(t)34 b Fq(to)g(GM-securit)m (y)-8 b(.)50 b(Supp)s(ose)32 b(\()p Fm(G;)15 b(E)5 b(;)15 b(D)s Fq(\))35 b(is)d(a)i(GM-)150 3008 y(secure)d(cryptosystem.)41 b(Let)31 b Fm(A)g Fq(b)s(e)f(an)g(arbitrary)g(adv)m(ersary)-8 b(,)31 b(and)f(let)h Fm(f)39 b Fq(b)s(e)30 b(the)h(negligible)d (function)150 3121 y(suc)m(h)42 b(that)h(for)f(all)f Fm(k)48 b(>)c(K)7 b Fq(,)45 b(the)e(required)d(probabilit)m(y)g(holds.) 74 b(Let)43 b Fm(f)2781 3088 y Fl(0)2846 3121 y Fq(b)s(e)e(the)i (function)d(where)150 3234 y Fm(f)205 3201 y Fl(0)228 3234 y Fq(\()p Fm(k)s Fq(\))27 b(=)f Fm(f)10 b Fq(\()p Fm(k)s Fq(\))31 b(for)g(all)f Fm(k)g(>)c(K)7 b Fq(,)31 b(and)f Fm(f)1491 3201 y Fl(0)1514 3234 y Fq(\()p Fm(k)s Fq(\))d(=)f(1)p Fm(=)p Fq(2)33 b(for)e(all)e Fm(k)h Fj(\024)c Fm(K)7 b Fq(.)43 b(No)m(w,)32 b(if)e(w)m(e)h(can)h(pro)m(v)m(e)g(that)f Fm(f)3635 3201 y Fl(0)3689 3234 y Fq(is)150 3347 y(negligible,)g(then)h (w)m(e)h(are)g(done,)g(since)f(it)g(is)g(clear)g(that)h(for)g(all)e Fm(k)s Fq(,)j(the)e(required)f(probabilit)m(y)f(is)i(less)150 3460 y(than)e(or)g(equal)g(to)h(1)p Fm(=)p Fq(2)22 b(+)e Fm(f)1127 3427 y Fl(0)1150 3460 y Fq(\()p Fm(k)s Fq(\).)291 3573 y(Let)39 b Fm(c)g(>)f Fq(0)h(b)s(e)f(an)g(arbitrary)f(in)m(teger.) 66 b(W)-8 b(e)39 b(kno)m(w)g(there)f(is)g(some)g Fm(N)2813 3587 y Fk(c)2887 3573 y Fq(suc)m(h)g(that)h(for)f(all)f Fm(n)h(>)150 3686 y(N)223 3700 y Fk(c)258 3686 y Fm(;)15 b(f)10 b Fq(\()p Fm(n)p Fq(\))25 b Fm(<)g Fq(1)p Fm(=n)744 3653 y Fk(c)779 3686 y Fq(.)41 b(No)m(w)31 b(I)e(claim)h(it)f(is)h(ob)m (vious)f(that)i(for)f(all)f Fm(n)c(>)g(N)2588 3653 y Fl(0)2578 3708 y Fk(c)2638 3686 y Fq(=max\()p Fm(N)2986 3700 y Fk(c)3021 3686 y Fm(;)15 b(K)7 b Fq(\))p Fm(;)15 b(f)3275 3653 y Fl(0)3298 3686 y Fq(\()p Fm(n)p Fq(\))26 b Fm(<)f Fq(1)p Fm(=n)3690 3653 y Fk(c)3725 3686 y Fq(,)150 3799 y(since)36 b(for)f(all)h(suc)m(h)f Fm(n)p Fq(,)j Fm(f)1039 3766 y Fl(0)1061 3799 y Fq(\()p Fm(n)p Fq(\))e(=)e Fm(f)10 b Fq(\()p Fm(n)p Fq(\))35 b Fm(<)f Fq(1)p Fm(=n)1792 3766 y Fk(c)1828 3799 y Fq(.)58 b(Th)m(us,)37 b(in)d(place)j(of)f Fm(N)2706 3813 y Fk(c)2741 3799 y Fq(,)i(the)e(p)s(oin)m(t)f(after)i (whic)m(h)e Fm(f)150 3912 y Fq(is)c(less)h(than)g(1)p Fm(=n)772 3879 y Fk(c)807 3912 y Fq(,)h(w)m(e)g(ha)m(v)m(e)h Fm(N)1296 3879 y Fl(0)1286 3934 y Fk(c)1321 3912 y Fq(.)46 b(Th)m(us,)32 b Fm(f)1705 3879 y Fl(0)1760 3912 y Fq(is)g(negligible,)f (so)h(w)m(e)h(ha)m(v)m(e)h(that)f(GM-securit)m(y)g(implies)150 4025 y(all-parameter-securit)m(y)-8 b(.)291 4220 y(T)g(o)33 b(sho)m(w)f(that)i(all-parameter-securit)m(y)e(implies)e(GM-securit)m (y)j(requires)e(only)h(the)g(observ)-5 b(ation)150 4333 y(that)42 b(if)e(w)m(e)h(c)m(ho)s(ose)h Fo(any)g Fm(K)49 b Fj(2)43 b Fi(N)53 b Fq(then)41 b(for)g(all)f Fm(k)46 b(>)c(K)7 b Fq(,)44 b(the)d(required)e(probabilit)m(y)g(is)h(less)g (than)150 4446 y(1)p Fm(=)p Fq(2)19 b(+)f Fm(f)10 b Fq(\()p Fm(k)s Fq(\),)29 b(since)g(this)f(is)g(true)h(for)g(all)f Fm(k)33 b Fq(generally)-8 b(.)40 b(In)28 b(particular,)g(this)g(is)g (true)h(for)g Fm(K)j Fq(=)25 b(0.)41 b(Th)m(us,)150 4559 y(GM-securit)m(y)31 b(is)f(equiv)-5 b(alen)m(t)29 b(to)i (all-parameter-securit)m(y)-8 b(.)291 4754 y Fr(\(d\))21 b Fo(A)n(l)5 b(l-message)24 b(se)-5 b(curity.)38 b Fq(Do)22 b(w)m(e)g(need)g(to)g(pic)m(k)f(the)h(message)h(randomly)d(during)f (the)j(proto)s(col?)150 4941 y Fj(8)j(f)p Fm(A)339 4955 y Fk(n)386 4941 y Fj(g)h(2)f Fm(P)13 b(P)g(T)38 b Fj(9)p Fm(f)c Fj(2)25 b Fm(neg)s(;)15 b(K)33 b Fj(2)25 b Fi(N)63 b Fj(8)p Fm(k)28 b(>)d(K)32 b Fj(8)p Fm(b)24 b Fj(2)h(f)p Fq(0)p Fm(;)15 b Fq(1)p Fj(g)p Fm(;)355 5054 y(P)e(r)s Fq([\()p Fm(P)g(K)q(;)i(S)5 b(K)i Fq(\))26 b Fj( )f Fm(G)p Fq(\(1)1192 5021 y Fk(k)1236 5054 y Fq(\);)355 5167 y Fm(c)g Fj( )g Fm(E)5 b Fq(\()p Fm(P)13 b(K)q(;)i(b)p Fq(\);)g Fm(b)984 5134 y Fl(0)1035 5167 y Fj( )26 b Fm(A)1220 5182 y Fk(k)1262 5167 y Fq(\()p Fm(P)13 b(K)q(;)i(c)p Fq(\))28 b(:)d Fm(b)1677 5134 y Fl(0)1726 5167 y Fq(=)g Fm(b)p Fq(])g Fj(\024)g Fq(1)p Fm(=)p Fq(2)d(+)e Fm(f)10 b Fq(\()p Fm(k)s Fq(\))1889 5416 y(6-3)p eop %%Page: 4 4 4 3 bop 291 159 a Fr(Solution:)49 b Fq(This)33 b(de\014nition)f(is)h Fr(unattainable.)52 b Fq(Consider)33 b(the)h(adv)m(ersary)h Fm(A)3144 173 y Fk(n)3225 159 y Fq(whic)m(h)e(alw)m(a)m(ys)150 271 y(outputs)k(0.)62 b(F)-8 b(or)38 b(this)e(adv)m(ersary)-8 b(,)40 b(if)c Fm(b)h Fq(=)f(0,)k(the)e(probabilit)m(y)c(w)m(e're)39 b(in)m(terested)e(in)f(is)g(1,)k(whic)m(h)c(is)150 384 y(greater)c(than)e(1)p Fm(=)p Fq(2)h(plus)e(an)m(y)i(negligible)d (function.)291 497 y(This)23 b(simple)f(pro)s(of)i(disguises)f(the)i (real)f(di\016cult)m(y)f(of)i(this)e(problem.)37 b(It)25 b(is)f(clear)g(that)i(all-message)150 610 y(securit)m(y)35 b(implies)e(GM-securit)m(y)-8 b(.)57 b(Ho)m(w)m(ev)m(er,)39 b(trying)c(to)h(pro)m(v)m(e)g(the)g(other)g(direction)e(w)m(e)i(run)e (in)m(to)i(a)150 723 y(pitfall.)i(W)-8 b(e)30 b(kno)m(w)f(that)g(if)e (a)j(system)e(is)g(secure)h(then)f(for)h(all)e Fm(A)p Fq(,)j(there)f(is)e(an)i Fm(f)34 b Fj(2)25 b Fm(neg)32 b Fq(suc)m(h)d(that)g Fj(8)p Fm(k)s Fq(,)150 836 y(Pr)o([)p Fm(S)328 850 y Fk(GM)462 836 y Fq(\()p Fm(k)s(;)15 b(A)p Fq(\))p Fj(j)p Fm(b)34 b Fq(=)d(0])p Fm(=)p Fq(2)24 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y(system)c(is)e(no)m(w)h(sho)m(wn)g(to)h(b)s(e)f(not)h (all-message-secure.)291 1495 y Fr(\(e\))36 b Fo(Ciphertext)k(game)f (se)-5 b(curity.)61 b Fq(Can)36 b(w)m(e)i(de\014ne)e(securit)m(y)h (di\013eren)m(tly:)52 b(sa)m(y)38 b(a)f(cryptosystem)150 1608 y(is)44 b(secure)g(if)g(the)g(c)m(hance)i(an)e(adv)m(ersary)h (determines)f(a)h(message)g(giv)m(en)g(its)f(encryption)f(is)h(only)150 1721 y(negligibly)33 b(b)s(etter)k(than)e(the)i(probabilit)m(y)c(that)k (the)f(adv)m(ersary)h(determines)e(the)h(message)h(giv)m(en)g(a)150 1834 y(random)30 b(encryption)f(of)h(some)h(kind?)150 2039 y Fj(8)25 b(f)p Fm(A)339 2053 y Fk(n)386 2039 y Fj(g)h(2)f Fm(P)13 b(P)g(T)38 b Fj(9)p Fm(f)c Fj(2)25 b Fm(neg)s(;)15 b(K)33 b Fj(2)25 b Fi(N)63 b Fj(8)p Fm(k)28 b(>)d(K)q(;)355 2152 y Fq(Pr)o([\()p Fm(P)13 b(K)q(;)i(S)5 b(K)i Fq(\))27 b Fj( )e Fm(G)p Fq(\(1)1175 2119 y Fk(k)1219 2152 y Fq(\);)15 b Fm(b)1333 2166 y Fh(0)1373 2152 y Fm(;)g(b)1452 2166 y Fh(1)1517 2152 y Fj( )26 b(f)p Fq(0)p Fm(;)15 b Fq(1)p Fj(g)355 2265 y Fm(c)25 b Fj( )g Fm(E)5 b Fq(\()p Fm(P)13 b(K)q(;)i(b)870 2279 y Fh(0)912 2265 y Fq(\);)g Fm(b)1026 2232 y Fl(0)1075 2265 y Fj( )25 b Fm(A)1259 2280 y Fk(k)1302 2265 y Fq(\()p Fm(P)13 b(K)q(;)i(c)p Fq(\))27 b(:)f Fm(b)1717 2232 y Fl(0)1765 2265 y Fq(=)f Fm(b)1900 2279 y Fh(0)1940 2265 y Fq(])p Fj(\000)355 2378 y Fq(Pr)o([\()p Fm(P)13 b(K)q(;)i(S)5 b(K)i Fq(\))27 b Fj( )e Fm(G)p Fq(\(1)1175 2345 y Fk(k)1219 2378 y Fq(\);)15 b Fm(b)1333 2392 y Fh(0)1373 2378 y Fm(;)g(b)1452 2392 y Fh(1)1517 2378 y Fj( )26 b(f)p Fq(0)p Fm(;)15 b Fq(1)p Fj(g)355 2491 y Fm(c)25 b Fj( )g Fm(E)5 b Fq(\()p Fm(P)13 b(K)q(;)i(b)870 2505 y Fh(1)912 2491 y Fq(\);)g Fm(b)1026 2458 y Fl(0)1075 2491 y Fj( )25 b Fm(A)1259 2506 y Fk(k)1302 2491 y Fq(\()p Fm(P)13 b(K)q(;)i(c)p Fq(\))27 b(:)f Fm(b)1717 2458 y Fl(0)1765 2491 y Fq(=)f Fm(b)1900 2505 y Fh(0)1940 2491 y Fq(])g Fj(\024)g Fm(f)10 b Fq(\()p Fm(k)s Fq(\))291 2791 y Fr(Solution:)41 b Fq(This)29 b(de\014nition)f(is)h Fr(equiv)-6 b(alen)m(t)31 b 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Fq(;)i(the)g(probabilit)m(y)c(that)k Fm(b)3235 3425 y Fh(0)3302 3411 y Fq(=)28 b Fm(b)3440 3378 y Fl(0)3495 3411 y Fq(is)j(1)p Fm(=)p Fq(2.)150 3524 y(Th)m(us,)f(w)m(e)g(ha)m(v)m(e)355 3729 y(Pr)o([\()p Fm(P)13 b(K)q(;)i(S)5 b(K)i Fq(\))27 b Fj( )e Fm(G)p Fq(\(1)1175 3696 y Fk(k)1219 3729 y Fq(\);)15 b Fm(b)1333 3743 y Fh(0)1373 3729 y Fm(;)g(b)1452 3743 y Fh(1)1517 3729 y Fj( )26 b(f)p Fq(0)p Fm(;)15 b Fq(1)p Fj(g)355 3842 y Fm(c)25 b Fj( )g Fm(E)5 b Fq(\()p Fm(P)13 b(K)q(;)i(b)870 3856 y Fh(0)912 3842 y Fq(\);)g Fm(b)1026 3809 y Fl(0)1075 3842 y Fj( )25 b Fm(A)1259 3857 y Fk(k)1302 3842 y Fq(\()p Fm(P)13 b(K)q(;)i(c)p Fq(\))27 b(:)f Fm(b)1717 3809 y Fl(0)1765 3842 y Fq(=)f Fm(b)1900 3856 y Fh(0)1940 3842 y Fq(])p Fj(\000)355 3955 y Fq(Pr)o([\()p Fm(P)13 b(K)q(;)i(S)5 b(K)i Fq(\))27 b Fj( )e Fm(G)p Fq(\(1)1175 3922 y Fk(k)1219 3955 y Fq(\);)15 b Fm(b)1333 3969 y Fh(0)1373 3955 y Fm(;)g(b)1452 3969 y Fh(1)1517 3955 y Fj( )26 b(f)p Fq(0)p Fm(;)15 b Fq(1)p Fj(g)355 4068 y Fm(c)25 b Fj( )g Fm(E)5 b Fq(\()p Fm(P)13 b(K)q(;)i(b)870 4082 y Fh(1)912 4068 y Fq(\);)g Fm(b)1026 4035 y Fl(0)1075 4068 y Fj( )25 b Fm(A)1259 4083 y Fk(k)1302 4068 y Fq(\()p Fm(P)13 b(K)q(;)i(c)p Fq(\))27 b(:)f Fm(b)1717 4035 y Fl(0)1765 4068 y Fq(=)f Fm(b)1900 4082 y Fh(0)1940 4068 y Fq(])g(=)355 4181 y(Pr)o([\()p Fm(P)13 b(K)q(;)i(S)5 b(K)i Fq(\))27 b Fj( )e Fm(G)p Fq(\(1)1175 4148 y Fk(k)1219 4181 y Fq(\);)15 b Fm(b)1333 4195 y Fh(0)1373 4181 y Fm(;)g(b)1452 4195 y Fh(1)1517 4181 y Fj( )26 b(f)p Fq(0)p Fm(;)15 b Fq(1)p Fj(g)355 4294 y Fm(c)25 b Fj( )g Fm(E)5 b Fq(\()p Fm(P)13 b(K)q(;)i(b)870 4308 y Fh(0)912 4294 y Fq(\);)g Fm(b)1026 4261 y Fl(0)1075 4294 y Fj( )25 b Fm(A)1259 4309 y Fk(k)1302 4294 y Fq(\()p Fm(P)13 b(K)q(;)i(c)p Fq(\))27 b(:)f Fm(b)1717 4261 y Fl(0)1765 4294 y Fq(=)f Fm(b)1900 4308 y Fh(0)1940 4294 y Fq(])20 b Fj(\000)g Fq(1)p Fm(=)p Fq(2)27 b Fj(\024)e Fm(f)10 b Fq(\()p Fm(k)s Fq(\))291 4499 y(so)355 4704 y(Pr)o([\()p Fm(P)j(K)q(;)i(S)5 b(K)i Fq(\))27 b Fj( )e Fm(G)p Fq(\(1)1175 4671 y Fk(k)1219 4704 y Fq(\);)15 b Fm(b)1333 4718 y Fh(0)1373 4704 y Fm(;)g(b)1452 4718 y Fh(1)1517 4704 y Fj( )26 b(f)p Fq(0)p Fm(;)15 b Fq(1)p Fj(g)355 4817 y Fm(c)25 b Fj( )g Fm(E)5 b Fq(\()p Fm(P)13 b(K)q(;)i(b)870 4831 y Fh(0)912 4817 y Fq(\);)g Fm(b)1026 4784 y Fl(0)1075 4817 y Fj( )25 b Fm(A)1259 4832 y Fk(k)1302 4817 y Fq(\()p Fm(P)13 b(K)q(;)i(c)p Fq(\))27 b(:)f Fm(b)1717 4784 y Fl(0)1765 4817 y Fq(=)f Fm(b)1900 4831 y Fh(0)1940 4817 y Fq(])g Fj(\024)g Fq(1)p Fm(=)p Fq(2)d(+)e Fm(f)10 b Fq(\()p Fm(k)s Fq(\))291 5023 y(whic)m(h)29 b(is)h(just)f(the)i(same) g(as)p 150 5082 1440 4 v 254 5135 a Fg(1)288 5167 y Ff(Here,)26 b(w)n(e)g(assume)g(the)f(adv)n(ersary)g(alw)n(a)n(ys)i(outputs)e (either)h(0)g(or)g(1.)1889 5416 y Fq(6-4)p eop %%Page: 5 5 5 4 bop 355 159 a Fq(Pr)o([\()p Fm(P)13 b(K)q(;)i(S)5 b(K)i Fq(\))27 b Fj( )e Fm(G)p Fq(\(1)1175 126 y Fk(k)1219 159 y Fq(\);)15 b Fm(b)1333 173 y Fh(0)1399 159 y Fj( )25 b(f)p Fq(0)p Fm(;)15 b Fq(1)p Fj(g)355 271 y Fm(c)25 b Fj( )g Fm(E)5 b Fq(\()p Fm(P)13 b(K)q(;)i(b)870 285 y Fh(0)912 271 y Fq(\);)g Fm(b)1026 238 y Fl(0)1075 271 y Fj( )25 b Fm(A)1259 286 y Fk(k)1302 271 y Fq(\()p Fm(P)13 b(K)q(;)i(c)p Fq(\))27 b(:)f Fm(b)1717 238 y Fl(0)1765 271 y Fq(=)f Fm(b)1900 285 y Fh(0)1940 271 y Fq(])g Fj(\024)g Fq(1)p Fm(=)p Fq(2)d(+)e Fm(f)10 b Fq(\()p Fm(k)s Fq(\))291 477 y(since)28 b Fm(b)551 491 y Fh(1)620 477 y Fq(is)g(actually)h(nev)m (er)g(used)f(again)h(after)h(it)e(is)g(generated,)j(and)d(this)g(last)h (expression)f(is)g(the)150 590 y(same)k(as)f(in)f(GM-securit)m(y)-8 b(,)33 b(except)g(with)d(us)g(writing)g Fm(b)2107 604 y Fh(0)2178 590 y Fq(instead)g(of)i Fm(b)p Fq(.)43 b(Th)m(us,)31 b(w)m(e)h(ha)m(v)m(e)h(sho)m(wn)d(that)150 702 y(if)c(the)i(required)d (probabilit)m(y)g(holds)h(for)h(ciphertext)g(game)h(securit)m(y)-8 b(,)28 b(then)f(the)h(probabilit)m(y)c(holds)i(for)150 815 y(GM-securit)m(y)-8 b(.)42 b(W)-8 b(e)31 b(ha)m(v)m(e)h(also)e(sho) 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Fq(Pro)m(v)m(e)40 b(that)f(if)f(for)h(an)m(y)g(0)g Fm(<)g(q)j Fj(\024)d Fq(1)p Fm(=)p Fq(2,)k(the)c(Goldw)m(asser-Micali)e(cryptosystem)j(is)d Fm(q)s Fq(-GM-secure,)150 1981 y(then)30 b(it)g(is)f(GM-secure.)291 2189 y Fr(Solution:)46 b Fq(W)-8 b(e)34 b(will)d(pro)m(v)m(e)i(this)f (b)m(y)h(pro)m(ving)f(the)h(con)m(trap)s(ositiv)m(e.)49 b(Supp)s(ose)31 b(the)i(Goldw)m(asser-)150 2302 y(Micali)23 b(cryptosystem)h(\()p Fm(G;)15 b(E)5 b(;)15 b(D)s Fq(\))27 b(is)c(not)h(GM-secure.)40 b(Then,)24 b(there)g(is)f(an)h(adv)m(ersary) h(that)f(breaks)g(it.)150 2415 y(And,)32 b(b)m(y)g(the)g(reduction)g (giv)m(en)g(in)f(lecture,)h(there)h(is)e(an)h(algorithm)f Fj(f)p Fm(A)2727 2429 y Fk(n)2775 2415 y Fj(g)h Fq(suc)m(h)g(that)h (for)f(in\014nitely)150 2527 y(man)m(y)e Fm(k)s Fq(,)h(there)g(is)e (some)i(p)s(olynomial)d Fm(p)i Fq(suc)m(h)g(that)150 2733 y(Pr)o([)p Fm(p)c Fj( )f Fq(PRIMES)817 2748 y Fk(k)860 2733 y Fq(;)15 b Fm(q)28 b Fj( )e Fq(PRIMES)1443 2748 y Fk(k)1486 2733 y Fq(;)15 b Fm(a)25 b Fj( )g Fm(J)1765 2747 y Fh(1)1805 2733 y Fq(\()p Fm(pq)s Fq(\);)15 b Fm(b)26 b Fj( )f Fm(A)2254 2748 y Fk(k)2297 2733 y Fq(\()p Fm(x;)15 b(pq)s Fq(\))26 b(:)g Fm(b)f Fq(=)g Fm(QC)7 b Fq(\()p Fm(x;)15 b(pq)s Fq(\)])25 b Fm(>)g Fq(1)p Fm(=)p Fq(2)d(+)e(1)p Fm(=p)p Fq(\()p Fm(k)s Fq(\))291 2938 y(That)32 b(is,)f Fj(f)p Fm(A)751 2952 y Fk(n)798 2938 y Fj(g)h Fq(solv)m(es)g(the)g (quadratic)f(residuosit)m(y)f(problem.)43 b(W)-8 b(e)33 b(wish)c(to)k(construct)f(an)f(algo-)150 3051 y(rithm)e(that)i(breaks)f Fm(q)s Fq(-GM)h(securit)m(y)-8 b(.)41 b(W)-8 b(e)31 b(do)g(this)e(as)i (follo)m(ws.)150 3256 y Fm(A)218 3223 y Fl(0)218 3284 y Fk(k)261 3256 y Fq(\(\()p Fm(n;)15 b(x)p Fq(\))p Fm(;)g(c)p Fq(\):)355 3369 y(1.)41 b(Let)31 b Fm(i)f Fq(=)g(1.)355 3482 y(2.)41 b(Select)30 b Fm(y)e Fj( )e Fi(Z)983 3449 y Fl(\003)983 3504 y Fk(n)1025 3482 y Fq(.)355 3595 y(3.)41 b(Select)30 b Fm(b)c Fj( )f(f)p Fq(0)p Fm(;)15 b Fq(1)p Fj(g)p Fq(.)355 3708 y(4.)41 b(Compute)30 b Fm(b)899 3675 y Fl(0)947 3708 y Fq(=)25 b Fm(A)1111 3723 y Fk(k)1154 3708 y Fq(\()p Fm(n;)15 b(cy)1371 3675 y Fh(2)1411 3708 y Fm(x)1463 3675 y Fk(b)1497 3708 y Fq(\).)355 3823 y(5.)41 b(Record)30 b(one)p 942 3749 63 4 v 31 w Fm(b)981 3797 y Fl(0)1024 3823 y Fj(\010)20 b Fm(b)31 b Fq(v)m(ote.)355 3936 y(6.)41 b(If)30 b Fm(i)25 b Fq(=)g Fm(g)s Fq(\()p Fm(k)s Fq(\))32 b(then)e(return)f(whic)m(hev)m(er)h(of)h(0)g(or)f(1)h (has)f(more)g(v)m(otes.)355 4049 y(7.)41 b(Otherwise,)29 b(incremen)m(t)h Fm(i)h Fq(and)e(go)j(to)f(step)f(2.)291 4254 y(It)35 b(should)d(b)s(e)i(clear)h(that)g(if)e Fm(A)1392 4269 y Fk(k)1469 4254 y Fq(is)h(correct)h(more)g(often)g(than)f(it)g (is)g(incorrect,)h(w)m(e)g(will)d(get)k(the)150 4367 y(righ)m(t)c(answ)m(er.)48 b(Then,)33 b(if)e Fm(b)e Fq(=)g(0,)34 b Fm(cy)1426 4334 y Fh(2)1466 4367 y Fm(x)1518 4334 y Fk(b)1585 4367 y Fq(will)c(ha)m(v)m(e)k(the)f(same)g(quadratic)f(c)m (haracter)i(as)f Fm(c)g Fq(-)g(so)g(if)f Fm(c)h Fq(is)150 4480 y(a)g(quadratic)f(residue)f(then)h(the)h(result)e Fm(A)1639 4495 y 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Fg(2)288 5076 y Ff(If)28 b Fe(x)f Ff(is)g(not)h(a)f(quadratic)h (non-residue,)f(the)g(algorithm)h(will)h(b)r(eha)n(v)n(e)e(strangely)-6 b(,)28 b(but)f(w)n(e)g(only)h(care)g(what)f Fe(A)3581 5044 y Fd(0)3631 5076 y Ff(will)150 5167 y(do)f(when)f(its)h(input)f (is)h(v)l(alid!)1889 5416 y Fq(6-5)p eop %%Page: 6 6 6 5 bop 150 159 a Fm(A)218 174 y Fk(k)297 159 y Fq(is)35 b(correct)i(is)e(at)h(least)g(1)p Fm(=)p Fq(2)25 b(+)f Fm(p)p Fq(\()p Fm(k)s Fq(\).)58 b(No)m(w,)38 b(if)d(w)m(e)h(let)g Fm(g)j Fq(gro)m(w)e(fast)f(enough)3013 126 y Fh(3)3052 159 y Fq(,)i(w)m(e)e(can)g(get)h(that)150 271 y(the)e(probabilit)m(y)d (that)j Fm(A)1049 238 y Fl(0)1049 299 y Fk(k)1126 271 y Fq(w)m(orks)f(correctly)h(is)f(arbitrarily)e(close)i(to)i(1.)53 b(That)34 b(is,)h(no)g(matter)g(what)150 384 y Fm(q)h Fq(is,)d(for)g(su\016cien)m(tly)f(large)h Fm(k)s Fq(,)h(the)f (probabilit)m(y)e(that)i Fm(A)2175 351 y Fl(0)2175 412 y Fk(k)2251 384 y Fq(giv)m(es)h(the)f(righ)m(t)g(answ)m(er)g(is)f (greater)i(than)150 497 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y(cryptosystems)f(\()p Fm(G;)15 b(E)5 b(;)15 b(D)s Fq(\))36 b(exists,)d(then)g(I)g(can)g(exhibit)f(a)h(cryptosystem)h (that)f(is)f Fm(q)s Fq(-secure)i(for)f(an)m(y)150 1407 y Fm(q)i(<)338 1371 y Fh(1)p 338 1386 36 4 v 338 1439 a(2)417 1407 y Fq(but)f(not)g(GM-secure.)1205 1374 y Fh(4)1298 1407 y Fq(Let)h(me)f(outline)f(the)h(structure)g(of)g(this)g (argumen)m(t.)52 b(Giv)m(en)34 b(a)h(GM-)150 1520 y(secure)h (cryptosystem,)h(I)e(will)e(sho)m(w)j(ho)m(w)f(to)h(construct)g(one)g (that)g(is)f(not)h(GM-secure,)i(but)c(at)j(the)150 1633 y(same)27 b(time)e(still)f Fm(q)s Fq(-GM)j(secure.)40 b(In)25 b(order)h(to)h(sho)m(w)f(that)g(it)g(is)f Fm(q)s Fq(-GM)i(secure)f(I)g(will)e(assume)i(that)g(it)g(is)150 1746 y(not)g(and)f(arriv)m(e)g(at)i(a)f(statemen)m(t)h(that)f(con)m (tradicts)h(the)e(GM)i(securit)m(y)e(of)h(the)g(original)d (cryptosystem)150 1859 y(\()p Fm(G;)15 b(E)5 b(;)15 b(D)s Fq(\).)46 b(This)30 b(is)g(a)i(rather)f(tric)m(ky)h(pro)s(of)f (outline,)f(so)i(please)f(pause)g(for)g(a)h(second)g(and)f(read)g(this) 150 1972 y(paragraph)c(again)h(to)g(mak)m(e)h(sure)e(y)m(ou)h (understand)d(the)j(direction)e(in)h(whic)m(h)f(I)h(am)h(going)g(to)g (pro)s(ceed)150 2085 y(with)h(this)g(pro)s(of.)291 2198 y(Next,)45 b(I)c(will)d(giv)m(e)k(some)f(in)m(tuition)e(for)i(this)f (argumen)m(t.)73 b(The)40 b(crucial)g(problem)f(with)h Fm(q)s Fq(-GM)150 2310 y(securit)m(y)20 b(is)g(that)h(it)g(only)e(prev) m(en)m(ts)i(adv)m(ersaries)g(whic)m(h)e(are)i(v)m(ery)g(successful.)37 b(Basically)-8 b(,)22 b(an)f(adv)m(ersary)150 2423 y(w)m(ould)g(only)h (violate)h Fm(q)s Fq(-GM)g(securit)m(y)g(for)f(small)f Fm(q)k Fq(if)d(its)g(probabilit)m(y)e(of)j(b)s(eing)e(correct)j(w)m(as) f(essen)m(tially)150 2536 y(1.)55 b(In)34 b(this)f(argumen)m(t,)k(w)m (e)e(not)g(only)f(sho)m(w)h(there)g(is)f(some)h Fm(q)j Fq(for)c(whic)m(h)g Fm(q)s Fq(-GM)i(securit)m(y)e(do)s(es)h(not)150 2649 y(imply)f(GM-securit)m(y)-8 b(,)38 b(but)d(actually)h(w)m(e)g(sho) m(w)g(a)g(single)f(cryptosystem)h(whic)m(h)e(pro)m(v)m(es)j(that)f(for) g(all)150 2762 y(0)26 b Fm(<)f(q)j(<)d Fq(1)p Fm(=)p Fq(2,)j Fm(q)s Fq(-GM)e(securit)m(y)g(do)s(es)f(not)h(imply)d (GM-securit)m(y)-8 b(.)40 b(The)26 b(basic)f(idea)g(of)h(the)f (construction)150 2875 y(is)34 b(that)i(w)m(e)f(will)e(leak)i(a)g(p)s (olynomially)d(v)-5 b(anishing)32 b(amoun)m(t)k(of)f(information)e(ab)s (out)i(the)g(encrypted)150 2988 y(bit,)28 b(whic)m(h)f(is)g(enough)h (to)g(violate)h(GM-securit)m(y)g(but)e(not)h(enough)g(to)h(mak)m(e)g (the)f(sc)m(heme)h(vulnerable)150 3101 y(under)g Fm(q)s Fq(-GM)i(securit)m(y)-8 b(.)291 3214 y(Let)28 b Fm(q)h Fq(b)s(e)e(giv)m(en,)h(and)e(supp)s(ose)f(w)m(e)j(are)f(giv)m(en)g(a)g (cryptosystem)h(\()p Fm(G;)15 b(E)5 b(;)15 b(D)s Fq(\))29 b(whic)m(h)d(is)g(GM-secure.)150 3327 y(Then)36 b(in)g(particular,)i (if)f(w)m(e)g(restrict)h(the)f(message)i(space)f(to)g(t)m(w)m(o)g (messages,)j(call)36 b(them)i(0)g(and)e(1,)150 3440 y(it)i(remains)g (GM)h(secure.)66 b(In)38 b(our)g(new)g(cryptosystem,)j Fm(G)2274 3407 y Fl(0)2298 3440 y Fq(\(1)2378 3407 y Fk(k)2422 3440 y Fq(\))e(just)f(returns)f(the)i(same)g(result)f(as)150 3552 y Fm(G)p Fq(\(1)301 3519 y Fk(k)345 3552 y Fq(\).)i(No)m(w)27 b(let)g(us)f(mak)m(e)i(the)f(follo)m(wing)e(mo)s(di\014cation)g(to)j (the)f(encryption)f(algorithm:)38 b(in)25 b(our)h(new)150 3665 y(cryptosystem,)37 b(the)d(encryption)g(algorithm)g Fm(E)1846 3632 y Fl(0)1904 3665 y Fq(w)m(orks)h(as)g(follo)m(ws.)53 b(Let)35 b Fj(\016)g Fq(denote)g(concatenation.)150 3778 y(De\014ne)g Fm(E)508 3745 y Fl(0)532 3778 y Fq(\()p Fm(P)13 b(K)q(;)i Fq(0\))35 b(=)d Fm(E)5 b Fq(\()p Fm(P)13 b(K)q(;)i Fq(0\))26 b Fj(\016)e Fm(X)42 b Fq(and)34 b Fm(E)1815 3745 y Fl(0)1839 3778 y Fq(\()p Fm(P)13 b(K)q(;)i Fq(1\))35 b(=)d Fm(E)5 b Fq(\()p Fm(P)13 b(K)q(;)i Fq(1\))26 b Fj(\016)e Fm(Y)c Fq(,)36 b(where)e Fm(X)43 b Fq(is)33 b(a)j(random)150 3891 y(v)-5 b(ariable)26 b(whic)m(h)f(is)h(0)i(with)d (probabilit)m(y)g(1)p Fm(=)p Fq(2)13 b(+)1815 3855 y Fh(1)p 1813 3870 39 4 v 1813 3923 a Fk(k)1862 3891 y Fq(,)28 b(and)e(1)h(otherwise,)h(and)e Fm(Y)20 b Fq('s)27 b(distribution)c(is)j(biased)150 4004 y(the)31 b(other)f(w)m(a)m(y)i (around.)291 4117 y(Clearly)e(this)h(new)g(cryptosystem)h(is)f(not)g (GM)i(secure,)f(since)f(the)h(last)f(bit)g(of)h(the)f(encryption)g(is) 150 4230 y(going)f(to)i(leak)e(information)f(ab)s(out)h(the)g(plain)m (text.)291 4343 y(No)m(w)j(w)m(e)g(claim)f(that)h(this)e(new)h (cryptosystem)h(\()p Fm(G)2113 4310 y Fl(0)2137 4343 y Fm(;)15 b(E)2249 4310 y Fl(0)2273 4343 y Fm(;)g(D)2391 4310 y Fl(0)2414 4343 y Fq(\))33 b(is)e(still)g Fm(q)s Fq(-GM)i(secure.)47 b(Supp)s(ose)30 b(it)150 4456 y(is)j(not.)50 b(Then)33 b(there)h(exists)f(an)g(adv)m(ersary)h Fj(f)p Fm(A)1840 4470 y Fk(n)1888 4456 y Fj(g)g Fq(suc)m(h)f(that)h(for)g (in\014nitely)c(man)m(y)k Fm(k)j Fq(the)c(follo)m(wing)150 4569 y(holds:)308 4773 y Fm(P)13 b(r)s Fq([)p Fm(P)g(K)32 b Fj( )25 b Fm(G)p Fq(\(1)895 4735 y Fk(k)939 4773 y Fq(\);)15 b Fm(b)26 b Fj( )g(f)p Fm(q)i Fq(:)e(0)p Fm(;)15 b Fq(1)p Fj(g)p Fq(;)g Fm(x)27 b Fj( )e Fm(E)1843 4735 y Fl(0)1867 4773 y Fq(\()p Fm(P)13 b(K)q(;)i(b)p Fq(\);)g Fm(b)27 b Fj( )e Fm(A)2455 4788 y Fk(k)2498 4773 y Fq(\()p Fm(P)13 b(K)q(;)i(c)p Fq(\))28 b(:)d Fm(b)g Fq(=)g Fm(b)3073 4735 y Fl(0)3097 4773 y Fq(])g Fj(\025)g Fq(1)c Fj(\000)f Fm(q)j Fq(+)d Fm(\017)p 150 4834 1440 4 v 254 4888 a Fg(4)288 4920 y Ff(Actually)30 b(this)g(result)h(is)f(stronger)h(than)f (what)h(is)f(needed.)48 b(W)-6 b(e)29 b(are)i(pro)n(ving)f(that)g(for)h (all)g(0)e Fe(<)g(q)j(<)c Ff(1)p Fe(=)p Ff(2,)33 b Fe(q)s Ff(-GM)150 5011 y(securit)n(y)c(do)r(es)h(not)f(imply)f(GM-securit)n(y) -6 b(.)46 b(F)-6 b(or)29 b(the)g(problem,)h(w)n(e)g(w)n(ould)f(only)h (need)f(to)g(pro)n(v)n(e)g(that)g(there)g(is)h Fc(some)g Fe(q)150 5102 y Ff(suc)n(h)25 b(that)h Fe(q)s Ff(-GM)f(securit)n(y)h (do)r(es)g(not)g(imply)e(GM-securit)n(y)-6 b(.)1889 5416 y Fq(6-7)p eop %%Page: 8 8 8 7 bop 291 159 a Fq(where)30 b Fm(\017)g Fq(is)g(non-negligible,)e (e.g.)41 b(1)p Fm(=p)p Fq(\()p Fm(n)p Fq(\))32 b(for)e(some)h(p)s (olynomial)c Fm(p)p Fq(.)291 284 y(In)35 b(particular)f(this)g(holds)g (for)h(some)h Fm(k)j Fq(suc)m(h)c(that)2145 239 y Fh(1)p Fk(=)p Fh(2)p Fl(\000)p Fh(1)p Fk(=k)p 2145 263 270 4 v 2145 317 a Fh(1)p Fk(=)p Fh(2+1)p Fk(=k)2458 284 y Fm(>)2618 243 y Fk(q)p 2573 263 125 4 v 2573 315 a Fh(1)p Fl(\000)p Fk(q)2707 284 y Fq(.)56 b(This)33 b(is)i(p)s(ossible)e(b)s (ecause)160 391 y Fh(1)p Fk(=)p Fh(2)p Fl(\000)p Fh(1)p Fk(=k)p 160 415 270 4 v 160 469 a Fh(1)p Fk(=)p Fh(2+1)p Fk(=k)475 436 y Fq(=)591 400 y Fk(k)r Fl(\000)p Fh(2)p 591 415 129 4 v 591 468 a Fk(k)r Fh(+2)765 436 y Fj(!)i Fq(1.)59 b(Note)37 b(that)g(if)e Fm(q)j(<)d Fq(1)p Fm(=)p Fq(2)j(then)2163 395 y Fk(q)p 2118 415 125 4 v 2118 468 a Fh(1)p Fl(\000)p Fk(q)2287 436 y Fm(<)d Fq(1.)59 b(Ho)m(w)m(ev)m(er,) 40 b(if)35 b Fm(q)j Fq(=)d(1)p Fm(=)p Fq(2,)3449 395 y Fk(q)p 3404 415 V 3404 468 a Fh(1)p Fl(\000)p Fk(q)3573 436 y Fq(=)g(1;)150 562 y(that)h(is)d(to)j(sa)m(y)-8 b(,)38 b(this)33 b(argumen)m(t)j(only)e(w)m(orks)h(for)f Fm(q)i(<)d Fq(1)p Fm(=)p Fq(2.)55 b(This)33 b(is)h(a)i(go)s(o)s(d)f (thing)e(to)j(notice)f(as)h(a)150 675 y(sanit)m(y)d(c)m(hec)m(k)h(-)f (after)g(all,)g(1/2-GM)i(securit)m(y)d Fo(is)h Fq(GM-securit)m(y)-8 b(,)35 b(so)e(w)m(e'd)g(b)s(etter)g(not)g(dispro)m(v)m(e)f(their)150 788 y(equiv)-5 b(alence!)291 901 y(First)23 b(of)h(all,)h(let)f(us)f (describ)s(e)f(our)h(\\maxim)m(um)h(lik)m(eliho)s(o)s(d")2381 868 y Fh(5)2441 901 y Fq(strategy)-8 b(.)41 b(Then)22 b(w)m(e)j(will)c(argue)j(that)150 1013 y(in)35 b(fact)j(the)f(adv)m (ersary)g Fj(f)p Fm(A)1143 1027 y Fk(n)1191 1013 y Fj(g)g Fq(is)f(using)f(something)h(more)h(clev)m(er)g(than)g(the)g(maxim)m(um) e(lik)m(eliho)s(o)s(d)150 1126 y(strategy)-8 b(.)53 b(Our)33 b(maxim)m(um)g(lik)m(eliho)s(o)s(d)d(strategy)36 b(will)31 b(alw)m(a)m(ys)j(output)g(1,)h(and)f(here)f(is)g(wh)m(y:)48 b(Ev)m(ery)150 1239 y(time)23 b(w)m(e)g(see)h(a)f(ciphertext)g(that)h 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1917 y Fq(\()p Fm(b)p Fq(\))g(as)g(c.)769 2095 y(Pr[)p Fm(b)25 b Fq(=)g(0)p Fj(j)p Fm(c)h Fq(=)f(0])h(=)f(Pr[)p Fm(c)h Fq(=)f(0)p Fj(j)p Fm(b)h Fq(=)e(0])15 b(Pr)q([)p Fm(b)25 b Fq(=)g(0])p Fm(=)15 b Fq(Pr)q([)p Fm(c)26 b Fq(=)f(0])h(=)2031 2219 y(\(1)p Fm(=)p Fq(2)c(+)e(1)p Fm(=k)s Fq(\))p Fm(q)p 1610 2260 1345 4 v 1610 2343 a Fq(\(1)p Fm(=)p Fq(2)i(+)e(1)p Fm(=k)s Fq(\))p Fm(q)25 b Fq(+)19 b(\(1)p Fm(=)p Fq(2)j Fj(\000)e Fq(1)p Fm(=k)s Fq(\)\(1)j Fj(\000)d Fm(q)s Fq(\))769 2573 y(Pr[)p Fm(b)25 b Fq(=)g(1)p Fj(j)p Fm(c)h Fq(=)f(0])h(=)f(Pr[)p Fm(c)h Fq(=)f(0)p Fj(j)p Fm(b)h Fq(=)e(1])15 b(Pr)q([)p Fm(b)25 b Fq(=)g(1])p Fm(=)15 b Fq(Pr)q([)p Fm(c)26 b Fq(=)f(1])h(=)1917 2697 y(\(1)p Fm(=)p Fq(2)c Fj(\000)e Fq(1)p Fm(=k)s Fq(\)\(1)j Fj(\000)d Fm(q)s Fq(\))p 1610 2738 V 1610 2821 a(\(1)p Fm(=)p Fq(2)i(+)e(1)p Fm(=k)s Fq(\))p Fm(q)25 b Fq(+)19 b(\(1)p Fm(=)p Fq(2)j Fj(\000)e Fq(1)p Fm(=k)s Fq(\)\(1)j Fj(\000)d Fm(q)s Fq(\))150 2986 y(and)30 b(so)g(the)h(statemen)m(t)h 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b(otherwise)e(where)f(w)m(ould)g(its)h(adv)-5 b(an)m(tage)37 b(come)f(from?)55 b(Therefore,)36 b(at)150 3776 y(least)31 b(one)f(of)h(the)f(statemen)m(ts)i(b)s(elo)m(w)e(is)f(true:)261 3942 y(1.)46 b(A)m(t)39 b(least)f(half)f(of)h(the)g(adv)m(ersary's)g (adv)-5 b(an)m(tage,)42 b Fm(\017=)p Fq(2)d(is)e(coming)h(from)f(the)h (adv)m(ersary)g(b)s(eing)377 4055 y(correct)32 b(b)s(ecause)e(it)g (guesses)h(0)f(when)g Fm(c)25 b Fq(=)g(0.)261 4234 y(2.)46 b(A)m(t)39 b(least)f(half)f(of)h(the)g(adv)m(ersary's)g(adv)-5 b(an)m(tage,)42 b Fm(\017=)p Fq(2)d(is)e(coming)h(from)f(the)h(adv)m (ersary)g(b)s(eing)377 4347 y(correct)32 b(b)s(ecause)e(it)g(guesses)h (0)f(when)g Fm(c)25 b Fq(=)g(1.)291 4513 y(Assume)f(for)g(no)m(w)h (that)g(\(1\))h(is)d(the)i(true)f(statemen)m(t.)41 b(Then,)25 b(w)m(e)g(can)f(use)h(this)e(adv)m(ersary)i(to)g(break)150 4626 y(the)31 b(encryption)e(sc)m(heme)j(\()p Fm(G;)15 b(E)5 b(;)15 b(D)s Fq(\).)43 b(If)30 b(w)m(e)h(are)g(giv)m(en)g Fm(x)p Fq(,)g(w)m(e)g(simply)d(run)h(the)i Fm(q)s Fq(-GM)g(adv)m (ersary)g(on)150 4751 y Fm(x)20 b Fj(\016)h Fq(0.)41 b(Let)31 b Fm(m)p Fq(\()p Fm(k)s Fq(\))26 b(=)1173 4707 y Fh(\(1)p Fk(=)p Fh(2+1)p Fk(=k)r Fh(\))p Fk(q)p 894 4731 918 4 v 894 4784 a Fh(\(1)p Fk(=)p Fh(2+1)p Fk(=k)r Fh(\))p Fk(q)r Fh(+\(1)p Fk(=)p Fh(2)p Fl(\000)p Fh(1)p Fk(=k)r Fh(\)\(1)p Fl(\000)p Fk(q)r Fh(\))1821 4751 y Fq(.)40 b(No)m(w)32 b(w)m(e)e(kno)m(w)h(that)p 150 4808 1440 4 v 254 4861 a Fg(5)288 4893 y Ff(If)18 b(w)n(e)g(w)n(ere)h (talking)f(ab)r(out)g(unconditional)g(securit)n(y)-6 b(,)19 b(w)n(e)f(w)n(ould)h(ha)n(v)n(e)e(solv)n(ed)h(the)f(problem)g (already)h(just)g(b)n(y)f(pro)n(ving)150 4985 y(that)26 b(the)f(maxim)n(um)e(lik)n(eliho)r(o)r(d)k(strategy)f(cannot)g(do)g(b)r (etter)g(than)g(1)17 b Fb(\000)g Fe(q)s Ff(.)35 b(Ho)n(w)n(ev)n(er,)27 b(our)f(setting)g(is)h(computational,)150 5076 y(and)36 b(therefore)g(w)n(e)h(cannot)f(use)g(a)g(probabilit)n(y)g(theorem)f 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b(A)p Fq(\)].)3482 619 y Fh(6)3573 652 y Fq(No)m(w)150 765 y(b)m(y)34 b(the)g(argumen)m(t)g(w)m(e)g(used)f(to)i(sho)m(w)f (that)g Fm(q)s Fq(-GM)h(securit)m(y)e(implies)e(GM-securit)m(y)-8 b(,)36 b(w)m(e)f(kno)m(w)e(that)150 878 y(Pr)o([)p Fm(S)328 892 y Fk(GM)462 878 y Fq(\()p Fm(k)s(;)15 b(A)p Fq(\)],)32 b(the)d(probabilit)m(y)d(that)k Fm(A)f Fq(is)f(successful)f(if)h(w)m(e) i(just)e(app)s(end)f(0)i(to)h Fm(x)f Fq(and)f(hand)g(it)h(o\013)150 991 y(to)i Fm(A)p Fq(,)g(is)e(at)i(least)g(1)p Fm(=)p Fq(2)22 b(+)d Fm(\017=)p Fq(4\(1)j Fj(\000)e Fm(m)p Fq(\),)31 b(so)g(w)m(e)f(ha)m(v)m(e)i(brok)m(en)e(the)h(GM-securit)m(y)g(of)f(\() p Fm(G;)15 b(E)5 b(;)15 b(D)s Fq(\).)291 1104 y(If)22 b(at)g(least)h(half)d(the)j(adv)m(ersary's)f(adv)-5 b(an)m(tage)24 b(comes)f(from)e(the)h(case)h(when)e(the)h(adv)m(ersary)h(guesses)150 1217 y(0)k(when)f Fm(c)f Fq(=)g(1,)k(w)m(e)e(do)f(the)h(same)h(thing,)e (except)i(w)m(e)f(ask)g(ab)s(out)g Fm(x)13 b Fj(\016)g Fq(1,)28 b(and)e(the)h(argumen)m(t)g(is)f(exactly)150 1330 y(the)31 b(same.)291 1443 y(Th)m(us,)c(w)m(e)h(are)g(done.)40 b(W)-8 b(e)29 b(ha)m(v)m(e)g(sho)m(wn)e(that)h(\()p Fm(G)2010 1410 y Fl(0)2034 1443 y Fm(;)15 b(E)2146 1410 y Fl(0)2170 1443 y Fm(;)g(D)2288 1410 y Fl(0)2311 1443 y Fq(\))28 b(is)f(not)h(GM-secure,)h(and)e(furthermore)150 1556 y(that)34 b(if)f(it)g(is)g(not)h Fm(q)s Fq(-GM)g(secure,)h(then)f(\()p Fm(G;)15 b(E)5 b(;)15 b(D)s Fq(\))36 b(is)c(not)i(GM-secure.)52 b(Therefore,)35 b Fm(q)s Fq(-GM)f(securit)m(y)150 1669 y(do)s(es)c(not)h(imply)d(GM-securit)m(y)-8 b(.)291 1872 y Fr(Common)44 b(errors:)60 b Fq(The)40 b(most)g(common)g(error)g(p)s (eople)f(made)g(w)m(as)i(the)f(error)f(of)h(assuming)150 1985 y(that)h(all)e(an)m(y)h(adv)m(ersary)h(can)f(do)g(is)f(try)h(to)h (guess)f(the)h(secret)g(k)m(ey)-8 b(,)44 b(and)39 b(if)h(they)g(fail,)h (just)f(guess)150 2098 y(according)33 b(to)h(the)g(distribution)29 b(of)34 b(the)f(input.)48 b(If)32 b(this)g(w)m(as)i(an)f(assumption)f (w)m(e)i(could)e(mak)m(e)i(then)150 2211 y(just)g(ab)s(out)g(an)m(y)h (cryptosystem)g(w)m(ould)e(b)s(e)h(secure)g(-)h(this)e(adv)m(ersary)i (guaran)m(tees)h(to)f(nev)m(er)g(ha)m(v)m(e)h(a)150 2323 y(go)s(o)s(d)g(probabilit)m(y)d(of)j(breaking)f(GM-securit)m(y)-8 b(.)57 b(The)36 b(really)e(useful)g(adv)m(ersaries)i(are)g(ones)g(that) g(do)150 2436 y(other)f(things.)54 b(F)-8 b(or)36 b(instance,)h(if)d (factoring)h(w)m(as)h(solv)-5 b(able)34 b(in)g(p)s(olynomial)e(time,)37 b(an)e(adv)m(ersary)g(for)150 2549 y(the)f(GM)g(cryptosystem)f(w)m (ould)g(simply)e(factor)j Fm(n)f Fq(and)g(then)g(compute)h(the)f (legendre)g(sym)m(b)s(ols)f(of)i Fm(c)150 2662 y Fq(mo)s(dulo)28 b Fm(p)h Fq(and)f(mo)s(dulo)g Fm(q)k Fq(to)e(determing)e(if)h Fm(c)g Fq(is)f(the)i(encryption)e(of)h(a)h(0)g(or)f(a)h(1.)40 b(Y)-8 b(ou)30 b(m)m(ust)f(assume)150 2775 y Fo(nothing)i Fq(ab)s(out)f(ho)m(w)h(an)f(adv)m(ersary)h(w)m(orks.)291 2888 y(The)39 b(w)m(a)m(y)i(this)d(w)m(as)i(t)m(ypically)f(used)g(w)m (as)h(to)g(pro)m(v)m(e)h(that)f(a)g(construction)f(w)m(as)h Fm(q)s Fq(-GM)h(secure.)150 3001 y(The)f(righ)m(t)h(idea)f(is)g(the)h (one)g(used)f(ab)s(o)m(v)m(e:)63 b(pro)m(v)m(e)42 b(that)f(if)f(an)h (adv)m(ersary)g(breaks)g Fm(q)s Fq(-GM)g(securit)m(y)150 3114 y(for)g(the)g(construction)f(then)h(they're)h(doing)e(b)s(etter)h (than)f(guessing)h(-)g(they're)g(actually)g(breaking)150 3227 y(the)32 b(underlying)d(secure)i(cryptosystem)i(in)d(some)i(w)m(a) m(y)-8 b(.)46 b(Th)m(us,)31 b(w)m(e)i(get)f(a)h(con)m(tradiction)e(b)m (y)h(pro)m(ving)150 3340 y(the)i(adv)m(ersary)f(that)i(defeats)f Fm(q)s Fq(-GM)g(securit)m(y)f(implies)e(one)j(that)g(defeats)g(the)g (GM-securit)m(y)g(of)g(the)150 3453 y(cryptosystem)d(w)m(e)g(assumed)e (w)m(as)i(GM-secure.)150 3737 y Fn(Problem)45 b(4)150 3940 y Fq(Pro)m(v)m(e)23 b(that)f(if)f(there)h(exists)f(a)h(GM-secure)h (cryptosystem)f(\()p Fm(G;)15 b(E)5 b(;)15 b(D)s Fq(\))23 b(then)f Fm(P)38 b Fj(6)p Fq(=)25 b Fm(N)10 b(P)j Fq(.)38 b(F)-8 b(or)22 b(simplicit)m(y)-8 b(,)150 4053 y(y)m(ou)31 b(ma)m(y)g(assume)f(that)h(\()p Fm(G;)15 b(E)5 b(;)15 b(D)s Fq(\))32 b(has)e(faithfulness)e(1.)291 4256 y Fr(Solution:)44 b Fq(Let)32 b Fm(L)967 4271 y Fk(b)1033 4256 y Fq(b)s(e)f(the)h (language)h(of)f(all)e(pairs)h(\()p Fm(P)13 b(K)q(;)i(c)p Fq(\))34 b(suc)m(h)d(that)i(there)f(is)e(some)j(random)150 4369 y(string)d Fm(r)k Fq(suc)m(h)d(that)h Fm(E)5 b Fq(\()p Fm(P)13 b(K)q(;)i(b)p Fq(\))34 b(run)c(with)g(randomness)g Fm(r)k Fq(pro)s(duces)c Fm(c)h Fq(as)h(an)f(output.)44 b Fm(L)3332 4384 y Fk(b)3397 4369 y Fq(is)31 b(clearly)150 4482 y(an)24 b Fm(N)10 b(P)38 b Fq(language,)26 b(since)e(w)m(e)h(can)f (nondeterministically)d(write)j(do)m(wn)g Fm(r)j Fq(and)d(then)g(v)m (erify)g(the)g(output)150 4595 y(of)38 b Fm(E)43 b Fq(with)36 b(randomness)h Fm(r)s Fq(.)62 b(Th)m(us,)39 b(if)d Fm(P)51 b Fq(=)37 b Fm(N)10 b(P)j Fq(,)39 b(there)f(is)f(an)h(algorithm)e Fm(A)i Fq(that)g(decides)f Fm(L)3597 4609 y Fh(0)3674 4595 y Fq(in)150 4708 y(p)s(olynomial)25 b(time.)39 b(So,)28 b(on)f(input)e Fm(P)13 b(K)q(;)i(c)29 b Fq(w)m(e)f(run)e Fm(A)p Fq(\()p Fm(P)13 b(K)q(;)i(c)p Fq(\))29 b(and)e(if)f(it)h (accepts,)i(w)m(e)f(output)f(0,)h(and)f(if)150 4821 y(it)j(rejects,)h (w)m(e)f(output)g(1.)41 b(This)28 b(will)g(break)i(GM-securit)m(y)-8 b(.)42 b(Th)m(us,)29 b(if)g Fm(P)38 b Fq(=)25 b Fm(N)10 b(P)43 b Fq(then)30 b(no)g(GM-secure)150 4934 y(cryptosystem)h(exists.) 40 b(Therefore,)31 b(if)e(a)i(GM-secure)g(cryptosystem)g(exists,)f (then)g Fm(P)39 b Fj(6)p Fq(=)24 b Fm(N)10 b(P)j Fq(.)p 150 4990 1440 4 v 254 5044 a Fg(6)288 5076 y Ff(The)28 b(o)n(v)n(erlap)f(of)h(this)g(notation)g(with)f Fe(S)1489 5084 y Fa(q)r(GM)1634 5076 y Ff(\()p Fe(k)r(;)13 b(A)p Ff(\))27 b(mak)n(e)f(sense)i(since)g(this)f(is)h(really)g(just)g(the)f (same)g(as)h(the)f(ev)n(en)150 5167 y(for)f Fe(m)p Ff(-GM)g(securit)n (y)-6 b(.)1889 5416 y Fq(6-9)p eop %%Page: 10 10 10 9 bop 291 159 a Fq(The)22 b(solution)f(is)g(only)h(sligh)m(tly)e (more)j(complex)f(if)f(w)m(e)i(consider)e(longer)h(message)h(spaces.)39 b(Supp)s(ose)150 271 y(all)28 b(messages)i(w)m(e)g(consider)e(for)h (securit)m(y)g(parameter)h Fm(k)j Fq(ha)m(v)m(e)d(length)f Fm(p)p Fq(\()p Fm(k)s Fq(\).)41 b(Let)29 b Fm(m)3093 285 y Fh(0)3162 271 y Fq(and)g Fm(m)3418 285 y Fh(1)3486 271 y Fq(b)s(e)g(an)m(y)150 384 y(t)m(w)m(o)23 b(messages)g(that)g (di\013er)d(in)h(their)g(\014rst)g(bit,)i(where)e(the)h(\014rst)f(bit)g (of)h Fm(m)2640 398 y Fh(0)2701 384 y Fq(\(without)f(loss)g(of)i (generalit)m(y\))150 497 y(is)h(0.)39 b(Let)26 b Fm(L)565 464 y Fl(0)565 525 y Fk(b)624 497 y Fq(b)s(e)e(the)h(language)g(of)g (all)f(pairs)g(\()p Fm(P)13 b(K)q(;)i(c)p Fq(\))27 b(suc)m(h)d(that)i (there)f(is)f(some)h(random)f(string)g Fm(r)k Fq(and)150 610 y(some)i(string)f Fm(u)h Fq(of)g(length)f Fm(p)p Fq(\()p Fm(k)s Fq(\))20 b Fj(\000)f Fq(1)30 b(suc)m(h)g(that)g Fm(E)5 b Fq(\()p Fm(P)13 b(K)q(;)i(b)21 b Fj(\016)f Fm(u)p Fq(\))30 b(run)e(with)h(randomness)g Fm(r)j Fq(pro)s(duces)d Fm(c)150 723 y Fq(as)h(an)g(output.)40 b Fm(L)782 690 y Fl(0)782 751 y Fk(b)846 723 y Fq(is)29 b(also)h(an)g Fm(N)10 b(P)43 b Fq(language,)31 b(so)f(let)g Fm(A)2143 690 y Fl(0)2196 723 y Fq(decide)g Fm(L)2536 690 y Fl(0)2536 747 y Fh(0)2605 723 y Fq(in)e(p)s(olynomial)g(time.)40 b(No)m(w,)31 b(on)150 836 y(input)d Fm(P)13 b(K)q(;)i(c)32 b Fq(w)m(e)f(run)e Fm(A)1021 803 y Fl(0)1044 836 y Fq(\()p Fm(P)13 b(K)q(;)i(c)p Fq(\))33 b(and)d(if)f(it)h(accepts,)i(w)m(e)f (output)f(0,)h(and)e(if)h(it)g(rejects)h(w)m(e)g(output)e(1.)291 949 y(Another)k(approac)m(h)h(is)f(to)h(try)f(to)h(use)f(the)h(p)s(o)m (w)m(er)f(of)h(assuming)d Fm(P)44 b Fq(=)30 b Fm(N)10 b(P)46 b Fq(to)34 b(try)f(to)h(guess)g(the)150 1062 y(randomness)29 b(used)h(to)h(generate)h Fm(P)13 b(K)7 b Fq(,)30 b(and)g(th)m(us)g (obtain)g Fm(S)5 b(K)i Fq(,)30 b(but)g(this)f(is)g(a)i(messier)e (solution.)291 1274 y Fr(Common)34 b(errors:)42 b Fq(Remem)m(b)s(er)30 b(that)h Fm(N)10 b(P)44 b Fq(is)30 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y(1)p 1745 215 46 4 v 1753 298 a Fk(l)1826 122 y Fg(l)q Fj(\000)p Fi(1)1816 150 y Ff(X)1820 345 y Fg(j)t Fi(=0)1972 175 y Fo(1)p 1972 215 V 1972 298 a(2)2043 236 y(\(1)21 b(+)f Fk(P)2293 250 y Fg(j)2350 236 y Fh(\000)g Fk(P)2499 250 y Fg(j)t Fi(+1)2626 236 y Fo(\))1582 520 y(=)1745 459 y(1)p 1745 499 V 1745 583 a(2)1821 520 y(+)1922 459 y(1)p 1922 499 V 1930 583 a Fk(l)1992 434 y Ff(X)1997 629 y Fg(j)t Fi(=0)2139 520 y Fk(P)2197 534 y Fg(j)2254 520 y Fh(\000)g Fk(P)2403 534 y Fg(j)t Fi(+1)1582 805 y Fo(=)1745 743 y(1)p 1745 784 V 1745 867 a(2)1821 805 y(+)1922 743 y(1)p 1922 784 V 1930 867 a Fk(l)1977 805 y Fo(\()p Fk(P)2070 819 y Fi(0)2131 805 y Fh(\000)g Fk(P)2280 820 y Fg(l)2306 805 y Fo(\))1582 1020 y Fh(\025)1745 959 y Fo(1)p 1745 1000 V 1745 1083 a(2)1821 1020 y(+)2030 959 y(1)p 1922 1000 262 4 v 1922 1083 a Fk(l)i Fh(\001)e Fk(p)p Fo(\()p Fk(k)s Fo(\))291 1275 y(No)m(w)31 b(this)e(pro)m(v)m(es)h(that)h(Pr)o([)p Fk(S)1330 1294 y Fi(\(1)p Fg(;GM)7 b Fi(\))1574 1275 y Fo(\()p Fk(k)s(;)15 b(R)1769 1242 y Fj(0)1793 1275 y Fo(\)])26 b Fh(\025)f Fo(1)p Fk(=)p Fo(2)20 b(+)f(1)p Fk(=pol)r(y)s Fo(,)31 b(so)f(w)m(e're)h(done.)40 b(This)28 b(breaks)i(the)150 1388 y(GM-securit)m(y)h(of)g(\()p Fk(G)884 1355 y Fj(0)908 1388 y Fk(;)15 b(E)1020 1355 y Fj(0)1044 1388 y Fk(;)g(D)1162 1355 y Fj(0)1185 1388 y Fo(\).)291 1600 y Fp(Common)35 b(errors:)43 b Fo(One)31 b(argumen)m(t)h(that)g(w)m(as)g(app)s(ealing)e(and)h(simple)e(w)m(as)j (to)g(sa)m(y)h(\(roughly\))150 1713 y(that)27 b(\\there's)g(at)g(least) g(one)f Fk(i)p Fo(,)i(sa)m(y)-8 b(,)29 b Fk(i)1443 1727 y Fi(0)1482 1713 y Fo(,)f(for)e(whic)m(h)f(our)h(adv)-5 b(an)m(tage)28 b(is)e(1)p Fk(=l)14 b Fh(\001)e Fk(p)p Fo(\()p Fk(k)s Fo(\),)29 b(from)d(lecture,)h(so)g(the)150 1826 y(probabilit)m(y)h(w)m(e)k(pic)m(k)e Fk(i)973 1840 y Fi(0)1043 1826 y Fo(is)g(1)p Fk(=l)r Fo(,)i(so)f(our)g(adv)-5 b(an)m(tage)32 b(is)e(only)g(1)p Fk(=l)2426 1793 y Fi(2)2487 1826 y Fh(\001)21 b Fk(p)p Fo(\()p Fk(k)s Fo(\),)31 b(whic)m(h)f(is)g (still)e(p)s(olynomial.)150 1939 y(The)f(problem)f(with)g(this)g(is)g (that)i(it)f(mak)m(es)i(an)e(unstated)g(assumption,)g(namely)f(that)i (if)f(w)m(e)h(pic)m(k)f(an)m(y)150 2052 y Fk(i)36 b Fo(other)h(than)f Fk(i)710 2066 y Fi(0)750 2052 y Fo(,)h(our)f(probabilit)m(y)e(of)i(b)s (eing)f(correct)i(is)f(at)h(least)f(1/2.)59 b(Equiv)-5 b(alen)m(tly)d(,)37 b(this)e(mak)m(es)150 2165 y(the)c(assumption)e (that)j(for)e(an)m(y)h Fk(i)g Fo(other)g(than)g Fk(i)1841 2179 y Fi(0)1881 2165 y Fo(,)g Fk(P)1995 2179 y Fg(i)2049 2165 y Fh(\025)26 b Fk(P)2204 2179 y Fg(i)p Fi(+1)2353 2165 y Fo(but)k(this)g(is)g(not)h(necessarily)f(the)h(case.)150 2278 y(Imagine)i(a)h(sequence)g(of)g(v)-5 b(alues)33 b(lik)m(e)g Fk(P)1567 2292 y Fi(0)1607 2278 y Fk(;)15 b(P)1705 2292 y Fi(0)1768 2278 y Fo(+)22 b Fk(\017;)15 b(P)1996 2292 y Fi(0)2058 2278 y Fo(+)22 b(2)p Fk(\017;)15 b(:)g(:)g(:)i(;)e(P)2493 2292 y Fi(0)2556 2278 y Fo(+)22 b(\()p Fk(l)j Fh(\000)d Fo(2\))p Fk(\017;)15 b(P)3044 2292 y Fi(0)3107 2278 y Fh(\000)22 b Fo(1)p Fk(=p)p Fo(\()p Fk(k)s Fo(\),)36 b(where)150 2391 y(this)30 b(assumption)f(do)s(es)i (not)g(hold.)41 b(What)32 b(turns)d(out)i(to)h(b)s(e)e(the)h(case)h(is) e(that)i(the)f(disadv)-5 b(an)m(tage)31 b(w)m(e)150 2504 y(get)h(from)e(this)f(strange)j(b)s(eha)m(vior)d(is)h(coun)m(teracted)j (b)m(y)d(the)h(adv)-5 b(an)m(tage)32 b(w)m(e)g(get)f(when)f(the)h(b)s (eha)m(vior)150 2617 y(is)e(normal.)39 b(Ho)m(w)m(ev)m(er,)32 b(it's)e(m)m(uc)m(h)f(easier)h(to)h(pro)m(v)m(e)f(the)g(theorem)g(b)m (y)g(an)f(argumen)m(t)h(suc)m(h)g(as)g(the)g(one)150 2730 y(giv)m(en)g(ab)s(o)m(v)m(e)i(than)e(b)m(y)g(\014xing)f(the)i (\015a)m(w)m(ed)g(argumen)m(t.)150 3016 y Fl(Problem)45 b(2:)61 b(Moses)45 b(Securit)l(y)150 3219 y Fo(Recall)31 b(that)h(the)g(Hamming)f(w)m(eigh)m(t)h(of)f(a)h(binary)e(string)g(is)h (the)h(n)m(um)m(b)s(er)e(of)h(1s)h(in)e(that)i(string.)43 b(Let)150 3332 y Fh(H)228 3299 y Fg(i)227 3360 y(k)300 3332 y Fo(b)s(e)29 b(the)i(set)g(of)f(all)g(strings)f(of)h(length)g (exactly)h Fk(k)j Fo(and)c(of)g(Hamming)g(w)m(eigh)m(t)h(exactly)g Fk(i)p Fo(.)291 3445 y(W)-8 b(e)32 b(sa)m(y)f(that)g(a)g(cryptosystem)f (\()p Fk(G;)15 b(E)5 b(;)15 b(D)s Fo(\))33 b(is)c Fk(i)p Fo(-Moses)j(secure)e(if)150 3657 y Fh(8f)p Fk(A)314 3671 y Fg(n)361 3657 y Fh(g)h(9)p Fk(f)j Fh(2)25 b Fk(neg)34 b Fo(suc)m(h)c(that)h Fh(8)p Fk(k)d(>)d(i)g Fh(2)g Fe(N)43 b Fh(8)p Fk(m)1820 3671 y Fi(0)1859 3657 y Fk(;)15 b(m)1979 3671 y Fi(1)2044 3657 y Fh(2)24 b(f)p Fo(0)p Fk(;)15 b Fo(1)p Fh(g)2349 3624 y Fg(k)2425 3657 y Fo(suc)m(h)30 b(that)h Fk(m)2907 3671 y Fi(0)2966 3657 y Fh(\010)20 b Fk(m)3137 3671 y Fi(1)3202 3657 y Fh(2)k(H)3365 3624 y Fg(i)3364 3685 y(k)150 3770 y Fo(Pr)o([\()p Fk(P)13 b(K)q(;)i(S)5 b(K)i Fo(\))27 b Fh( )f Fk(G)p Fo(\(1)971 3737 y Fg(k)1015 3770 y Fo(\);)15 b Fk(b)26 b Fh( )f(f)p Fo(0)p Fk(;)15 b Fo(1)p Fh(g)p Fo(;)g Fk(c)28 b Fh( )d Fk(E)5 b Fo(\()p Fk(P)13 b(K)q(;)i(m)2090 3785 y Fg(b)2126 3770 y Fo(\);)273 3883 y Fk(b)312 3850 y Fj(0)360 3883 y Fh( )25 b Fk(A)544 3898 y Fg(k)587 3883 y Fo(\()p Fk(P)13 b(K)q(;)i(c;)g(m)970 3897 y Fi(0)1012 3883 y Fk(;)g(m)1132 3897 y Fi(1)1171 3883 y Fo(\))26 b(:)f Fk(b)1321 3850 y Fj(0)1370 3883 y Fo(=)g Fk(b)p Fo(])h Fh(\024)e Fo(1)p Fk(=)p Fo(2)e(+)e Fk(f)10 b Fo(\()p Fk(k)s Fo(\))291 4096 y(The)28 b(in)m(tuition)f(b)s(ehind)f(this)i(de\014nition)e(is)i (that)i(for)e(t)m(w)m(o)j(strings)c(whic)m(h)h(di\013er)g(in)f(exactly) j Fk(i)f Fo(bits,)150 4209 y(it)h(is)f(imp)s(ossible)e(to)k (distinguish)c(the)k(encryption)e(of)h(one)h(from)f(the)g(encryption)g (of)g(the)h(other.)291 4421 y Fp(\(a\))f Fo(Pro)m(v)m(e)h(or)g(dispro)m (v)m(e:)40 b(F)-8 b(or)31 b(all)e(o)s(dd)g Fk(i)p Fo(,)i Fk(i)p Fo(-Moses)h(securit)m(y)e(is)g(equiv)-5 b(alen)m(t)29 b(to)j(GM-securit)m(y)-8 b(.)291 4634 y Fp(Solution:)40 b Fo(First,)28 b(I)f(will)f(pro)m(v)m(e)i(that)h(for)e(an)m(y)h Fk(i)p Fo(,)h(if)e(\()p Fk(G;)15 b(E)5 b(;)15 b(D)s Fo(\))30 b(is)d(GM-secure)i(then)e(it)h(is)e Fk(i)p Fo(-Moses)150 4747 y(secure.)41 b(This)27 b(turns)i(out)g(to)i(b)s(e)d(v)m(ery)i (easy)-8 b(.)42 b(Since)28 b(\()p Fk(G;)15 b(E)5 b(;)15 b(D)s Fo(\))32 b(is)d(GM-secure,)i(for)e(an)m(y)h(adv)m(ersary)g Fk(A)p Fo(,)150 4860 y(there)e(is)f(a)h(negligible)e(function)h Fk(f)37 b Fo(suc)m(h)27 b(that)i(for)e(all)g Fk(k)k Fo(and)d(for)f(all) g(messages)i Fk(m)3013 4874 y Fi(0)3080 4860 y Fo(and)e Fk(m)3334 4874 y Fi(1)3401 4860 y Fo(of)i(length)150 4972 y Fk(k)s Fo(,)37 b(the)f(condion)e(holds.)55 b(In)35 b(particular,)g(for)h(an)m(y)g(adv)m(ersary)f Fk(A)p Fo(,)j(there)d(is)g(a)h(\(the)g(same\))g(negligible)150 5085 y(function)c Fk(f)42 b Fo(suc)m(h)33 b(that)h(for)f(all)f Fk(k)i(>)29 b(i)34 b Fo(and)f(for)g(all)f(messages)i Fk(m)2440 5099 y Fi(0)2512 5085 y Fo(and)f Fk(m)2772 5099 y Fi(1)2845 5085 y Fo(of)g(length)g Fk(k)j Fo(di\013ering)31 b(in)1889 5416 y(8-2)p eop %%Page: 3 3 3 2 bop 150 159 a Fo(exactly)28 b Fk(i)f Fo(bits,)g(the)g(condition)f (holds.)38 b(In)26 b(other)i(w)m(ords,)f(the)h Fk(i)p Fo(-Moses)g(securit)m(y)f(condition)f(is)g(a)h(strict)150 271 y(sub)s(condition)g(of)k(GM-securit)m(y;)g(no)g(reduction)e(is)g (necessary)i(to)g(pro)m(v)m(e)g(this)f(part.)291 484 y(No)m(w,)35 b(let)e Fk(i)h Fo(b)s(e)e(some)i(o)s(dd)e(n)m(um)m(b)s (er.)49 b(Supp)s(ose)31 b(\()p Fk(G;)15 b(E)5 b(;)15 b(D)s Fo(\))35 b(is)e(not)g(GM-secure.)51 b(Then,)33 b(w)m(e)h(ha)m(v)m(e)150 597 y(t)m(w)m(o)j(messages)f Fk(m)797 611 y Fi(0)872 597 y Fo(and)f Fk(m)1134 611 y Fi(1)1208 597 y Fo(suc)m(h)g(that)h(some)g(adv)m(ersary)g Fk(A)f Fo(can)h(distinguish)c(encryptions)i(of)i(one)150 710 y(from)30 b(the)g(other.)41 b(W)-8 b(e)32 b(construct)f(h)m(ybrids) d Fk(m)1755 724 y Fi(0)1819 710 y Fo(=)d Fk(m)1995 677 y Fi(0)2034 710 y Fk(;)15 b(m)2154 677 y Fi(1)2194 710 y Fk(;)g(:)g(:)g(:)i(;)e(m)2476 677 y Fg(l)q Fi(\()p Fg(k)r(;i)p Fi(\))2664 710 y Fo(=)25 b Fk(m)2840 724 y Fi(1)2910 710 y Fo(and)30 b(de\014ne)405 1027 y Fk(P)463 1041 y Fg(j)525 1027 y Fo(=)25 b(Pr)o([\()p Fk(P)13 b(K)q(;)i(S)5 b(K)i Fo(\))27 b Fh( )e Fk(G)p Fo(\(1)1441 989 y Fg(k)1485 1027 y Fo(\);)15 b Fk(c)27 b Fh( )e Fk(E)5 b Fo(\()p Fk(P)13 b(K)q(;)i(m)2118 989 y Fg(j)2156 1027 y Fo(\);)g Fk(b)2270 989 y Fj(0)2319 1027 y Fh( )25 b Fk(A)2503 1042 y Fg(k)2546 1027 y Fo(\()p Fk(P)13 b(K)q(;)i(c;)g(m)2929 1041 y Fi(0)2970 1027 y Fk(;)g(m)3090 1041 y Fi(1)3130 1027 y Fo(\))26 b(:)f Fk(b)3280 989 y Fj(0)3329 1027 y Fo(=)g(0])291 1231 y(and)j(note)i(that)f(b)m(y)g(the)g(same)g (argumen)m(t)g(in)f(class,)h Fk(P)2173 1245 y Fi(0)2230 1231 y Fh(\000)17 b Fk(P)2376 1250 y Fg(l)q Fi(\()p Fg(k)r(;i)p Fi(\))2565 1231 y Fh(\025)24 b Fo(1)p Fk(=p)p Fo(\()p Fk(k)s Fo(\))31 b(for)d(some)i(p)s(olynomial)150 1344 y Fk(p)p Fo(,)f(and)g(therefore)h(there)f(is)f(some)i Fk(j)1392 1358 y Fi(0)1461 1344 y Fo(suc)m(h)f(that)g Fk(P)1918 1358 y Fg(j)1947 1367 y Fd(0)2004 1344 y Fh(\000)18 b Fk(P)2151 1358 y Fg(j)2180 1367 y Fd(0)2214 1358 y Fi(+1)2334 1344 y Fh(\025)25 b Fo(2)p Fk(=)p Fo(\()p Fk(l)r Fo(\()p Fk(k)s(;)15 b(i)p Fo(\))20 b Fh(\001)e Fk(p)p Fo(\()p Fk(k)s Fo(\)\).)42 b(No)m(w,)30 b(if)e(for)h(all)f Fk(j)5 b Fo(,)150 1457 y Fk(m)230 1424 y Fg(j)287 1457 y Fh(\010)19 b Fk(m)457 1424 y Fg(j)t Fi(+1)609 1457 y Fh(2)25 b(H)773 1424 y Fg(i)772 1485 y(k)815 1457 y Fo(,)30 b(w)m(e)h(can)g(use)f(the)g(follo)m(wing)f(algorithm)g(to)j (break)e Fk(i)p Fo(-Moses)i(securit)m(y:)150 1670 y Fk(A)218 1637 y Fj(0)218 1697 y Fg(j;k)309 1670 y Fo(\()p Fk(m)424 1684 y Fi(0)464 1670 y Fk(;)15 b(m)584 1684 y Fi(1)623 1670 y Fk(;)g(A)731 1685 y Fg(k)774 1670 y Fk(;)g(P)e(K)q(;)i(c;)g(m) 1162 1637 y Fj(0)1162 1694 y Fi(0)1204 1670 y Fk(;)g(m)1324 1637 y Fj(0)1324 1694 y Fi(1)1364 1670 y Fo(\))25 b(:)355 1789 y(1.)41 b(If)30 b Fk(m)637 1756 y Fj(0)637 1813 y Fi(0)701 1789 y Fh(6)p Fo(=)25 b Fk(m)877 1756 y Fg(j)944 1789 y Fo(or)30 b Fk(m)1135 1756 y Fj(0)1135 1813 y Fi(1)1200 1789 y Fh(6)p Fo(=)25 b Fk(m)1376 1756 y Fg(j)t Fi(+1)1502 1789 y Fo(,)31 b(output)f(a)g(random)g(bit.)355 1902 y(2.)41 b(Output)29 b Fk(A)857 1917 y Fg(k)900 1902 y Fo(\()p Fk(P)13 b(K)q(;)i(c;)g(m)1283 1916 y Fi(0)1324 1902 y Fk(;)g(m)1444 1916 y Fi(1)1484 1902 y Fo(\).)291 2115 y(W)-8 b(e)46 b(kno)m(w)f(that)h(the)f(probabilit)m(y)d(that)j Fk(A)1855 2082 y Fj(0)1855 2142 y Fg(j;k)1991 2115 y Fo(is)f(successful)g(is)g(1)p Fk(=)p Fo(2\(1)32 b(+)d Fk(P)3036 2129 y Fg(j)3103 2115 y Fh(\000)h Fk(P)3262 2129 y Fg(j)t Fi(+1)3389 2115 y Fo(\),)49 b(so)c(the)150 2227 y(probabilit)m(y)28 b(that)j Fk(A)880 2195 y Fj(0)880 2255 y Fg(j)909 2264 y Fd(0)943 2255 y Fg(;k)1036 2227 y Fo(is)e(successful)g(is)h(at)h(least)f(1)p Fk(=)p Fo(2)22 b(+)e(1)p Fk(=)p Fo(\()p Fk(l)r Fo(\()p Fk(k)s(;)15 b(i)p Fo(\))23 b Fh(\001)d Fk(p)p Fo(\()p Fk(k)s Fo(\)\).)291 2453 y(No)m(w,)42 b(w)m(e)e(m)m(ust)f(still)d(sho)m(w)j(ho)m(w)g(to)h (construct)f(the)g(h)m(ybrids)e Fk(m)2634 2420 y Fg(j)2709 2453 y Fo(and)i(pro)m(v)m(e)g(that)h(for)f(an)m(y)g Fk(j)5 b Fo(,)150 2566 y Fk(m)230 2533 y Fg(j)287 2566 y Fh(\010)19 b Fk(m)457 2533 y Fg(j)t Fi(+1)609 2566 y Fh(2)25 b(H)773 2533 y Fg(i)772 2594 y(k)815 2566 y Fo(.)291 2679 y(First,)e(w)m(e)e (will)e(sho)m(w)h(ho)m(w)i(to)f(h)m(ybridize)e(b)s(et)m(w)m(een)j(t)m (w)m(o)g(messages)g Fk(m)2660 2693 y Fi(0)2721 2679 y Fo(and)e Fk(m)2968 2693 y Fi(1)3028 2679 y Fo(whic)m(h)g(are)h (di\013eren)m(t)150 2792 y(in)31 b(only)g(1)h(bit,)g(with)f Fk(i)h Fo(di\013eren)m(t)g(h)m(ybrids.)43 b(Supp)s(ose)31 b Fk(\016)36 b Fo(is)31 b(the)h(index)f(of)h(the)g(bit)f(whic)m(h)g(is) g(di\013eren)m(t.)150 2905 y(Let)24 b Fk(I)346 2920 y Fg(\016)408 2905 y Fo(b)s(e)f(some)h(set)h(of)f Fk(i)g Fo(indices)e(other)i(than)f Fk(\016)s Fo(.)1830 2872 y Fi(1)1909 2905 y Fo(No)m(w,)j(let)e Fk(m)2341 2872 y Fi(0)2405 2905 y Fo(=)h Fk(m)2581 2919 y Fi(0)2621 2905 y Fo(,)g(and)e(to)i(get)g(from)e Fk(m)3379 2872 y Fg(j)3439 2905 y Fo(to)h Fk(m)3623 2872 y Fg(j)t Fi(+1)150 3017 y Fo(w)m(e)h(\015ip)f(bit)g Fk(\016)k Fo(and)d(all)e(the)j(bits)d (at)j(indices)d(in)h Fk(I)1789 3032 y Fg(\016)1852 3017 y Fo(except)h(the)h Fk(j)2325 2984 y Fg(th)2420 3017 y Fo(index)e(in)g Fk(I)2801 3032 y Fg(\016)2838 3017 y Fo(.)39 b(No)m(w,)27 b(in)d(h)m(ybrid)f Fk(m)3594 2984 y Fg(i)3646 3017 y Fo(w)m(e)150 3130 y(will)28 b(ha)m(v)m(e)j(\015ipp)s (ed)c(bit)i Fk(\016)34 b(i)d Fo(times,)f(an)g(o)s(dd)f(n)m(um)m(b)s(er) g(of)h(times,)g(so)h(bit)e Fk(\016)34 b Fo(will)27 b(b)s(e)j(\015ipp)s (ed)d(from)j Fk(m)3600 3097 y Fi(0)3669 3130 y Fo(to)150 3243 y Fk(m)230 3210 y Fg(i)258 3243 y Fo(.)41 b(F)-8 b(urthermore,)30 b(ev)m(ery)h(bit)e(in)g Fk(I)1403 3258 y Fg(\016)1471 3243 y Fo(will)f(b)s(e)i(\015ipp)s(ed)d Fk(i)20 b Fh(\000)g Fo(1)31 b(times)e(\(since)h(one)h(of)f(the)h (\015ips)d(excluded)150 3356 y(it\),)37 b(so)f(it)f(will)d(b)s(e)j (\015ipp)s(ed)e(an)i(ev)m(en)h(n)m(um)m(b)s(er)f(of)g(times,)i(and)e (th)m(us)g(all)f(those)i(bits)f(will)d(remain)j(the)150 3469 y(same)e(from)e Fk(m)676 3436 y Fi(0)748 3469 y Fo(to)i Fk(m)941 3436 y Fg(i)969 3469 y Fo(.)46 b(Finally)-8 b(,)31 b(ev)m(ery)i(other)f(bit)g(remains)f(the)h(same.)47 b(Th)m(us,)32 b Fk(m)3100 3436 y Fg(i)3156 3469 y Fo(=)c Fk(m)3335 3483 y Fi(1)3406 3469 y Fo(since)k Fk(m)3711 3483 y Fi(1)150 3582 y Fo(w)m(as)f(assumed)e(to)j(b)s(e)d(the)i(same)g (as)f Fk(m)1500 3596 y Fi(0)1570 3582 y Fo(in)f(ev)m(ery)i(bit)e (except)j(bit)d Fk(\016)s Fo(,)j(where)d(they)i(w)m(ere)g(di\013eren)m (t.)291 3695 y(Here)g(is)e(an)g(example)h(of)g(this)f(pro)s(cess.)40 b(Let)30 b Fk(m)1952 3709 y Fi(0)2017 3695 y Fo(=)25 b(01010110)33 b(and)d(let)f Fk(m)2893 3709 y Fi(1)2958 3695 y Fo(=)c(01110110,)34 b(and)29 b(let)150 3808 y Fk(i)d Fo(=)f(5.)41 b Fk(\016)29 b Fo(=)c(3,)31 b(and)e(let)i Fk(I)1028 3823 y Fg(\016)1091 3808 y Fo(=)25 b Fh(f)p Fo(1)p Fk(;)15 b Fo(2)p Fk(;)g Fo(4)p Fk(;)g Fo(5)p Fk(;)g Fo(6)p Fh(g)p Fo(.)45 b(Then,)30 b(the)g(h)m(ybrids)e(are:)990 4125 y Fk(m)1070 4087 y Fi(0)1192 4125 y Fo(=)83 b(01010110)990 4263 y Fk(m)1070 4225 y Fi(1)1192 4263 y Fo(=)g(01010110)24 b Fh(\010)c Fo(01111100)29 b(=)c(00101010)990 4401 y Fk(m)1070 4363 y Fi(2)1192 4401 y Fo(=)83 b(00101010)24 b Fh(\010)c Fo(10111100)29 b(=)c(10010110)990 4538 y Fk(m)1070 4501 y Fi(3)1192 4538 y Fo(=)83 b(10010110)24 b Fh(\010)c Fo(11101100)29 b(=)c(01111010)990 4676 y Fk(m)1070 4639 y Fi(4)1192 4676 y Fo(=)83 b(01111010)24 b Fh(\010)c Fo(11110100)29 b(=)c(10001110)990 4814 y Fk(m)1070 4777 y Fi(5)1192 4814 y Fo(=)83 b(10001110)24 b Fh(\010)c Fo(11111000)29 b(=)c(01110110)k(=)c Fk(m)2871 4828 y Fi(1)p 150 4875 1440 4 v 254 4929 a Fd(1)288 4961 y Fc(W)-6 b(e)25 b(kno)n(w)f(w)n(e)h(can)g(select)h Fb(I)1114 4970 y Fa(\016)1172 4961 y Fc(suc)n(h)f(that)f(no)h(index)f(in)h Fb(I)1945 4970 y Fa(\016)2003 4961 y Fc(is)g(greater)h(than)e Fb(k)j Fc(b)r(ecause)e(w)n(e)h(are)f(allo)n(w)n(ed)h(to)f(presume)150 5052 y(that)g Fb(k)f(>)d(i)p Fc(.)1889 5416 y Fo(8-3)p eop %%Page: 4 4 4 3 bop 291 159 a Fo(No)m(w,)45 b(let)c(us)f(b)s(e)h(more)g(general.)73 b(If)41 b Fk(m)1742 173 y Fi(0)1822 159 y Fo(and)f Fk(m)2089 173 y Fi(1)2170 159 y Fo(di\013er)f(in)h Fk(d)h Fo(bits,)i(where)e Fk(d)i Fh(\025)g Fo(1,)i(then)40 b(w)m(e)150 271 y(simply)30 b(rep)s(eat)j(this)e Fk(i)p Fo(-h)m(ybrid)g(pro)s(cess)h Fk(d)g Fo(times)g(to)h(c)m(hange)h(one)f(bit)e(at)i(a)g(time.)47 b(More)33 b(precisely)-8 b(,)32 b(if)150 384 y(\001)25 b(=)g Fh(f)p Fk(\016)432 398 y Fi(0)472 384 y Fk(;)15 b(:)g(:)g(:)i(;)e(\016)714 399 y Fg(d)p Fj(\000)p Fi(1)845 384 y Fh(g)29 b Fo(is)e(the)i(set)g(of)f(indices)f(in)g(whic)m(h)g Fk(m)2143 398 y Fi(0)2210 384 y Fo(and)h Fk(m)2465 398 y Fi(1)2533 384 y Fo(di\013er,)f(then,)i(for)f(eac)m(h)i Fk(\016)f Fh(2)c Fo(\001,)j(w)m(e)150 497 y(c)m(ho)s(ose)33 b(a)f(set)g Fk(I)700 512 y Fg(\016)770 497 y Fo(as)g(ab)s(o)m(v)m(e.)47 b(Then,)31 b(w)m(e)h(let)g Fk(m)1793 464 y Fi(0)1860 497 y Fo(=)27 b Fk(m)2038 511 y Fi(0)2078 497 y Fo(,)32 b(and)f(to)i(get)g(from)e Fk(m)2875 464 y Fg(si)p Fi(+)p Fg(j)3055 497 y Fo(to)i Fk(m)3248 464 y Fg(si)p Fi(+)p Fg(j)t Fi(+1)3517 497 y Fo(where)150 610 y(0)26 b Fh(\024)f Fk(j)31 b(<)25 b(i)p Fo(,)30 b(w)m(e)h(\015ip)e(bit)g Fk(\016)1039 624 y Fg(s)1107 610 y Fo(and)g(all)h(bits)f(at)i(indices)e (in)g Fk(I)2143 625 y Fg(\016)2174 633 y Fa(s)2242 610 y Fo(except)i(at)g(the)g Fk(j)2837 577 y Fg(th)2938 610 y Fo(index)e(from)h Fk(I)3438 625 y Fg(\016)3469 633 y Fa(s)3506 610 y Fo(.)291 723 y(Note)h(that)f(the)g(total)h(n)m(um)m (b)s(er)d(of)i(h)m(ybrids)d(is)i Fk(di)p Fo(,)h(whic)m(h)f(is)g(at)h (most)g Fk(k)s(i)p Fo(.)41 b(Since)29 b Fk(i)g Fo(is)g(constan)m(t,)i Fk(k)s(i)150 836 y Fo(is)e(p)s(olynomial)f(in)h Fk(k)s Fo(,)i(and)f(so)h Fk(l)r Fo(\()p Fk(k)s(;)15 b(i)p Fo(\))27 b Fh(\024)e Fk(k)s(i)p Fo(.)41 b(Th)m(us,)30 b(the)g(probabilit)m(y)e (that)j Fk(A)2844 851 y Fg(j)2873 860 y Fd(0)2908 851 y Fg(;k)3000 836 y Fo(succeeds)g(is)e(at)j(least)150 949 y(1)p Fk(=)p Fo(2)22 b(+)d(1)p Fk(=k)s(i)j Fh(\001)f Fk(p)p Fo(\()p Fk(k)s Fo(\),)31 b(whic)m(h)e(is)g(nonnegligible.)38 b(This)29 b(completes)h(the)h(pro)s(of.)291 1160 y Fp(Common)40 b(errors:)53 b Fo(The)36 b(most)g(common)h(error)e(in)g(this)g(section) i(w)m(as)f(for)g(p)s(eople)g(to)h(b)s(ecome)150 1273 y(confused)f(ab)s(out)g(cryptosystems)h(whic)m(h)f(handle)f(messages)i (longer)g(than)f(a)h(single)e(bit.)59 b Fp(When)42 b(a)150 1386 y(cryptosystem)31 b(is)g(sp)s(eci\014ed)h(that)e(handles)h(longer) h(messages,)f(y)m(ou)h(cannot)f(assume)f(it)h(is)150 1499 y(a)j(bit-b)m(y-bit)f(encryption)i(sc)m(heme.)40 b Fo(T)-8 b(rue,)29 b(this)g(do)s(es)g(mak)m(e)h(up)f(some)h (cryptosystems,)g(but)f(not)150 1612 y(all.)291 1725 y(Another)g(common)f(error)g(w)m(as)h(to)g(use)f(the)h(wrong)f (high-lev)m(el)f(argumen)m(t.)41 b(The)28 b(form)g(the)g(h)m(ybrid)150 1838 y(argumen)m(t)37 b Fm(should)i Fo(tak)m(e)f(is)e(something)g(lik)m (e)h(this.)59 b(\(1\))38 b(Assume)e(some)h(adv)m(ersary)h(can)f (distinguish)150 1951 y(b)s(et)m(w)m(een)30 b(t)m(w)m(o)g(distibutions) c(with)i(probabilit)m(y)f(nonnegligibly)f(b)s(etter)j(than)g(1/2.)42 b(\(2\))30 b(Pro)m(vide)f(some)150 2064 y(n)m(um)m(b)s(er)i(of)i(h)m (ybrids)e(where)h(the)h(\014rst)f(is)g(the)h(\014rst)f(distribution)d (and)j(the)h(last)g(is)e(the)i(last)g(distribu-)150 2177 y(tion.)42 b(\(3\))32 b(Argue)f(that)h(there)f(m)m(ust)g(b)s(e)f(some)i (pair)d(of)j(adjacen)m(t)g(h)m(ybrids)c(whic)m(h)i(the)h(adv)m(ersary)h (can)150 2290 y(distinguish)24 b(nonnegligibly)g(b)s(etter)j(than)h (1/2.)41 b(\(4\))28 b(Use)g(this,)f(regardless)g(of)h(whic)m(h)e(pair)g (of)i(adjacen)m(t)150 2402 y(h)m(ybrids)g(the)j(adv)m(ersary)f(can)h (distinguish,)c(to)k(pro)m(v)m(e)g(some)g(fact)g(y)m(ou)g(w)m(an)m(t.) 291 2515 y(Also,)40 b(some)e(p)s(eople)f(automatically)h(leapt)f(to)m (w)m(ards)i(using)d(an)i(induction)e(pro)s(of)h(\(presumably)150 2628 y(b)s(ecause)f(asking)f(to)h(pro)m(v)m(e)g(something)f(\\for)h (all)f(o)s(dd)f Fk(i)p Fo(")i(lo)s(oks)f(lik)m(e)g(it)g(needs)h(that)g (kind)e(of)h(thing\).)150 2741 y(Y)-8 b(ou)28 b(should)d(b)s(e)h(a)m(w) m(are)j(that)e(induction)e(is)h(just)h(one)g(to)s(ol)g(y)m(ou)h(can)f (use)g(to)h(pro)m(v)m(e)g(theorems,)g(but)f(y)m(ou)150 2854 y(shouldn't)32 b(automatically)j(assume)f(it's)g(the)g(w)m(a)m(y)i (to)f(go.)53 b(Sev)m(eral)35 b(p)s(eople)e(who)h(tried)f(to)i(solv)m(e) g(this)150 2967 y(problem)29 b(b)m(y)h(using)f(induction)f(ended)i(up)f (with)g(v)m(ery)i(confused,)f(incorrect)g(pro)s(ofs.)291 3179 y Fp(\(b\))g Fo(Pro)m(v)m(e)h(or)g(dispro)m(v)m(e:)40 b(F)-8 b(or)31 b(all)e(ev)m(en)i Fk(i)p Fo(,)g Fk(i)p Fo(-Moses)h(securit)m(y)e(is)g(equiv)-5 b(alen)m(t)29 b(to)j(GM-securit)m(y)-8 b(.)291 3390 y Fp(Solution:)38 b Fo(T)-8 b(o)24 b(dispro)m(v)m(e)g(this,)g(consider)f(the)h(case)h Fk(i)h Fo(=)f(2.)39 b(W)-8 b(e)25 b(will)c(pro)m(v)m(e)k(that)g (2-Moses)g(securit)m(y)150 3503 y(do)s(es)30 b(not)h(imply)d (GM-securit)m(y)-8 b(,)32 b(whic)m(h)d(will)e(pro)m(v)m(e)32 b(that)f(they)f(are)h(not)f(equiv)-5 b(alen)m(t.)291 3616 y(Supp)s(ose)34 b(w)m(e)i(ha)m(v)m(e)h(\()p Fk(G;)15 b(E)5 b(;)15 b(D)s Fo(\),)40 b(a)c(GM-secure)g(cryptosystem,)i(whic)m (h)d(op)s(erates)h(on)f Fk(k)s Fo(-bit)h(mes-)150 3729 y(sages.)50 b(Let)33 b(\()p Fk(G)698 3696 y Fj(0)722 3729 y Fk(;)15 b(E)834 3696 y Fj(0)858 3729 y Fk(;)g(D)976 3696 y Fj(0)1000 3729 y Fo(\))33 b(b)s(e)g(as)g(follo)m(ws,)g(where)g Fk(par)s Fo(\()p Fk(s)p Fo(\))f(is)g(the)i(parit)m(y)e(\(Hamming)h(w)m (eigh)m(t)h(mo)s(dulo)150 3842 y(2\))d(of)g(the)f(string)g Fk(s)p Fo(:)150 4053 y Fk(G)221 4020 y Fj(0)245 4053 y Fo(\(1)325 4020 y Fg(k)368 4053 y Fo(\):)355 4166 y(Return)f Fk(G)p Fo(\(1)815 4133 y Fg(k)859 4166 y Fo(\))150 4279 y Fk(E)222 4246 y Fj(0)246 4279 y Fo(\()p Fk(P)13 b(K)q(;)i(m)p Fo(\):)355 4392 y(Return)29 b Fk(E)5 b Fo(\()p Fk(P)13 b(K)q(;)i(m)p Fo(\))22 b Fh(\016)f Fk(par)s Fo(\()p Fk(m)p Fo(\))150 4504 y Fk(D)228 4472 y Fj(0)251 4504 y Fo(\()p Fk(S)5 b(K)q(;)15 b(c)22 b Fh(\016)f Fk(d)p Fo(\):)355 4617 y(Return)29 b Fk(D)s Fo(\()p Fk(S)5 b(K)q(;)15 b(c)p Fo(\))291 4828 y(It)29 b(is)f(clear)h(that)h(\()p Fk(G)996 4795 y Fj(0)1020 4828 y Fk(;)15 b(E)1132 4795 y Fj(0)1156 4828 y Fk(;)g(D)1274 4795 y Fj(0)1298 4828 y Fo(\))29 b(is)f(not)h(GM-secure.)41 b(The)29 b(adv)m(ersary)g(that)h(on)f(input) e Fk(c)17 b Fh(\016)h Fk(d;)d(m)3550 4842 y Fi(0)3590 4828 y Fk(;)g(m)3710 4842 y Fi(1)150 4941 y Fo(alw)m(a)m(ys)31 b(returns)e Fk(d)i Fo(will)c(alw)m(a)m(ys)k(b)s(e)f(correct)i(if)d Fk(par)s Fo(\()p Fk(m)2059 4955 y Fi(0)2098 4941 y Fo(\))c(=)g(0)31 b(and)f Fk(par)s Fo(\()p Fk(m)2760 4955 y Fi(1)2799 4941 y Fo(\))25 b(=)g(1.)291 5054 y(F)-8 b(urthermore,)32 b(\()p Fk(G)951 5021 y Fj(0)975 5054 y Fk(;)15 b(E)1087 5021 y Fj(0)1111 5054 y Fk(;)g(D)1229 5021 y Fj(0)1253 5054 y Fo(\))32 b(is)e(2-Moses)k(secure.)44 b(If)31 b Fk(m)2245 5068 y Fi(0)2316 5054 y Fo(and)g Fk(m)2574 5068 y Fi(1)2645 5054 y Fo(are)h(suc)m(h)f(that)i Fk(m)3283 5068 y Fi(0)3343 5054 y Fh(\010)21 b Fk(m)3515 5068 y Fi(1)3581 5054 y Fo(=)27 b(2,)150 5167 y(then)39 b(the)h(Hamming)f(w)m (eigh)m(t)h(of)f Fk(m)1447 5181 y Fi(1)1526 5167 y Fo(m)m(ust)g(b)s(e)g 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Fs(\(1)971 1821 y Fm(k)1015 1854 y Fs(\);)15 b Fo(m)25 b Fn( )h Fo(M)1400 1869 y Fm(k)1442 1854 y Fs(;)15 b Fo(c)1521 1868 y Fk(0)1587 1854 y Fn( )25 b Fo(E)5 b Fs(\()p Fo(P)13 b(K)q(;)i(m)p Fs(\);)g Fo(c)2193 1868 y Fk(1)2260 1854 y Fn( )26 b Fo(S)5 b Fs(\(1)2518 1821 y Fm(k)2561 1854 y Fo(;)15 b(P)e(K)7 b Fs(\);)273 1967 y Fo(b)25 b Fn( )g(f)p Fs(0)p Fo(;)15 b Fs(1)p Fn(g)p Fs(;)g Fo(b)752 1934 y Fj(0)803 1967 y Fn( )26 b Fo(A)988 1982 y Fm(k)1030 1967 y Fs(\()p Fo(P)13 b(K)q(;)i(c)1293 1982 y Fm(b)1329 1967 y Fo(;)g(m)p Fs(\))26 b(:)g Fo(b)1600 1934 y Fj(0)1648 1967 y Fs(=)f Fo(b)p Fs(])h Fn(\024)f Fs(1)p Fo(=)p Fs(2)c(+)f Fo(f)10 b Fs(\()p Fo(k)s Fs(\))291 2179 y(The)28 b(in)m(tuition)f(b)s(ehind)e (this)j(de\014nition)e(is)i(that)h(a)g(cryptosystem)g(is)e(secure)i(if) e(no)i(adv)m(ersary)g(can)150 2292 y(distinguish)j(the)37 b(encryption)e(of)h(a)h(random)e(message)j(from)d(a)i(random)f(sample)f (from)h(the)g(domain)150 2405 y(sampled)29 b(b)m(y)h Fo(S)5 b Fs(.)41 b(Pro)m(v)m(e)32 b(or)e(dispro)m(v)m(e)g(eac)m(h)h(of) g(the)f(follo)m(wing.)291 2618 y Ft(\(a\))g Fs(If)f Fo(M)632 2632 y Fm(n)705 2618 y Fs(=)c Fn(f)p Fs(0)p Fo(;)15 b Fs(1)p Fn(g)32 b Fs(for)f(ev)m(ery)g Fo(n)p Fs(,)f(then)g(GM-securit)m (y)h(implies)d(SD)i(securit)m(y)-8 b(.)291 2830 y Ft(\(c\))30 b Fs(If)g Fo(M)628 2844 y Fm(n)701 2830 y Fs(=)25 b Fn(f)p Fs(0)p Fo(;)15 b Fs(1)p Fn(g)1017 2797 y Fm(n)1096 2830 y Fs(for)30 b(ev)m(ery)h Fo(n)p Fs(,)f(then)g(GM-securit)m(y)i(implies) 27 b(SD)j(securit)m(y)-8 b(.)291 3043 y Ft(Solution:)41 b Fs(I)29 b(will)e(pro)m(v)m(e)j(these)g(t)m(w)m(o)g(parts)f(with)g (one)g(pro)s(of,)g(b)m(y)h(not)f(relying)f(on)h(the)h(size)f(of)h(the) 150 3156 y(message)e(space.)40 b(My)27 b(pro)s(of)f(will)e(rely)i(on)h (m)m(y)g(pro)s(of)f(for)g(problem)f(2;)k(I)e(will)d(pro)m(v)m(e)j(that) h(RM-securit)m(y)150 3269 y(implies)d(SD-securit)m(y)j(here,)g(and)f (then)h(pro)m(v)m(e)g(that)h(GM-securit)m(y)f(implies)d(RM-securit)m(y) j(in)f(problem)150 3382 y(2.)291 3594 y(Supp)s(ose)33 b(\()p Fo(G;)15 b(E)5 b(;)15 b(D)s Fs(\))38 b(is)c(not)h(SD)g(secure.) 55 b(Then,)35 b(for)g Fq(any)h Fs(sampling)d(algorithm)h Fo(S)5 b Fs(,)37 b(there)e(is)f(an)150 3707 y(adv)m(ersary)40 b Fo(A)g Fs(that)g(breaks)f(SD)h(securit)m(y)-8 b(.)69 b(So,)42 b(in)c(particular,)j(for)e(the)h Fo(S)k Fs(that)d(pic)m(ks)e (a)h(constan)m(t)150 3820 y(message)i Fo(m)590 3834 y Fk(0)671 3820 y Fs(and)e(encrypts)h(it)g(under)f Fo(P)13 b(K)7 b Fs(,)43 b(there)f(is)e(an)h(adv)m(ersary)h Fo(A)f Fs(whic)m(h)f(can)h(distinguish.)150 3933 y(No)m(w,)31 b(let)g(us)e(describ)s(e)g(an)h(adv)m(ersary)h Fo(A)1589 3900 y Fj(0)1643 3933 y Fs(to)g(break)f(RM)h(securit)m(y:)150 4145 y(1.)41 b(On)30 b(input)e Fo(P)13 b(K)7 b Fs(,)30 b(output)g Fo(\013)c Fs(=)f Fo(P)13 b(K)37 b Fs(and)30 b Fo(m)25 b Fs(=)g Fo(m)1986 4159 y Fk(0)2025 4145 y Fs(.)150 4258 y(2.)41 b(On)30 b(input)e Fo(\013)e Fs(=)f Fo(P)13 b(K)q(;)i(c;)g(m)1183 4272 y Fk(0)1224 4258 y Fo(;)g(r)s Fs(,)30 b(run)f Fo(A)i Fs(on)f(input)e(\()p Fo(P)13 b(K)q(;)i(c;)g(r)s Fs(\),)33 b(and)d(output)g(its)g(answ)m(er.) 291 4471 y(No)m(w)24 b(note)f(that)h Fo(A)f Fs(gets)g(as)h(input)c(a)k (randomly)d(c)m(hosen)i Fo(P)13 b(K)7 b Fs(,)25 b(and)d Fo(r)j Fs(whic)m(h)d(is)g(selected)h(at)h(random)150 4584 y(from)29 b Fo(M)452 4599 y Fm(k)495 4584 y Fs(,)h(and)f Fo(c)h Fs(whic)m(h)f(has)h(probabilit)m(y)d(1)p Fo(=)p Fs(2)k(of)f(b)s(eing)e(an)i(encryption)f(of)h Fo(r)s Fs(,)f(and)g(probabilit)m(y)f(1)p Fo(=)p Fs(2)150 4697 y(of)j(b)s(eing)e(an)i(encryption)e(of)i Fo(m)1262 4711 y Fk(0)1301 4697 y Fs(.)42 b(Ho)m(w)m(ev)m(er,)33 b(in)d(the)g(latter)h (case,)i(this)c(is)h(exactly)h(a)g(random)f(output)150 4810 y(of)h Fo(S)5 b Fs(\(1)395 4777 y Fm(k)438 4810 y Fo(;)15 b(P)e(K)7 b Fs(\).)43 b(Th)m(us,)30 b Fo(A)h Fs(will)d(b)s(e)i(correct)i(with)e(probabilit)m(y)e(at)k(least)f(1)p Fo(=)p Fs(2)21 b(+)g(1)p Fo(=pol)r(y)s Fs(,)31 b(so)g Fo(A)3430 4777 y Fj(0)3484 4810 y Fs(will)e(b)s(e)150 4922 y(correct)j(with)d(probabilit)m(y)f(at)j(least)f(1)p Fo(=)p Fs(2)22 b(+)e(1)p Fo(=pol)r(y)s Fs(,)31 b(b)s(ecause)f Fo(A)2414 4889 y Fj(0)2468 4922 y Fs(is)f(correct)j(exactly)f(when)e Fo(A)i Fs(is.)291 5135 y Ft(\(b\))f Fs(If)g Fo(M)640 5149 y Fm(n)712 5135 y Fs(=)25 b Fn(f)p Fs(0)p Fo(;)15 b Fs(1)p Fn(g)33 b Fs(for)d(ev)m(ery)h Fo(n)p Fs(,)f(then)g(SD)g (securit)m(y)h(implies)c(GM-securit)m(y)-8 b(.)1867 5416 y(10-1)p eop %%Page: 2 2 2 1 bop 291 159 a Ft(Solution:)39 b Fs(Supp)s(ose)25 b(\()p Fo(G;)15 b(E)5 b(;)15 b(D)s Fs(\))28 b(for)e(a)h(1-bit)g (message)g(space)g(is)f(not)g(GM)h(secure.)40 b(Then)25 b(there)i(is)150 271 y(an)e(adv)m(ersary)g Fo(A)h Fs(whic)m(h)e(breaks) g(GM)i(securit)m(y)-8 b(.)39 b(No)m(w,)27 b(there)f(are)f(t)m(w)m(o)i (cases)f(in)e(whic)m(h)g Fo(A)h Fs(can)g(op)s(erate)150 384 y(-)33 b(either)g(it)g(can)g(initially)c(c)m(ho)s(ose)35 b(0)e(and)g(1)g(as)g(its)g(t)m(w)m(o)h(messages)g(or)f(it)g(can)g(c)m (ho)s(ose)i(b)s(oth)d(messages)150 497 y(the)f(same.)42 b(Since)30 b(in)f(the)i(latter)g(case,)i Fo(A)p Fs('s)e(input)d(is)i (indep)s(enden)m(t)f(of)i(the)f(bit)g(it)h(is)e(trying)h(to)i(guess,) 150 610 y(the)e(probabilit)m(y)e(that)j Fo(A)f Fs(is)f(correct)i(is)e (1/2.)42 b(Th)m(us,)30 b(all)f(the)h(adv)-5 b(an)m(tage)32 b(is)d(when)g Fo(A)h Fs(pro)s(duces)f(0)i(and)150 723 y(1)g(as)f(its)g(t)m(w)m(o)i(messages.)42 b(Therefore,)30 b(this)f(m)m(ust)h(happ)s(en)f(with)g(nonnegligible)f(probabilit)m(y)-8 b(.)291 836 y(In)31 b(order)f(to)i(pro)m(v)m(e)g(that)g(\()p Fo(G;)15 b(E)5 b(;)15 b(D)s Fs(\))33 b(is)d(not)i(SD)f(secure,)g(w)m(e) h(need)f(to)h(pro)m(vide)e(an)h(adv)m(ersary)g Fo(A)3701 803 y Fj(0)3725 836 y Fs(,)150 949 y(giv)m(en)h(the)h(sampling)e (algorithm)g Fo(S)37 b Fs(that)c(w)m(e)g(w)m(an)m(t)h(to)f(use)f(to)h (distinguish.)43 b(Here)33 b(is)f(our)g(algorithm)150 1062 y Fo(A)218 1029 y Fj(0)241 1062 y Fs(:)150 1265 y Fo(A)218 1232 y Fj(0)241 1265 y Fs(\()p Fo(P)13 b(K)q(;)i(c;)g(m)p Fs(\):)150 1378 y(1.)41 b(Run)29 b Fo(A)i Fs(on)f(input)f Fo(P)13 b(K)37 b Fs(to)31 b(reco)m(v)m(er)h Fo(\013;)15 b(m)1712 1392 y Fk(0)1752 1378 y Fo(;)g(m)1872 1392 y Fk(1)1912 1378 y Fo(:)150 1491 y Fs(2.)41 b(If)30 b Fo(m)432 1505 y Fk(0)497 1491 y Fs(=)24 b Fo(m)672 1505 y Fk(1)712 1491 y Fs(,)30 b(output)g(a)h(random)f(bit.)150 1604 y(3.)41 b(Let)31 b Fo(b)463 1571 y Fj(0)512 1604 y Fs(=)25 b Fo(A)p Fs(\()p Fo(\013;)15 b(c)p Fs(\))p Fo(:)150 1719 y Fs(4.)41 b(If)30 b Fo(m)25 b Fs(=)g Fo(m)633 1733 y Fk(0)672 1719 y Fs(,)31 b(output)f Fo(b)1065 1686 y Fj(0)1088 1719 y Fs(,)h(otherwise)f(output)p 1846 1645 63 4 v 29 w Fo(b)1885 1693 y Fj(0)1909 1719 y Fs(.)291 1923 y(Let)i Fo(p)f Fs(b)s(e)f(the)i(probabilit)m(y)c(that)k(w)m(e)g(get)g(to)g (step)f(3)h(of)f(the)h(algorithm.)42 b(This)30 b(is)g(indep)s(enden)m (t)f(of)150 2036 y(the)i(correct)g(answ)m(er)f(since)g(the)h (computation)f(to)h(that)g(p)s(oin)m(t)e(dep)s(ends)g(only)g(on)i Fo(P)13 b(K)7 b Fs(.)291 2149 y(Giv)m(en)39 b(that)h(w)m(e)f(get)i(to)f (step)f(3,)j(the)d(probabilit)m(y)e(w)m(e)i(are)h(successful)e(can)h(b) s(e)g(represen)m(ted)g(as)150 2262 y Fo(p)196 2276 y Fm(m)262 2262 y Fo(=)p Fs(2)17 b(+)f Fo(p)502 2276 y Fm(s)539 2262 y Fo(=)p Fs(2,)30 b(where)e Fo(p)991 2276 y Fm(m)1085 2262 y Fs(is)f(the)i(probabilit)m(y)d(w)m(e)j(are)g (successful)e(giv)m(en)h(that)h Fo(c)g Fs(is)e(the)i(encryption)e(of) 150 2374 y Fo(m)p Fs(,)j(and)g Fo(p)508 2388 y Fm(s)575 2374 y Fs(is)f(the)i(probabilit)m(y)d(w)m(e)j(are)f(successful)g(giv)m (en)g(that)h Fo(c)g Fs(is)e(sampled)g(from)h Fo(S)5 b Fs(.)291 2487 y(If)35 b Fo(c)h Fs(is)e(an)h(encryption)g(of)g Fo(m)p Fs(,)i(then)e(the)g(input)f(w)m(e)i(giv)m(e)g(to)g Fo(A)f Fs(will)e(b)s(e)i(exactly)h(of)f(the)h(form)f Fo(A)150 2600 y Fs(exp)s(ects:)41 b Fo(\013)29 b Fs(and)g Fo(c)g Fs(where)g Fo(c)h Fs(is)e(the)h(encryption)g(of)g(either)g Fo(m)2307 2614 y Fk(0)2375 2600 y Fs(or)g Fo(m)2565 2614 y Fk(1)2605 2600 y Fs(.)2630 2567 y Fk(1)2709 2600 y Fs(Th)m(us,)g(the)h(probabilit)m(y)c(that)150 2713 y Fo(A)k Fs(will)e(giv)m(e)j(us)f(the)g(correct)i(answ)m(er)e(is)g Fo(p)1623 2727 y Fm(m)1714 2713 y Fs(=)25 b(1)p Fo(=)p Fs(2)d(+)e(1)p Fo(=pol)r(y)s Fs(.)291 2826 y(If)28 b Fo(c)g Fs(is)f(a)h(random)f(sample)g(from)g Fo(S)5 b Fs(,)29 b(then)e(the)h(input)e(w)m(e)j(giv)m(e)f(to)h Fo(A)f Fs(is)f Fq(indep)-5 b(endent)29 b Fs(of)f Fo(m)p Fs(.)40 b(So,)28 b Fo(m)150 2939 y Fs(could)h(b)s(e)h(generated)h (after)g Fo(A)f Fs(pro)s(duces)f(its)g(answ)m(er,)i(and)e(th)m(us,)h (whether)g(w)m(e)h(use)e Fo(b)3157 2906 y Fj(0)3211 2939 y Fs(or)p 3322 2865 V 30 w Fo(b)3361 2913 y Fj(0)3414 2939 y Fs(could)h(b)s(e)150 3052 y(c)m(hosen)g(after)f Fo(A)h Fs(runs.)38 b(Th)m(us,)29 b(the)g(probabilit)m(y)e Fo(A)1930 3019 y Fj(0)1982 3052 y Fs(is)h(correct)j(is)d(exactly)i Fo(p)2823 3066 y Fm(s)2885 3052 y Fs(=)24 b(1)p Fo(=)p Fs(2.)42 b(Therefore,)29 b(our)150 3165 y(total)i(probabilit)m(y)d(of)j (success)f(is)854 3437 y(1)21 b Fn(\000)f Fo(p)p 854 3477 203 4 v 933 3560 a Fs(2)1087 3498 y(+)g Fo(p)1239 3397 y Fi(\020)1303 3437 y Fo(p)1349 3451 y Fm(m)p 1303 3477 113 4 v 1336 3560 a Fs(2)1445 3498 y(+)1546 3437 y Fo(p)1592 3451 y Fm(s)p 1546 3477 83 4 v 1565 3560 a Fs(2)1639 3397 y Fi(\021)1776 3498 y Fs(=)1940 3437 y(1)g Fn(\000)g Fo(p)p 1940 3477 203 4 v 2018 3560 a Fs(2)2172 3498 y(+)g Fo(p)2319 3437 y Fs(1)p Fo(=)p Fs(2)h(+)f(1)p Fo(=pol)r(y)p 2319 3477 505 4 v 2548 3560 a Fs(2)2854 3498 y(+)g Fo(p)3001 3437 y Fs(1)p 3001 3477 46 4 v 3001 3560 a(4)1776 3714 y(=)1940 3652 y(1)p 1940 3693 V 1940 3776 a(2)2015 3714 y(+)2176 3652 y Fo(p)p 2116 3693 167 4 v 2116 3776 a(pol)r(y)291 3954 y Fs(and)43 b(since)h Fo(p)f Fs(is)g(nonnegligible,)i Fo(p=pol)r(y)h Fs(is)d(also)h (nonnegligible,)h(so)f Fo(A)2893 3921 y Fj(0)2960 3954 y Fs(breaks)g(SD)f(securit)m(y)-8 b(.)150 4067 y(Note)35 b(that)f(the)g(adv)m(ersary)f Fo(A)1216 4034 y Fj(0)1273 4067 y Fs(w)m(e)h(constructed)g(w)m(orks)f Fq(indep)-5 b(endent)35 b Fs(of)f(the)g(sim)m(ulator)e Fo(S)5 b Fs(.)50 b(This)32 b(is)150 4180 y(crucial,)j(since)f(if)g(it)h(only)f(w)m(ork)m (ed)h(for)g(certain)g(sim)m(ulators)e(it)i(w)m(ould)e(not)j(pro)m(v)m (e)f(that)h(\()p Fo(G;)15 b(E)5 b(;)15 b(D)s Fs(\))37 b(is)150 4293 y(not)31 b(SD-secure,)g(since)e(there)i(migh)m(t)f(b)s(e) g(a)h(sim)m(ulator)e(that)i(w)m(as)f(successful.)291 4500 y Ft(\(d\))g Fs(If)g Fo(M)640 4514 y Fm(n)712 4500 y Fs(=)25 b Fn(f)p Fs(0)p Fo(;)15 b Fs(1)p Fn(g)1028 4467 y Fm(n)1108 4500 y Fs(for)30 b(ev)m(ery)h Fo(n)p Fs(,)f(then)g(SD)g(securit)m(y)g(implies)e(GM-securit)m(y)-8 b(.)291 4706 y Ft(Solution:)59 b Fs(This)37 b(one)i(is)f(false.)67 b(The)39 b(problem)e(with)h(SD)h(securit)m(y)g(is)f(basically)f(that)j (it)f(only)150 4819 y(protects)33 b(us)e(from)h(the)g(encryption)f(of)h Fq(r)-5 b(andom)34 b Fs(messages)f(b)s(eing)e(break)-5 b(able.)45 b(If)31 b(w)m(e)h(set)h(up)e(a)h(cryp-)150 4932 y(tosystem)45 b(where)f(there)h(is)f(a)h(w)m(eak)g(message,)k(the) c(c)m(hance)h(that)f(w)m(e)g(pic)m(k)f(it)g(should)f(b)s(e)g(so)i(lo)m (w)p 150 4990 1440 4 v 254 5044 a Fh(1)288 5076 y Fg(Since)25 b Ff(m)556 5084 y Fh(0)612 5076 y Fe(6)p Fg(=)c Ff(m)761 5084 y Fh(1)820 5076 y Fg(w)n(e)26 b(kno)n(w)e Ff(m)d Fg(=)g Ff(m)1370 5084 y Fh(0)1430 5076 y Fg(or)k Ff(m)c Fg(=)g Ff(m)1761 5084 y Fh(1)1820 5076 y Fg(since)26 b Ff(M)2082 5085 y Fd(k)2142 5076 y Fg(=)21 b Fe(f)p Fg(0)p Ff(;)14 b Fg(1)p Fe(g)p Fg(.)35 b(This)25 b(is)h(the)f(k)n(ey)f (di\013erence)h(b)r(et)n(w)n(een)g(this)150 5167 y(part)h(and)f(1d,)h (where)g(implication)g(do)r(es)g(not)g(hold)1867 5416 y Fs(10-2)p eop %%Page: 3 3 3 2 bop 150 159 a Fs(that)34 b(the)g(sc)m(heme)g(w)m(ould)e(still)g(b)s (e)h(SD)g(secure.)50 b(So,)35 b(supp)s(ose)d(w)m(e)i(ha)m(v)m(e)h(\()p Fo(G;)15 b(E)5 b(;)15 b(D)s Fs(\))35 b(whic)m(h)e(is)f(a)i(GM-)150 271 y(secure)29 b(cryptosystem,)g(and)f(de\014ne)g(\()p Fo(G)1544 238 y Fj(0)1568 271 y Fo(;)15 b(E)1680 238 y Fj(0)1704 271 y Fo(;)g(D)1822 238 y Fj(0)1845 271 y Fs(\))29 b(where)f Fo(G)2241 238 y Fj(0)2290 271 y Fs(=)d Fo(G)p Fs(,)k(and)f(where)g Fo(E)3019 238 y Fj(0)3043 271 y Fs(\()p Fo(P)13 b(K)q(;)i Fs(1)3312 238 y Fm(k)3356 271 y Fs(\))26 b(=)f(0)17 b Fn(\016)g Fs(1)3682 238 y Fm(k)3725 271 y Fs(,)150 384 y Fo(E)222 351 y Fj(0)246 384 y Fs(\()p Fo(P)c(K)q(;)i(m)p Fs(\))26 b(=)f(0)c Fn(\016)g Fo(E)5 b Fs(\()p Fo(P)13 b(K)q(;)i(m)p Fs(\),)32 b(for)e(all)g(other)g Fo(m)p Fs(.)291 497 y(It)44 b(is)f(clear)h(that)g(\()p Fo(G)1055 464 y Fj(0)1079 497 y Fo(;)15 b(E)1191 464 y Fj(0)1215 497 y Fo(;)g(D)1333 464 y Fj(0)1357 497 y Fs(\))44 b(is)e(not)i(GM-secure.)82 b(The)43 b(adv)m(ersary)h(that)h (breaks)e(it)g(simply)150 610 y(outputs)c Fo(\013)j Fs(=)e Fo(";)15 b(m)866 624 y Fk(0)947 610 y Fs(=)41 b(0)1104 577 y Fm(k)1147 610 y Fo(;)15 b(m)1267 624 y Fk(1)1348 610 y Fs(=)40 b(1)1504 577 y Fm(k)1547 610 y Fs(,)j(and)c(if)g(its)g (input)f(is)g Fo(\013;)15 b Fs(0)28 b Fn(\016)f Fs(1)2670 577 y Fm(k)2753 610 y Fs(returns)39 b(1,)j(and)e(otherwise)150 723 y(returns)29 b(0.)291 836 y(Ho)m(w)m(ev)m(er,)35 b(it)c(is)g(SD-secure.)46 b(The)32 b Fo(S)37 b Fs(I)32 b(will)d(use)j(tak)m(es)h(a)g(particular)d(message)j Fo(m)f Fs(other)g(than)g(0)3707 803 y Fm(k)150 949 y Fs(or)f(1)307 916 y Fm(k)382 949 y Fs(and)g(outputs)g(an)g(encryption)f (of)i Fo(m)f Fs(under)f(the)i(k)m(ey)g Fo(P)13 b(K)7 b Fs(.)44 b(No)m(w,)32 b(if)f(there)g(is)g(an)g(adv)m(ersary)h Fo(A)150 1062 y Fs(that)i(breaks)f(SD)g(securit)m(y)g(under)f(this)g Fo(S)5 b Fs(,)34 b(that)g(adv)m(ersary)f(will)e(let)i(us)g(break)g(the) g(RM)h(securit)m(y)f(of)150 1175 y(\(G,)e(E,)g(D\).)g(W)-8 b(e)32 b(describ)s(e)c(the)j(adv)m(ersary)g Fo(A)1736 1142 y Fj(0)1789 1175 y Fs(as)g(follo)m(ws:)150 1387 y Fo(A)218 1354 y Fj(0)267 1387 y Fs(:)150 1500 y(1.)41 b(On)30 b(input)e Fo(P)13 b(K)7 b Fs(,)30 b(output)g Fo(\013)c Fs(=)f Fo(P)13 b(K)37 b Fs(and)30 b Fo(m)p Fs(.)150 1613 y(2.)41 b(On)30 b(input)e Fo(\013)e Fs(=)f Fo(P)13 b(K)q(;)i(c;)g(m;)g(r)30 b Fn(6)p Fs(=)25 b(1)1435 1580 y Fm(k)1478 1613 y Fs(,)30 b(output)g Fo(A)p Fs(\()p Fo(P)13 b(K)q(;)i Fs(1)22 b Fn(\016)f Fo(c;)15 b(r)s Fs(\).)150 1726 y(3.)41 b(On)30 b(input)e Fo(\013)e Fs(=)f Fo(P)13 b(K)q(;)i(c;)g(m;)g(r)30 b Fs(=)25 b(1)1435 1693 y Fm(k)1478 1726 y Fs(,)30 b(output)g(a)h(random)f(bit.)291 1939 y(The)f(imp)s(ortan)m(t)g(thing)g(to)h(notice)g(is)e(that)j Fo(A)e Fs(m)m(ust)h(ha)m(v)m(e)h(some)f(nonnegligible)c(adv)-5 b(an)m(tage)32 b(when)150 2052 y Fo(r)c Fn(6)p Fs(=)d(1)360 2019 y Fm(k)403 2052 y Fs(.)38 b(This)21 b(is)h(b)s(ecause)h(the)g(b)s (est)g(probabilit)m(y)d Fo(A)j Fs(can)h(attain)f(otherwise)f(is)g(1)p Fo(=)p Fs(2)3000 2019 y Fm(k)3050 2052 y Fs(+)5 b(\(1)g Fn(\000)g Fs(2)p Fo(=)p Fs(2)3422 2019 y Fm(k)3467 2052 y Fs(\)\(1)p Fo(=)p Fs(2)g(+)150 2164 y Fo(neg)s Fs(\),)32 b(whic)m(h)e(is)g(only)g(negligibly)e(b)s(etter)j(than)g(1/2.)1994 2131 y Fk(2)2077 2164 y Fs(Th)m(us,)f(the)h(probabilit)m(y)d(that)k Fo(A)3221 2131 y Fj(0)3275 2164 y Fs(is)e(successful)150 2277 y(giv)m(en)d(that)h Fo(r)g Fn(6)p Fs(=)c(0)787 2244 y Fm(k)858 2277 y Fs(or)j(1)1011 2244 y Fm(k)1081 2277 y Fs(is)f(at)i(least)f(1)p Fo(=)p Fs(2)14 b(+)g(1)p Fo(=pol)r(y)s Fs(,)29 b(so)e(our)g(probabilit)m(y)d(of)j(success)h(with)d(the)j(ab)s (o)m(v)m(e)150 2390 y(algorithm)h(is)447 2707 y(1)p Fo(=)p Fs(2)582 2670 y Fm(k)r Fk(+1)736 2707 y Fs(+)20 b(\(1)h Fn(\000)f Fs(1)p Fo(=)p Fs(2)1154 2670 y Fm(k)1198 2707 y Fs(\)\(1)p Fo(=)p Fs(2)i(+)e(1)p Fo(=pol)r(y)s Fs(\))84 b(=)f(1)p Fo(=)p Fs(2)21 b(+)f(1)p Fo(=pol)r(y)k Fs(+)c(1)p Fo(=)p Fs(2)2797 2670 y Fm(k)r Fk(+1)2951 2707 y Fn(\000)g Fs(1)p Fo(=)p Fs(2)3177 2670 y Fm(k)3241 2707 y Fn(\001)h Fo(pol)r(y)1892 2845 y Fn(\025)83 b Fs(1)p Fo(=)p Fs(2)21 b(+)f(\(1)h Fn(\000)f Fs(2)2530 2808 y Fj(\000)p Fm(k)2628 2845 y Fs(\))p Fo(=pol)r(y)291 3187 y Fs(whic)m(h)44 b(is)g(nonnegligibly)d(b)s(etter)k(than)g(1/2.)85 b(Th)m(us,)48 b(\()p Fo(G;)15 b(E)5 b(;)15 b(D)s Fs(\))47 b(w)m(ould)c(not)i(b)s(e)g (RM)g(secure.)150 3300 y(Ho)m(w)m(ev)m(er,)39 b(b)m(y)c(our)g(result)g (in)f(problem)f(2,)38 b(if)c(\()p Fo(G;)15 b(E)5 b(;)15 b(D)s Fs(\))38 b(is)c(not)i(RM)f(secure,)i(it)e(is)g(not)h(GM)g (secure,)150 3413 y(whic)m(h)42 b(is)h(a)g(con)m(tradiction.)80 b(Therefore,)47 b(with)42 b(the)h(giv)m(en)h Fo(S)5 b Fs(,)47 b(\()p Fo(G)2577 3380 y Fj(0)2601 3413 y Fo(;)15 b(E)2713 3380 y Fj(0)2737 3413 y Fo(;)g(D)2855 3380 y Fj(0)2878 3413 y Fs(\))44 b(is)f(SD)g(secure.)80 b(This)150 3526 y(concludes)30 b(the)g(pro)s(of.)150 3812 y Fp(Problem)45 b(2:)61 b(Random)45 b(Message)h(Securit)l(y)150 4015 y Fs(Consider)32 b(the)j(follo)m(wing)e(de\014nition)e(of)k(securit)m (y)-8 b(,)35 b(whic)m(h)e(w)m(e)i(call)f(random)f(message)i(securit)m (y)f(\(RM)150 4128 y(securit)m(y\).)k(W)-8 b(e)21 b(sa)m(y)g(that)g(a)g (cryptosystem)g(\()p Fo(G;)15 b(E)5 b(;)15 b(D)s Fs(\))22 b(on)e(a)h(sequence)g(of)f(message)i(spaces)f Fn(M)k Fs(=)g Fn(f)p Fo(M)3657 4142 y Fm(n)3705 4128 y Fn(g)150 4241 y Fs(is)k(RM)i(secure)f(if)150 4454 y Fn(8f)p Fo(A)314 4468 y Fm(n)361 4454 y Fn(g)h(9)p Fo(f)j Fn(2)25 b Fo(neg)34 b Fs(suc)m(h)c(that)h Fn(8)p Fo(k)d Fn(2)c Fl(N)150 4567 y Fs(Pr)o([\()p Fo(P)13 b(K)q(;)i(S)5 b(K)i Fs(\))27 b Fn( )f Fo(G)p Fs(\(1)971 4534 y Fm(k)1015 4567 y Fs(\);)15 b(\()p Fo(\013;)g(m)p Fs(\))27 b Fn( )e Fo(A)1549 4582 y Fm(k)1592 4567 y Fs(\()p Fo(P)13 b(K)q(;)i Fs(1)1861 4534 y Fm(k)1906 4567 y Fs(\);)g Fo(r)28 b Fn( )d Fo(M)2254 4582 y Fm(k)2297 4567 y Fs(;)273 4680 y Fo(c)312 4694 y Fk(0)377 4680 y Fn( )g Fo(E)5 b Fs(\()p Fo(P)13 b(K)q(;)i(m)p Fs(\);)g Fo(c)983 4694 y Fk(1)1050 4680 y Fn( )25 b Fo(E)5 b Fs(\()p Fo(P)13 b(K)q(;)i(r)s Fs(\);)g Fo(b)28 b Fn( )d(f)p Fs(0)p Fo(;)15 b Fs(1)p Fn(g)p Fs(;)273 4793 y Fo(b)312 4760 y Fj(0)360 4793 y Fn( )25 b Fo(A)544 4808 y Fm(k)587 4793 y Fs(\()p Fo(\013;)15 b(c)759 4808 y Fm(b)795 4793 y Fo(;)g(r)s Fs(\))26 b(:)f Fo(b)1029 4760 y Fj(0)1078 4793 y Fs(=)g Fo(b)p Fs(])g Fn(\024)g Fs(1)p Fo(=)p Fs(2)d(+)e Fo(f)10 b Fs(\()p Fo(k)s Fs(\))p 150 4854 1440 4 v 254 4910 a Fh(2)288 4942 y Fg(W)-6 b(e)24 b(get)g(this)h(expression)f(b)n (y)f(assuming)i(our)f(probabilit)n(y)g(of)h(success)g(if)g Ff(r)e Fg(=)e(1)2618 4910 y Fd(k)2681 4942 y Fg(is)j(1,)h(and)f(is)h(1) p Ff(=)p Fg(2)15 b(+)f Ff(neg)26 b Fg(otherwise.)1867 5416 y Fs(10-3)p eop %%Page: 4 4 4 3 bop 291 159 a Fs(The)35 b(in)m(tuition)f(b)s(ehind)f(this)h (de\014nition)g(is)h(that)h(for)f(an)m(y)h(\014xed)g(message)g Fo(m)g Fs(it)f(is)g(infeasible)e(to)150 271 y(distinguish)27 b(b)s(et)m(w)m(een)k(an)f(encryption)f(of)i Fo(m)f Fs(and)g(an)g (encryption)f(of)i(some)f(random)g(message)i Fo(r)s Fs(.)291 384 y(Pro)m(v)m(e)f(or)e(dispro)m(v)m(e)f(that)i(for)f(all)f(p)s (olynomials)f Fo(p)p Fs(,)i(for)g(the)g(sequence)h(of)f(message)i (spaces)e Fn(f)p Fo(M)3606 398 y Fm(n)3679 384 y Fs(=)150 497 y Fn(f)p Fs(0)p Fo(;)15 b Fs(1)p Fn(g)370 464 y Fm(p)p Fk(\()p Fm(n)p Fk(\))510 497 y Fn(g)p Fs(,)31 b(RM)f(securit)m(y)g(is)g (equiv)-5 b(alen)m(t)30 b(to)h(GM)g(securit)m(y)-8 b(.)291 699 y Ft(Solution:)41 b Fs(First)29 b(w)m(e)h(will)d(pro)m(v)m(e)k (that)f(GM)g(securit)m(y)g(implies)d(RM)j(securit)m(y)-8 b(.)41 b(Supp)s(ose)27 b(w)m(e)k(ha)m(v)m(e)150 812 y(an)j(adv)m (ersary)h Fo(A)f Fs(whic)m(h)f(breaks)h(the)h(RM)f(securit)m(y)h(of)f (a)h(cryptosystem)g(\()p Fo(G;)15 b(E)5 b(;)15 b(D)s Fs(\).)54 b(Consider)33 b(the)150 925 y(follo)m(wing)c(adv)m(ersary)h Fo(A)1014 892 y Fj(0)1038 925 y Fs(:)150 1121 y(1.)41 b(On)30 b(input)e Fo(P)13 b(K)7 b Fs(,)30 b(run)f Fo(A)p Fs(\()p Fo(P)13 b(K)7 b Fs(\))31 b(to)g(pro)s(duce)e Fo(\013;)15 b(m)p Fs(.)150 1234 y(2.)41 b(Generate)32 b Fo(r)c Fn( )d Fo(M)924 1249 y Fm(k)997 1234 y Fs(randomly)-8 b(.)150 1347 y(3.)41 b(Return)30 b Fo(\013)p Fs(,)h Fo(m)765 1361 y Fk(0)829 1347 y Fs(=)25 b Fo(m)p Fs(,)31 b(and)e Fo(m)1317 1361 y Fk(1)1382 1347 y Fs(=)c Fo(r)150 1460 y Fs(4.)41 b(On)30 b(input)e Fo(\013)713 1427 y Fj(0)737 1460 y Fo(;)15 b(c)p Fs(,)31 b(return)e Fo(A)p Fs(\()p Fo(\013;)15 b(c;)g(r)s Fs(\).)291 1656 y(W)-8 b(e)35 b(kno)m(w)f(the)h(input)d(to)j Fo(A)f Fs(is,)g(with)f(probabilit)m(y)f (1)p Fo(=)p Fs(2,)k(an)e(encryption)f(of)h Fo(m)p Fs(,)h(and)f(with)f (prob-)150 1769 y(abilit)m(y)f(1)p Fo(=)p Fs(2,)j(an)e(encryption)f(of) h(a)h(randomly)d(c)m(hosen)j(other)f(message)h Fo(r)s Fs(.)49 b(Th)m(us,)33 b Fo(A)g Fs(is)f(correct)i(with)150 1882 y(probabilit)m(y)f(1)p Fo(=)p Fs(2)24 b(+)g(1)p Fo(=pol)r(y)38 b Fs(when)c(w)m(e)i(call)e(it)h(in)f Fo(A)1993 1849 y Fj(0)2017 1882 y Fs(.)55 b(Ho)m(w)m(ev)m(er,)38 b Fo(A)2562 1849 y Fj(0)2621 1882 y Fs(is)c(correct)i(exactly)g(when)f Fo(A)g Fs(is)150 1995 y(correct.)40 b(Th)m(us,)25 b Fo(A)806 1962 y Fj(0)854 1995 y Fs(succeeds)g(with)e(probabilit)m(y)g(1)p Fo(=)p Fs(2)9 b(+)g(1)p Fo(=pol)r(y)28 b Fs(and)c(so)h(\()p Fo(G;)15 b(E)5 b(;)15 b(D)s Fs(\))27 b(is)d(not)h(GM-secure.)291 2196 y(Next,)j(supp)s(ose)23 b(\()p Fo(G;)15 b(E)5 b(;)15 b(D)s Fs(\))28 b(is)c(not)i(GM)g(secure,)g(and)f(that)h Fo(A)f Fs(is)g(an)g(adv)m(ersary)g(whic)m(h)f(breaks)h(GM)150 2309 y(securit)m(y)-8 b(.)41 b(Consider)28 b(the)j(follo)m(wing)e(adv)m (ersary)h Fo(A)1917 2276 y Fj(0)1941 2309 y Fs(:)150 2505 y(1.)41 b(On)30 b(input)e Fo(P)13 b(K)7 b Fs(,)30 b(run)f Fo(A)p Fs(\()p Fo(P)13 b(K)7 b Fs(\))31 b(to)g(pro)s(duce)e Fo(\013;)15 b(m)1991 2519 y Fk(0)2032 2505 y Fo(;)g(m)2152 2519 y Fk(1)2191 2505 y Fs(.)150 2618 y(2.)41 b(Pic)m(k)30 b(a)h(bit)f Fo(b)g Fs(at)h(random.)150 2731 y(3.)41 b(Return)30 b Fo(\013)629 2698 y Fj(0)678 2731 y Fs(=)25 b(\()p Fo(\013;)15 b(b)p Fs(\))32 b(and)d Fo(m)c Fs(=)g Fo(m)1470 2746 y Fm(b)1505 2731 y Fs(.)150 2844 y(4.)41 b(On)30 b(input)e Fo(\013)713 2811 y Fj(0)737 2844 y Fo(;)15 b(c;)g(r)s Fs(,)31 b(return)e Fo(A)p Fs(\()p Fo(\013;)15 b(c)p Fs(\))23 b Fn(\010)d Fo(b)p Fs(.)291 3040 y(In)30 b(order)g(to)h(pro)m(v)m(e)g (that)g(this)e(w)m(orks,)i(w)m(e)g(m)m(ust)f(de\014ne)g(the)g(follo)m (wing)f(three)h(probabilities:)252 3341 y Fo(P)310 3355 y Fk(0)432 3341 y Fs(=)83 b(Pr[\()p Fo(P)13 b(K)q(;)i(S)5 b(K)i Fs(\))27 b Fn( )e Fo(G)p Fs(\(1)1407 3304 y Fm(k)1451 3341 y Fs(\);)15 b(\()p Fo(\013;)g(m)1739 3355 y Fk(0)1780 3341 y Fo(;)g(m)1900 3355 y Fk(1)1940 3341 y Fs(\))25 b Fn( )g Fo(A)p Fs(\()p Fo(P)13 b(K)7 b Fs(\))26 b(:)g Fo(A)p Fs(\()p Fo(\013;)15 b(E)5 b Fs(\()p Fo(P)13 b(K)q(;)i(m)3063 3355 y Fk(0)3105 3341 y Fs(\)\))26 b(=)f(0])252 3479 y Fo(P)310 3493 y Fk(1)432 3479 y Fs(=)83 b(Pr[\()p Fo(P)13 b(K)q(;)i(S)5 b(K)i Fs(\))27 b Fn( )e Fo(G)p Fs(\(1)1407 3441 y Fm(k)1451 3479 y Fs(\);)15 b(\()p Fo(\013;)g(m)1739 3493 y Fk(0)1780 3479 y Fo(;)g(m)1900 3493 y Fk(1)1940 3479 y Fs(\))25 b Fn( )g Fo(A)p Fs(\()p Fo(P)13 b(K)7 b Fs(\);)15 b Fo(r)29 b Fn( )c Fo(M)2723 3494 y Fm(k)2791 3479 y Fs(:)h Fo(A)p Fs(\()p Fo(\013;)15 b(E)5 b Fs(\()p Fo(P)13 b(K)q(;)i(r)s Fs(\)\))29 b(=)c(0])252 3617 y Fo(P)310 3631 y Fk(2)432 3617 y Fs(=)83 b(Pr[\()p Fo(P)13 b(K)q(;)i(S)5 b(K)i Fs(\))27 b Fn( )e Fo(G)p Fs(\(1)1407 3579 y Fm(k)1451 3617 y Fs(\);)15 b(\()p Fo(\013;)g(m)1739 3631 y Fk(0)1780 3617 y Fo(;)g(m)1900 3631 y Fk(1)1940 3617 y Fs(\))25 b Fn( )g Fo(A)p Fs(\()p Fo(P)13 b(K)7 b Fs(\))26 b(:)g Fo(A)p Fs(\()p Fo(\013;)15 b(E)5 b Fs(\()p Fo(P)13 b(K)q(;)i(m)3063 3631 y Fk(1)3105 3617 y Fs(\)\))26 b(=)f(0])291 3805 y(The)e(idea)g(is)g(that)h Fo(P)989 3819 y Fk(0)1052 3805 y Fs(is)f(the)g(probabilit)m(y)e(that)j Fo(A)g Fs(outputs)f(zero)h(when)f(its)g(input)e(is)i(an)g(encryption) 150 3918 y(of)k Fo(m)330 3932 y Fk(0)369 3918 y Fs(;)h Fo(P)480 3932 y Fk(1)547 3918 y Fs(is)e(when)g(its)g(input)f(is)h(the)h (encryption)f(of)h(a)g(random)f(message,)j(and)d Fo(P)3026 3932 y Fk(2)3093 3918 y Fs(is)g(when)g(its)g(input)150 4031 y(is)i(an)h(encryption)f(of)h Fo(m)998 4045 y Fk(1)1037 4031 y Fs(.)41 b(The)28 b(probabilit)m(y)f(that)i Fo(A)g Fs(is)f(successful)g(is)g(1)p Fo(=)p Fs(2\(1)20 b(+)d Fo(P)3017 4045 y Fk(0)3074 4031 y Fn(\000)h Fo(P)3221 4045 y Fk(2)3260 4031 y Fs(\),)30 b(so)f(since)g Fo(A)150 4143 y Fs(is)c(successful)g(with)g(probabilit)m(y)f(nonnegligibly)e(b)s (etter)27 b(than)e(1)p Fo(=)p Fs(2,)k(w)m(e)e(kno)m(w)f(that)g Fo(P)3139 4157 y Fk(0)3191 4143 y Fn(\000)12 b Fo(P)3332 4157 y Fk(2)3396 4143 y Fn(\025)25 b Fs(1)p Fo(=p)p Fs(\()p Fo(k)s Fs(\))150 4256 y(for)30 b(some)h(p)s(olynomial)d Fo(p)p Fs(.)291 4369 y(The)i(probabilit)m(y)e(that)j Fo(A)1208 4336 y Fj(0)1262 4369 y Fs(is)e(successful)g(can)i(b)s(e)f (brok)m(en)g(do)m(wn)g(as)441 4670 y(Pr)o([)p Fo(S)619 4684 y Fm(RM)752 4670 y Fs(\()p Fo(k)s(;)15 b(A)945 4633 y Fj(0)969 4670 y Fs(\)])26 b(=)f(1)p Fo(=)p Fs(2)15 b(Pr)q([)p Fo(S)1481 4684 y Fm(RM)1613 4670 y Fs(\()p Fo(k)s(;)g(A)1806 4633 y Fj(0)1831 4670 y Fs(\))p Fn(j)p Fo(m)25 b Fs(=)g Fo(m)2172 4684 y Fk(0)2212 4670 y Fs(])20 b(+)g(1)p Fo(=)p Fs(2)15 b(Pr)q([)p Fo(S)2678 4684 y Fm(RM)2811 4670 y Fs(\()p Fo(k)s(;)g(A)3004 4633 y Fj(0)3028 4670 y Fs(\))p Fn(j)p Fo(m)26 b Fs(=)f Fo(m)3370 4684 y Fk(1)3409 4670 y Fs(])p Fo(;)291 4858 y Fs(Where)31 b Fo(S)638 4872 y Fm(RM)770 4858 y Fs(\()p Fo(k)s(;)15 b(A)963 4825 y Fj(0)988 4858 y Fs(\))30 b(is)g(the)g(ev)m(en)m(t)150 5054 y(\()p Fo(P)13 b(K)q(;)i(S)5 b(K)i Fs(\))27 b Fn( )e Fo(G)p Fs(\(1)848 5021 y Fm(k)892 5054 y Fs(\);)15 b(\()p Fo(\013;)g(m)p Fs(\))27 b Fn( )e Fo(A)1426 5021 y Fj(0)1450 5054 y Fs(\()p Fo(P)13 b(K)7 b Fs(\);)15 b Fo(b)26 b Fn( )f(f)p Fs(0)p Fo(;)15 b Fs(1)p Fn(g)p Fs(;)150 5167 y Fo(r)28 b Fn( )d Fo(M)423 5182 y Fm(k)466 5167 y Fs(;)15 b Fo(c)545 5181 y Fk(0)610 5167 y Fn( )25 b Fo(E)5 b Fs(\()p Fo(P)13 b(K)q(;)i(m)p Fs(\);)g Fo(c)1216 5181 y Fk(1)1284 5167 y Fn( )25 b Fo(E)5 b Fs(\()p Fo(P)13 b(K)q(;)i(r)s Fs(\))27 b(:)e Fo(A)1920 5134 y Fj(0)1944 5167 y Fs(\()p Fo(\013;)15 b(c)2116 5182 y Fm(b)2152 5167 y Fo(;)g(r)s Fs(\))25 b(=)g Fo(b)1867 5416 y Fs(10-4)p eop %%Page: 5 5 5 4 bop 291 159 a Fs(No)m(w,)32 b(what)e(is)g(Pr[)p Fo(S)1022 173 y Fm(RM)1154 159 y Fs(\()p Fo(k)s(;)15 b(A)1347 126 y Fj(0)1372 159 y Fs(\))p Fn(j)p Fo(m)26 b Fs(=)f Fo(m)1714 173 y Fk(0)1754 159 y Fs(]?)41 b(W)-8 b(ell,)31 b(if)f Fo(c)h Fs(is)e(the)i(encryption)f(of)h Fo(m)p Fs(,)g(w)m(e)g(are)g (correct)150 271 y(with)21 b(probabilit)m(y)f Fo(P)864 285 y Fk(0)904 271 y Fs(,)k(since)d(the)i(correct)g(answ)m(er)g(is)e (0.)38 b(If)22 b Fo(c)h Fs(is)e(the)i(encryption)e(of)h(a)h(random)f (message,)150 384 y(w)m(e)31 b(are)g(correct)g(with)e(probabilit)m(y)f (1)21 b Fn(\000)f Fo(P)1626 398 y Fk(1)1665 384 y Fs(.)41 b(Th)m(us,)30 b(this)f(probabilit)m(y)f(is)h(1)p Fo(=)p Fs(2\(1)23 b(+)d Fo(P)3107 398 y Fk(0)3167 384 y Fn(\000)f Fo(P)3315 398 y Fk(1)3355 384 y Fs(\).)291 497 y(Similarly)-8 b(,)27 b(Pr[)p Fo(S)874 511 y Fm(RM)1006 497 y Fs(\()p Fo(k)s(;)15 b(A)1199 464 y Fj(0)1224 497 y Fs(\))p Fn(j)p Fo(m)26 b Fs(=)f Fo(m)1566 511 y Fk(1)1605 497 y Fs(])g(=)g(1)p Fo(=)p Fs(2\(1)e(+)d Fo(P)2138 511 y Fk(1)2198 497 y Fn(\000)f Fo(P)2346 511 y Fk(2)2386 497 y Fs(\).)41 b(Therefore,)629 795 y(Pr[)p Fo(S)808 809 y Fm(RM)940 795 y Fs(\()p Fo(k)s(;)15 b(A)1133 758 y Fj(0)1158 795 y Fs(\)])83 b(=)g(1)p Fo(=)p Fs(2\(1)23 b(+)c Fo(P)1841 809 y Fk(0)1881 795 y Fo(=)p Fs(2)i Fn(\000)f Fo(P)2141 809 y Fk(1)2181 795 y Fo(=)p Fs(2\))i(+)d(1)p Fo(=)p Fs(2\(1)k(+)d Fo(P)2805 809 y Fk(1)2845 795 y Fo(=)p Fs(2)h Fn(\000)f Fo(P)3105 809 y Fk(2)3144 795 y Fo(=)p Fs(2\))1301 933 y(=)83 b(1)p Fo(=)p Fs(2\(1)23 b(+)c Fo(P)1841 947 y Fk(0)1881 933 y Fo(=)p Fs(2)i Fn(\000)f Fo(P)2141 947 y Fk(2)2181 933 y Fo(=)p Fs(2\))1301 1071 y(=)83 b(1)p Fo(=)p Fs(2)22 b(+)e(\()p Fo(P)1796 1085 y Fk(0)1856 1071 y Fn(\000)g Fo(P)2005 1085 y Fk(2)2045 1071 y Fs(\))p Fo(=)p Fs(2)1301 1209 y Fn(\025)83 b Fs(1)p Fo(=)p Fs(2)22 b(+)e(1)p Fo(=)p Fs(2)p Fo(p)p Fs(\()p Fo(k)s Fs(\))p Fo(;)291 1394 y Fs(since)33 b Fo(P)575 1408 y Fk(0)638 1394 y Fn(\000)22 b Fo(P)789 1408 y Fk(2)860 1394 y Fn(\025)31 b Fs(1)p Fo(=p)p Fs(\()p Fo(k)s Fs(\).)52 b(Th)m(us,)34 b Fo(A)1623 1361 y Fj(0)1681 1394 y Fs(breaks)f(RM)h(securit)m(y)-8 b(,)35 b(so)g(\()p Fo(G;)15 b(E)5 b(;)15 b(D)s Fs(\))35 b(is)e(not)i(RM)f(secure.)150 1507 y(Therefore,)c(RM)h(securit)m(y)f (is)f(equiv)-5 b(alen)m(t)30 b(to)h(GM)g(securit)m(y)-8 b(.)291 1620 y(T)g(o)32 b(state)g(this)e(last)h(argumen)m(t)h (di\013eren)m(tly)-8 b(,)30 b(w)m(e)i(are)f(making)g(a)g(h)m(ybrid)e (argumen)m(t.)43 b(Since)30 b Fo(A)i Fs(can)150 1733 y(distinguish)25 b(encryptions)i(of)h Fo(m)1276 1747 y Fk(0)1343 1733 y Fs(from)g(encryptions)f(of)i Fo(m)2224 1747 y Fk(1)2263 1733 y Fs(,)g(either)f Fo(A)g Fs(can)h(distinguish)24 b(encryptions)150 1846 y(of)j Fo(m)330 1860 y Fk(0)396 1846 y Fs(from)g(encryptions)e(of)i(a)h(random)e(message,)j(or)d Fo(A)h Fs(can)h(distinguish)23 b(encryptions)i(of)i(a)h(random)150 1958 y(message)40 b(from)e(encryptions)f(of)i Fo(m)1419 1972 y Fk(1)1458 1958 y Fs(.)65 b(W)-8 b(e)40 b(don't)e(kno)m(w)h(whic) m(h)e(of)i(these)g(is)e(true,)k(so)e(w)m(e)g(pic)m(k)f(one)150 2071 y(of)33 b(them)g(at)h(random)e(and)h(pro)s(ceed)g(as)g(if)f(the)h (adv)-5 b(an)m(tage)35 b(w)m(as)e(there.)49 b(In)33 b(the)g(end,)g (this)f(is)g(lik)m(e)h(the)150 2184 y(randomized)40 b(h)m(ybrid)e (argumen)m(t)k(from)e(problem)f(set)j(3)f(-)g(w)m(e)h(are)f(pic)m(king) e(the)j(index,)g(0)f(or)g(1,)j(at)150 2297 y(random.)150 2581 y Fp(Problem)h(3:)61 b(Securit)l(y)45 b(of)g(ElGamal-A)150 2784 y Fs(Assume)26 b(that)h(one)f(can,)i(in)d(probabilistic)e(p)s (olynomial)h(time,)j(compute,)h(on)e(input)e(1)3137 2751 y Fm(k)3180 2784 y Fs(,)k(a)f Fo(k)s Fs(-bit)e(prime)150 2896 y Fo(p)32 b Fs(suc)m(h)g(that)i Fo(p)28 b Fs(=)h(2)p Fo(q)c Fs(+)c(1)33 b(where)f Fo(q)j Fs(is)d(also)g(prime.)46 b(Consider)31 b(the)h(follo)m(wing)f(cryptosystem,)j(whic)m(h)150 3009 y(is)29 b(a)i(v)-5 b(arian)m(t)31 b(of)f(the)h(ElGamal)e (cryptosystem)i(\(to)h(b)s(e)d(attributed)h(later\),)h(call)f(it)f (ElGamal-A.)291 3122 y(Alice)24 b(selects)h(at)g(random)f(\(1\))h(a)g Fo(k)s Fs(-bit)f(prime)f Fo(p)i Fs(=)g(2)p Fo(q)12 b Fs(+)c(1)25 b(where)f Fo(q)j Fs(is)d(also)g(prime,)h(\(2\))g(an)f (elemen)m(t)150 3235 y Fo(g)35 b Fs(in)30 b Fo(Z)404 3202 y Fj(\003)397 3258 y Fm(p)474 3235 y Fs(of)i(order)f Fo(q)s Fs(,)g(and)g(\(3\))i(an)e Fo(x)g Fs(in)f Fl(Z)1626 3202 y Fj(\003)1626 3258 y Fm(q)1661 3235 y Fs(.)44 b(Then,)31 b(she)g(sets)h(her)f(public)d(k)m(ey)k(to)h(b)s(e)d(\()p Fo(p;)15 b(g)s(;)g(h)p Fs(\),)34 b(where)150 3348 y Fo(h)26 b Fs(=)f Fo(g)370 3315 y Fm(x)439 3348 y Fs(mo)s(d)g Fo(p)p Fs(,)30 b(and)g(her)g(secret)h(k)m(ey)g(to)g(b)s(e)f Fo(x)p Fs(.)291 3461 y(Alice's)22 b(public)d(k)m(ey)k(restricts)f(the)g (message)h(space)g(to)g(consist)e(of)i(all)e(elemen)m(ts)h(in)f Fo(Z)3197 3428 y Fj(\003)3190 3483 y Fm(p)3258 3461 y Fs(ha)m(ving)g(order)150 3574 y Fo(q)s Fs(.)52 b(T)-8 b(o)35 b(send)e(suc)m(h)h(a)h(message)g Fo(m)f Fs(to)h(Alice,)g(one)f (pic)m(ks)g(an)g Fo(r)g Fn( )e Fl(Z)2546 3588 y Fm(q)2613 3574 y Fs(and)i(computes)g(his)f(ciphertext)150 3687 y Fo(c)26 b Fs(=)f(\()p Fo(g)392 3654 y Fm(r)456 3687 y Fs(mo)s(d)f Fo(p;)15 b(h)793 3654 y Fm(r)832 3687 y Fo(m)25 b Fs(mo)s(d)f Fo(p)p Fs(\).)291 3800 y(T)-8 b(o)34 b(decrypt)f Fo(c)d Fs(=)g(\()p Fo(a;)15 b(b)p Fs(\),)36 b(Alice)c(computes)i Fo(b=)p Fs(\()p Fo(a)1997 3767 y Fm(x)2042 3800 y Fs(\))25 b(mo)s(d)g Fo(p)p Fs(.)49 b(\(Exercise:)e(v)m (erify)32 b(that)i(this)e(indeed)150 3913 y(pro)s(duces)d Fo(m)p Fs(.\))291 4026 y(Call)k(the)h(pro)s(cedure)f(for)h(generating)h Fo(p)f Fs(and)g Fo(g)j Fs(describ)s(ed)32 b(ab)s(o)m(v)m(e)k Fn(D)s(L)o Fs(.)53 b(The)33 b Fq(de)-5 b(cisional)38 b(Di\016e-)150 4138 y(Hel)5 b(lman)31 b Fs(assumption)e(is)g(as)i (follo)m(ws:)150 4332 y Fn(8f)p Fo(A)314 4346 y Fm(n)361 4332 y Fn(g)g(9)p Fo(f)j Fn(2)25 b Fo(neg)34 b Fs(suc)m(h)c(that)h Fn(8)p Fo(k)d Fn(2)c Fl(N)150 4445 y Fs(Pr)o([\()p Fo(p;)15 b(g)s Fs(\))27 b Fn( )f(D)s(L)o Fs(\(1)833 4412 y Fm(n)881 4445 y Fs(\);)15 b Fo(x)26 b Fn( )f Fl(Z)1215 4412 y Fj(\003)1215 4467 y Fm(q)1250 4445 y Fs(;)15 b Fo(y)28 b Fn( )e Fl(Z)1545 4412 y Fj(\003)1545 4467 y Fm(q)273 4559 y Fo(b)f Fn( )g(f)p Fs(0)p Fo(;)15 b Fs(1)p Fn(g)p Fs(;)g Fo(z)755 4573 y Fk(0)823 4559 y Fn( )25 b Fo(xy)j Fs(mo)s(d)c Fo(q)s Fs(;)15 b Fo(z)1389 4573 y Fk(1)1455 4559 y Fn( )25 b Fl(Z)1636 4526 y Fj(\003)1636 4582 y Fm(q)1671 4559 y Fs(;)15 b Fo(b)1750 4526 y Fj(0)1799 4559 y Fn( )25 b Fo(A)1983 4574 y Fm(k)2026 4559 y Fs(\()p Fo(p;)15 b(g)s(;)g(g)2279 4526 y Fm(x)2325 4559 y Fo(;)g(g)2411 4526 y Fm(y)2453 4559 y Fo(;)g(g)2539 4526 y Fm(z)2572 4538 y Fd(b)2609 4559 y Fs(\))25 b(:)h Fo(b)2759 4526 y Fj(0)2807 4559 y Fs(=)f Fo(b)p Fs(])h Fn(\024)f Fs(1)p Fo(=)p Fs(2)c(+)f Fo(f)10 b Fs(\()p Fo(k)s Fs(\))291 4753 y(where)30 b Fn(f)p Fo(A)667 4767 y Fm(n)715 4753 y Fn(g)g Fs(is)g(PPTF.)1156 4720 y Fk(3)p 150 4808 1440 4 v 254 4861 a Fh(3)288 4893 y Fg(In)22 b(other)g(w)n(ords,)j(it)d(is)h (hard,)g(giv)n(en)f(four)h(elemen)n(ts)f(of)h Fc(Z)1968 4861 y Fb(\003)1968 4906 y Fd(p)1998 4893 y Fg(,)h(\()p Ff(g)s(;)12 b(g)2183 4901 y Fh(1)2217 4893 y Ff(;)h(g)2288 4901 y Fh(2)2323 4893 y Ff(;)g(g)2394 4901 y Fh(3)2428 4893 y Fg(\),)23 b(eac)n(h)g(of)g(order)g Ff(q)s Fg(,)g(to)g(determine) e(whether)150 4985 y(the)28 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b(can)f(just)g(drop)f(the)i(\014rst)f(bit)f(of)h(that)h(preimage)f(to)h (get)h(a)f(preimage)f(of)150 1274 y Fh(F)221 1241 y Fe(0)244 1274 y Fl(\()p Fh(x)p Fl(\).)42 b(No)m(w,)31 b(though,)f(for)g(an)m(y)h Fh(x)p Fl(,)f Fh(F)13 b Fl(\(0)21 b Fg(\016)f Fh(x)p Fl(\))26 b(=)f Fh(F)13 b Fl(\(1)20 b Fg(\016)h Fh(x)p Fl(\))30 b(so)h Fh(P)13 b Fl(\()p Fh(F)g Fl(\(0)21 b Fg(\016)f Fh(x)p Fl(\)\))26 b(=)f Fh(P)13 b Fl(\()p Fh(F)g Fl(\(1)21 b Fg(\016)f Fh(x)p Fl(\)\))31 b(so)g Fh(P)i Fg(\016)20 b Fh(F)150 1387 y Fl(is)29 b(not)i(one-to-one.)291 1600 y Fm(\(e\))f Fl(Is)g Fh(G)20 b Fg(\016)h Fh(P)43 b Fl(a)31 b(PR)m(G?)291 1812 y Fm(Solution:)65 b Fl(Y)-8 b(es.)79 b(Since)41 b Fh(P)56 b Fl(preserv)m(es)42 b(length,)k(and)41 b Fh(G)i Fl(increases)g(length,)i Fh(G)29 b Fg(\016)g Fh(P)55 b Fl(is)42 b(clearly)150 1925 y(length-increasing.)48 b(F)-8 b(urthermore,)35 b(since)e Fh(P)46 b Fl(is)33 b(one-to-one,)j(the)e(distribution)c Fh(P)13 b Fl(\()p Fh(s)p Fl(\))33 b(for)h(random)e Fh(s)150 2038 y Fl(is)g(exactly)i(the) g(same)f(as)h(the)f(distribution)d(of)j(random)f Fh(s)p Fl(.)49 b(Therefore,)34 b Fh(G)22 b Fg(\016)h Fh(P)46 b Fl(is)32 b(indistinguishable)150 2151 y(from)h Fh(G)p Fl(,)i(so)f(if)e Fh(G)23 b Fg(\016)g Fh(P)46 b Fl(can)34 b(b)s(e)f(distinguished)d(from)j(random,)h(then)f Fh(G)h Fl(can)g(b)s(e)f(distinguished)c(from)150 2264 y(random,)h(whic)m(h)f (w)m(ould)g(b)s(e)h(a)h(con)m(tradiction.)40 b(Th)m(us,)30 b Fh(G)20 b Fg(\016)h Fh(P)43 b Fl(is)30 b(a)h(PR)m(G.)291 2476 y Fm(\(f)10 b(\))30 b Fl(Is)g Fh(G)20 b Fg(\016)h Fh(F)43 b Fl(a)31 b(PR)m(G?)291 2689 y Fm(Solution:)60 b Fl(Not)41 b(necessarily)-8 b(.)68 b(Supp)s(ose)38 b Fh(F)1892 2656 y Fe(0)1955 2689 y Fl(is)h(a)h(O)m(WF)g(and)f Fh(G)2658 2656 y Fe(0)2722 2689 y Fl(is)g(a)h(PR)m(G.)g(Then,)i Fh(F)13 b Fl(\()p Fh(x)p Fl(\))41 b(=)150 2802 y(0)21 b Fg(\016)g Fh(F)353 2769 y Fe(0)377 2802 y Fl(\()p Fh(x)p Fl(\))31 b(is)g(a)g(O)m(WF)h(\(if)f(w)m(e)g(can)h(\014nd)d(preimages)i (of)h Fh(F)13 b Fl(\()p Fh(x)p Fl(\))31 b(then)g(w)m(e)h(can)f(\014nd)f (preimages)h(of)g Fh(F)3604 2769 y Fe(0)3627 2802 y Fl(\()p Fh(x)p Fl(\))150 2915 y(b)m(y)g(prep)s(ending)d(a)k(zero\).)45 b(Also,)31 b Fh(G)p Fl(\()p Fh(b)22 b Fg(\016)f Fh(x)p Fl(\))27 b(=)f Fh(b)21 b Fg(\016)g Fh(G)1965 2882 y Fe(0)1989 2915 y Fl(\()p Fh(x)p Fl(\))32 b(is)e(a)i(PR)m(G.)g(Clearly)e Fh(G)h Fl(is)g(length-increasing)150 3028 y(since)g Fh(G)445 2995 y Fe(0)500 3028 y Fl(is.)42 b(If)31 b Fh(G)h Fl(is)e (distinguishable)d(from)k(random,)g(then)g(if)f(w)m(e)i(are)g(giv)m(en) f(an)g(input)e(of)j(the)f(form)150 3141 y Fh(G)221 3108 y Fe(0)245 3141 y Fl(\()p Fh(x)p Fl(\))k(for)f(random)g Fh(x)g Fl(and)g(w)m(e)h(prep)s(end)d(a)j(random)e(bit,)i(w)m(e)g(ha)m (v)m(e)g(a)g(string)f(of)g(the)h(form)f Fh(G)p Fl(\()p Fh(x)p Fl(\))h(for)150 3254 y(random)28 b Fh(x)p Fl(.)40 b(F)-8 b(urthermore,)30 b(if)e(w)m(e)i(are)f(giv)m(en)g(a)h(random)e (string)g(and)h(w)m(e)g(prep)s(end)e(a)j(random)e(bit,)g(w)m(e)150 3366 y(still)f(ha)m(v)m(e)j(a)f(random)f(string.)39 b(Th)m(us,)28 b(if)g Fh(A)h Fl(can)g(distinguish)c Fh(G)k Fl(from)f(random,)h(and)f (w)m(e)h(are)g(giv)m(en)g(an)150 3479 y(input)g(string)i Fh(r)s Fl(,)g(w)m(e)h(can)g(prep)s(end)d(a)j(random)e(bit)h(to)h Fh(r)s Fl(,)f(and)g(pass)g(this)f(to)i Fh(A)p Fl(,)g(and)f(b)s(e)g (correct)h(with)150 3592 y(probabilit)m(y)i(nonnegligibly)f(b)s(etter)k (than)f(1/2.)60 b(This)35 b(pro)m(v)m(es)i(that)g Fh(G)2675 3559 y Fe(0)2735 3592 y Fl(is)e(not)i(a)g(PR)m(G,)g(whic)m(h)e(is)h(a) 150 3705 y(con)m(tradiction.)291 3818 y(Ho)m(w)m(ev)m(er,)d Fh(G)21 b Fg(\016)g Fh(F)44 b Fl(alw)m(a)m(ys)31 b(starts)g(with)f(0.) 42 b(Th)m(us,)30 b(if)g(w)m(e)h(guess)g(\\pseudorandom")f(whenev)m(er)g (our)150 3931 y(input)e(starts)j(with)e(0)i(w)m(e)g(will)d(b)s(e)h (correct)j(3)p Fh(=)p Fl(4)g(of)e(the)h(time,)f(so)h Fh(G)20 b Fg(\016)h Fh(F)43 b Fl(is)30 b(not)g(a)h(PR)m(G.)150 4217 y Fi(Problem)45 b(2:)61 b(Hard-core)45 b(bits)150 4420 y Fl(\(20)32 b(p)s(oin)m(ts\))e(Let)h Fh(b)26 b Fl(:)g Fg(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)1115 4387 y Fe(\003)1183 4420 y Fg(!)26 b(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)33 b Fl(b)s(e)d(an)g(e\016cien)m(tly)h(computable,)g(non)m (trivial)d(predicate)j(\(with)150 4533 y(resp)s(ect)f(to)h(sampling)e (with)g(the)h(uniform)f(distribution\).)291 4746 y Fm(\(a\))34 b Fl(Pro)m(v)m(e)i(that)g(if)e(O)m(WP's)h(exist,)h(there)g(is)e(a)h(O)m (WP)g Fh(P)46 b Fl(:)33 b Fg(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)2628 4713 y Fe(\003)2703 4746 y Fg(!)33 b(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)3047 4713 y Fe(\003)3123 4746 y Fl(suc)m(h)35 b(that)h Fh(b)f Fl(is)f(a)150 4859 y(hard-core)f (predicate)g(of)g Fh(P)13 b Fl(.)48 b(Here)34 b(y)m(ou)f(ma)m(y)h (assume)e(that)i Fh(b)f Fl(is)f(of)h(the)g(form)g Fh(b)3000 4873 y Fd(r)3038 4859 y Fl(\()p Fh(x)p Fl(\))d(=)f Fh(r)24 b Fg(\001)f Fh(x)i Fl(mo)s(d)f(2,)150 4972 y(where)30 b Fg(j)p Fh(r)s Fg(j)25 b Fl(=)g Fg(j)p Fh(x)p Fg(j)p Fl(.)1867 5416 y(13-2)p eop %%Page: 3 3 3 2 bop 291 159 a Fm(Solution:)45 b Fl(Supp)s(ose)31 b Fh(P)1172 126 y Fe(0)1228 159 y Fl(is)g(a)i(O)m(WP)-8 b(,)33 b(and)f(that)h Fh(b)2089 174 y Fd(r)2123 155 y Fc(0)2182 159 y Fl(is)e(a)i(hard-core)g(predicate)f(of)h Fh(P)3333 126 y Fe(0)3356 159 y Fl(.)47 b(Let)33 b Fh(i)g Fl(b)s(e)150 271 y(the)j(index)e(of)h(the)h(\014rst)f(one)h(of)f Fh(r)1349 238 y Fe(0)1372 271 y Fl(,)i(and)e(let)g Fh(j)41 b Fl(b)s(e)35 b(the)h(index)e(of)i(the)f(\014rst)g(one)h(of)g Fh(r)s Fl(.)55 b(Consider)34 b(the)150 384 y(follo)m(wing)29 b(p)s(erm)m(utation)g Fh(\031)s Fl(:)150 597 y Fh(\031)s Fl(\()p Fh(x)292 611 y Ff(1)352 597 y Fg(\016)21 b Fh(:)15 b(:)g(:)21 b Fg(\016)g Fh(x)662 612 y Fd(k)704 597 y Fl(\):)150 710 y(1.)41 b(On)30 b(input)e Fh(x)707 724 y Ff(1)767 710 y Fg(\016)20 b Fh(:)15 b(:)g(:)22 b Fg(\016)e Fh(x)1076 725 y Fd(k)1119 710 y Fl(,)30 b(let)h Fh(x)1358 677 y Fe(0)1358 737 y Fd(l)1409 710 y Fl(=)25 b Fh(x)1557 725 y Fd(l)1613 710 y Fl(for)30 b(all)g Fh(l)d Fg(6)p Fl(=)e Fh(i;)15 b(j)5 b Fl(.)150 823 y(2.)41 b(Let)31 b Fh(x)476 790 y Fe(0)476 848 y Fd(j)538 823 y Fl(=)25 b Fh(x)686 837 y Fd(i)714 823 y Fl(.)150 940 y(3.)41 b(If)30 b Fh(b)391 954 y Fd(r)429 940 y Fl(\()p Fh(x)p Fl(\))c(=)f Fh(b)712 955 y Fd(r)746 936 y Fc(0)772 940 y Fl(\()p Fh(x)859 907 y Fe(0)859 964 y Ff(1)919 940 y Fg(\016)c Fh(:)15 b(:)g(:)21 b Fg(\016)g Fh(x)1229 907 y Fe(0)1229 966 y Fd(i)p Fe(\000)p Ff(1)1367 940 y Fg(\016)g Fl(0)g Fg(\016)f Fh(x)1616 907 y Fe(0)1616 966 y Fd(i)p Ff(+1)1755 940 y Fg(\016)g Fh(:)15 b(:)g(:)22 b Fg(\016)e Fh(x)2064 907 y Fe(0)2064 968 y Fd(k)2107 940 y Fl(\),)31 b(let)f Fh(x)2381 907 y Fe(0)2381 966 y Fd(i)2435 940 y Fl(=)24 b(0.)150 1053 y(4.)41 b(Otherwise,)29 b(let)i Fh(x)900 1020 y Fe(0)900 1078 y Fd(i)953 1053 y Fl(=)25 b(1.)41 b(Output)29 b Fh(x)1535 1020 y Fe(0)1559 1053 y Fl(.)291 1266 y Fh(\031)34 b Fl(is)29 b(clearly)h (length-preserving.)39 b(T)-8 b(o)30 b(pro)m(v)m(e)i(that)f Fh(\031)i Fl(is)c(one-to-one,)k(w)m(e)e(sho)m(w)f(ho)m(w)h(to)g(in)m(v) m(ert)f(it:)150 1478 y Fh(\031)205 1445 y Fe(\000)p Ff(1)299 1478 y Fl(\()p Fh(x)386 1445 y Fe(0)386 1502 y Ff(1)446 1478 y Fg(\016)21 b Fh(:)15 b(:)g(:)21 b Fg(\016)g Fh(x)756 1445 y Fe(0)756 1506 y Fd(k)799 1478 y Fl(\):)150 1591 y(1.)41 b(On)30 b(input)e Fh(x)707 1558 y Fe(0)707 1615 y Ff(1)767 1591 y Fg(\016)20 b Fh(:)15 b(:)g(:)22 b Fg(\016)e Fh(x)1076 1558 y Fe(0)1076 1619 y Fd(k)1119 1591 y Fl(,)30 b(let)h Fh(x)1358 1606 y Fd(l)1409 1591 y Fl(=)25 b Fh(x)1557 1558 y Fe(0)1557 1619 y Fd(l)1613 1591 y Fl(for)30 b(all)g Fh(l)d Fg(6)p Fl(=)e Fh(i;)15 b(j)5 b Fl(.)150 1704 y(2.)41 b(Let)31 b Fh(x)476 1718 y Fd(i)529 1704 y Fl(=)25 b Fh(x)677 1671 y Fe(0)677 1729 y Fd(j)714 1704 y Fl(.)150 1821 y(3.)41 b(If)30 b Fh(b)391 1837 y Fd(r)425 1818 y Fc(0)451 1821 y Fl(\()p Fh(x)538 1788 y Fe(0)562 1821 y Fl(\))c(=)f Fh(b)758 1835 y Fd(r)796 1821 y Fl(\()p Fh(x)883 1835 y Ff(1)942 1821 y Fg(\016)c Fh(:)15 b(:)g(:)21 b Fg(\016)g Fh(x)1252 1835 y Fd(j)t Fe(\000)p Ff(1)1399 1821 y Fg(\016)g Fl(0)f Fg(\016)h Fh(x)1648 1835 y Fd(j)t Ff(+1)1795 1821 y Fg(\016)f Fh(:)15 b(:)g(:)22 b Fg(\016)e Fh(x)2104 1836 y Fd(k)2147 1821 y Fl(\),)31 b(let)f Fh(x)2421 1835 y Fd(j)2483 1821 y Fl(=)25 b(0.)150 1934 y(4.)41 b(Otherwise,)29 b(let)i Fh(x)900 1948 y Fd(j)961 1934 y Fl(=)25 b(1.)42 b(Output)29 b Fh(x)p Fl(.)291 2147 y(F)-8 b(urthermore,)40 b(note)f(that)g Fh(b)1307 2161 y Fd(r)1345 2147 y Fl(\()p Fh(x)p Fl(\))g(=)f Fh(b)1654 2162 y Fd(r)1688 2143 y Fc(0)1714 2147 y Fl(\()p Fh(\031)s Fl(\()p Fh(x)p Fl(\)\).)66 b(No)m(w,)41 b(if)c Fh(P)13 b Fl(\()p Fh(x)p Fl(\))39 b(=)f Fh(P)2797 2114 y Fe(0)2820 2147 y Fl(\()p Fh(\031)2910 2114 y Fe(\000)p Ff(1)3005 2147 y Fl(\()p Fh(x)p Fl(\)\))h(then)f(if)f Fh(A)i Fl(can)150 2260 y(reco)m(v)m(er)29 b Fh(b)497 2274 y Fd(r)535 2260 y Fl(\()p Fh(x)p Fl(\))g(giv)m(en)e Fh(P)13 b Fl(\()p Fh(x)p Fl(\),)29 b(this)e(is)f(the)i(same)g(as)g(reco)m(v)m(ering)h Fh(b)2387 2274 y Fd(r)2425 2260 y Fl(\()p Fh(\031)2515 2227 y Fe(\000)p Ff(1)2610 2260 y Fl(\()p Fh(x)p Fl(\)\))f(giv)m(en)g Fh(P)3101 2227 y Fe(0)3124 2260 y Fl(\()p Fh(\031)3214 2227 y Fe(\000)p Ff(1)3309 2260 y Fl(\()p Fh(x)p Fl(\)\),)h(whic)m(h) 150 2373 y(is)k(the)i(same)g(as)f(reco)m(v)m(ering)i Fh(b)1229 2388 y Fd(r)1263 2369 y Fc(0)1289 2373 y Fl(\()p Fh(x)p Fl(\))f(giv)m(en)f Fh(P)1758 2340 y Fe(0)1781 2373 y Fl(\()p Fh(x)p Fl(\),)j(whic)m(h)c(w)m(e)h(assumed)g(to)h(b)s(e) f(hard.)51 b(Th)m(us,)35 b Fh(b)3537 2387 y Fd(r)3609 2373 y Fl(is)f(a)150 2485 y(hard-core)d(predicate)f(of)g Fh(P)13 b Fl(.)291 2598 y(T)-8 b(o)38 b(sho)m(w)g(that)g Fh(P)51 b Fl(is)37 b(a)h(p)s(erm)m(utation,)h(w)m(e)f(need)g(only)f (observ)m(e)h(that)g(the)g(comp)s(osition)f(of)h(t)m(w)m(o)150 2711 y(p)s(erm)m(utations)27 b(is)h(a)h(p)s(erm)m(utation.)39 b(Since)28 b Fh(b)1698 2725 y Fd(r)1764 2711 y Fl(is)g(a)h(hard-core)f (predicate,)h Fh(P)42 b Fl(m)m(ust)28 b(b)s(e)g(a)h(O)m(WP)-8 b(,)30 b(as)e(w)m(e)150 2824 y(pro)m(v)m(e)j(b)s(elo)m(w.)40 b(This)29 b(completes)h(the)h(pro)s(of.)291 3037 y Fm(\(b\))h Fl(Pro)m(v)m(e)h(that)g(if)e Fh(P)41 b Fl(:)29 b Fg(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)1378 3004 y Fe(\003)1448 3037 y Fg(!)28 b(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)1787 3004 y Fe(\003)1861 3037 y Fl(is)31 b(an)h(e\016cien)m(tly)g (computable)f(p)s(erm)m(utation,)h(and)g Fh(b)150 3150 y Fl(is)h(a)i(hard-core)g(predicate)f(of)h Fh(P)13 b Fl(,)35 b(then)f Fh(P)48 b Fl(is)33 b(a)i(O)m(WP)-8 b(.)35 b(Also,)h(pro)m(v)m(e)f(that)g(if)e Fh(F)46 b Fl(:)32 b Fg(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)3292 3117 y Fe(\003)3365 3150 y Fg(!)32 b(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)3708 3117 y Fe(\003)150 3263 y Fl(and)30 b Fh(b)p Fl(\()p Fh(x)p Fl(\))h(is)e(hard)h(to)h(compute)f(giv)m(en)h(only)e(access)j (to)f Fh(F)13 b Fl(\()p Fh(x)p Fl(\),)31 b(then)f Fh(F)44 b Fl(is)29 b(not)i(necessarily)e(a)i(O)m(WF.)291 3475 y Fm(Solution:)51 b Fl(Supp)s(ose)34 b Fh(A)i Fl(can)g(\014nd)e Fh(x)h Fl(giv)m(en)g Fh(P)13 b Fl(\()p Fh(x)p Fl(\).)57 b(Then,)36 b(the)g(follo)m(wing)e(algorithm)h(will)d(giv)m(e)150 3588 y Fh(b)p Fl(\()p Fh(x)p Fl(\))f(with)e(probabilit)m(y)f (non-negligbly)g(b)s(etter)j(than)f(1)p Fh(=)p Fl(2)c(:)150 3801 y Fh(A)218 3768 y Fe(0)241 3801 y Fl(\()p Fh(P)13 b Fl(\()p Fh(x)p Fl(\)\):)150 3913 y(1.)41 b(Compute)30 b Fh(x)707 3880 y Fe(0)756 3913 y Fg( )25 b Fh(A)p Fl(\()p Fh(P)13 b Fl(\()p Fh(x)p Fl(\)\).)150 4026 y(2.)41 b(If)30 b Fh(P)13 b Fl(\()p Fh(x)p Fl(\))26 b(=)f Fh(P)13 b Fl(\()p Fh(x)825 3993 y Fe(0)848 4026 y Fl(\),)31 b(return)f Fh(b)p Fl(\()p Fh(x)1344 3993 y Fe(0)1367 4026 y Fl(\).)150 4139 y(3.)41 b(Otherwise,)29 b(return)h(a)g(random)g(bit.)291 4352 y(Th)m(us,)f(whenev)m(er)h Fh(A)f Fl(is)g(correct,)i(w)m(e)f(are)g (correct)h(with)d(probabilit)m(y)f(1.)41 b(F)-8 b(urthermore,)30 b(whenev)m(er)150 4465 y Fh(A)41 b Fl(is)f(not)h(correct,)k(w)m(e)d (are)f(correct)h(with)e(probabilit)m(y)e(1/2.)74 b(Th)m(us,)43 b(if)d Fh(A)h Fl(is)f(correct)i Fh(\017)p Fl(\()p Fh(k)s Fl(\))g(of)f(the)150 4578 y(time,)29 b(w)m(e)g(are)h(correct)g Fh(\017)p Fl(\()p Fh(k)s Fl(\))17 b(+)g(\(1)h Fg(\000)f Fh(\017)p Fl(\()p Fh(k)s Fl(\)\))p Fh(=)p Fl(2)27 b(=)e(1)p Fh(=)p Fl(2)19 b(+)d Fh(\017)p Fl(\()p Fh(k)s Fl(\))p Fh(=)p Fl(2)31 b(of)e(the)g(time,)g(whic)m(h)e(is)h(non-negligible.)150 4691 y(Therefore,)35 b(it)e(is)g(not)h(hard)f(to)i(\014nd)d Fh(b)p Fl(\()p Fh(x)p Fl(\))j(giv)m(en)f Fh(P)13 b Fl(\()p Fh(x)p Fl(\),)35 b(so)f(w)m(e)h(ha)m(v)m(e)g(a)f(con)m(tradiction.)51 b(Th)m(us,)34 b Fh(P)13 b Fl(\()p Fh(x)p Fl(\))150 4803 y(m)m(ust)30 b(b)s(e)g(one-w)m(a)m(y)-8 b(.)43 b(Since)29 b(w)m(e)i(w)m(ere)g(giv)m(en)f(that)h Fh(P)43 b Fl(is)30 b(a)h(p)s(erm)m(utation,)e Fh(P)44 b Fl(is)29 b(a)i(O)m(WP)-8 b(.)291 4916 y(F)g(or)22 b(the)e(second)h(part,)i(note)e(that)g(since)f Fh(P)34 b Fl(is)20 b(a)h(p)s(erm)m(utation)f(is)f(guaran)m(tees)j(that) g(if)d Fh(P)13 b Fl(\()p Fh(x)p Fl(\))26 b(=)f Fh(P)13 b Fl(\()p Fh(x)3691 4883 y Fe(0)3715 4916 y Fl(\))150 5029 y(then)36 b Fh(b)p Fl(\()p Fh(x)p Fl(\))f(=)g Fh(b)p Fl(\()p Fh(x)791 4996 y Fe(0)814 5029 y Fl(\),)j(since)e Fh(x)e Fl(=)h Fh(x)1385 4996 y Fe(0)1408 5029 y Fl(.)58 b(The)36 b(k)m(ey)-8 b(,)38 b(therefore,)h(is)c(that)h(there)h(ma)m(y)f (b)s(e)g Fj(many)h Fh(x)3461 4996 y Fe(0)3520 5029 y Fl(whic)m(h)150 5142 y(map)25 b(to)h(the)g(same)f(v)-5 b(alue)25 b(as)h Fh(F)13 b Fl(\()p Fh(x)p Fl(\))26 b(but)e(whic)m(h)g (do)h(not)h(corresp)s(ond)e(to)i(the)g(same)f Fh(b)p Fl(\()p Fh(x)p Fl(\).)40 b(F)-8 b(or)26 b(example,)1867 5416 y(13-3)p eop %%Page: 4 4 4 3 bop 150 159 a Fl(let)31 b Fh(b)p Fl(\()p Fh(x)p Fl(\))g(return)e (the)i(\014rst)f(bit)g(of)g(the)h(string)f Fh(x)p Fl(.)41 b(Then,)30 b(the)h(function)e Fh(F)13 b Fl(\()p Fh(x)2810 173 y Ff(0)2870 159 y Fg(\016)21 b Fh(x)p Fl(\))26 b(=)f Fh(x)31 b Fl(where)f Fh(x)3543 173 y Ff(0)3613 159 y Fl(is)g(a)150 271 y(bit)g(mak)m(es)i(it)f(hard)f(to)i(compute)f Fh(b)p Fl(\()p Fh(x)p Fl(\))h(giv)m(en)f Fh(F)13 b Fl(\()p Fh(x)p Fl(\).)43 b(Actually)-8 b(,)32 b(it)e(mak)m(es)i(it)f(imp)s (ossible)d({)j(applying)150 384 y Fh(F)43 b Fl(destro)m(ys)31 b(all)f(information)e(ab)s(out)j Fh(b)p Fl(.)40 b(Ho)m(w)m(ev)m(er,)33 b(this)d Fh(F)43 b Fl(is)29 b(not)i(a)g(O)m(WF)g(-)g(giv)m(en)f Fh(F)13 b Fl(\()p Fh(x)p Fl(\),)32 b(the)e(v)-5 b(alue)150 497 y(0)27 b Fg(\016)f Fh(F)13 b Fl(\()p Fh(x)p Fl(\))40 b(is)e(a)i(preimage.)67 b(Another)39 b(elegan)m(t)i(example)e(is)f Fh(F)13 b Fl(\()p Fh(x)p Fl(\))41 b(=)f(0)f(for)g(all)g(inputs.)65 b(No)m(w,)42 b Fj(any)150 610 y Fl(non)m(trivial)28 b(predicate)j(is)e (hard)g(to)i(compute)g(but)f Fh(F)43 b Fl(is)30 b(not)g(a)h(O)m(WF.)291 823 y Fm(Common)38 b(errors:)49 b Fl(Man)m(y)35 b(p)s(eople)e(made)h (the)h(error)e(that)i(to)g(reco)m(v)m(er)h Fh(b)p Fl(\()p Fh(x)p Fl(\))f(giv)m(en)g Fh(P)13 b Fl(\()p Fh(x)p Fl(\),)36 b(if)d Fh(A)150 936 y Fl(in)m(v)m(erts)39 b Fh(P)52 b Fl(with)37 b(nonnegligible)g(probabilit)m(y)-8 b(,)39 b(then)g Fh(b)p Fl(\()p Fh(A)p Fl(\()p Fh(P)13 b Fl(\()p Fh(x)p Fl(\)\)\))41 b(returns)c(the)j(correct)g(v)-5 b(alue)38 b(with)150 1049 y(probabilit)m(y)d(nonnegligibly)g(b)s(etter) j(than)f(1/2.)64 b(Ho)m(w)m(ev)m(er,)42 b(while)36 b(it)h(is)g(true)h (that)g(if)f Fh(A)h Fl(is)f(correct,)150 1161 y(then)26 b(this)f(will)e(giv)m(e)k(the)f(righ)m(t)g(answ)m(er,)h(it)f(do)s(es)g Fj(not)h Fl(guaran)m(tee)h(that)e(if)f Fh(A)i Fl(is)e(incorrect,)i (then)f(w)m(e)h(will)150 1274 y(get)32 b(the)f(righ)m(t)f(answ)m(er)h (with)f(probabilit)m(y)e(1/2.)44 b(Imagine)30 b(that)h Fh(A)2469 1241 y Fe(0)2524 1274 y Fl(is)f(an)g(adv)m(ersary)h(that)h (can)f(in)m(v)m(ert)150 1387 y Fh(P)13 b Fl(\()p Fh(x)p Fl(\))31 b(with)e(probabilit)m(y)f Fh(\017)p Fl(\()p Fh(k)s Fl(\).)41 b(No)m(w,)32 b(supp)s(ose)d Fh(A)h Fl(w)m(orks)g(as)h (follo)m(ws:)150 1600 y Fh(A)p Fl(\()p Fh(P)13 b Fl(\()p Fh(x)p Fl(\)\):)150 1713 y(1.)41 b(Compute)30 b Fh(x)707 1680 y Fe(0)756 1713 y Fg( )25 b Fh(A)940 1680 y Fe(0)963 1713 y Fl(\()p Fh(P)13 b Fl(\()p Fh(x)p Fl(\)\).)150 1826 y(2.)41 b(If)30 b Fh(P)13 b Fl(\()p Fh(x)510 1793 y Fe(0)534 1826 y Fl(\))25 b(=)g Fh(P)13 b Fl(\()p Fh(x)p Fl(\),)31 b(then)f(with)g(probabilit)m(y)d(1/4,)32 b(return)e Fh(x)2341 1793 y Fe(0)2364 1826 y Fl(.)150 1946 y(3.)41 b(With)30 b(probabilit)m(y)e(3/4,)k(return)d(a)i(v)-5 b(alue)30 b Fh(y)j Fl(suc)m(h)d(that)h Fh(b)p Fl(\()p Fh(x)2348 1913 y Fe(0)2371 1946 y Fl(\))26 b(=)p 2528 1867 158 4 v 25 w Fh(b)p Fl(\()p Fh(y)s Fl(\))q(.)150 2059 y(4.)41 b(Otherwise,)29 b(return)h(a)g(random)g(v)-5 b(alue)30 b Fh(y)s Fl(.)291 2271 y(No)m(w,)f(with)c(probabilit)m(y)g Fh(\017)p Fl(\()p Fh(k)s Fl(\))p Fh(=)p Fl(4,)30 b(w)m(e)d(are)h (correct,)h(with)c(probabilit)m(y)g(3)p Fh(\017)p Fl(\()p Fh(k)s Fl(\))p Fh(=)p Fl(4,)30 b(w)m(e)d(are)h(de\014nitely)150 2384 y(wrong,)j(and)g(with)e(probabilit)m(y)g(\(1)22 b Fg(\000)e Fh(\017)p Fl(\()p Fh(k)s Fl(\)\))32 b(w)m(e)g(are)f (correct)i(with)c(probabilit)m(y)g(1/2,)k(whic)m(h)d(adds)g(up)150 2497 y(to)d(a)g(probabilit)m(y)e(of)h(1)p Fh(=)p Fl(2)13 b Fg(\000)g Fh(\017)p Fl(\()p Fh(k)s Fl(\))p Fh(=)p Fl(2)29 b({)e(w)m(e)h(are)f(actually)f(wrong)g(more)h(than)f(w)m(e)h(are)g (righ)m(t)g(in)e(this)h(case.)150 2610 y(This)33 b(is)g(wh)m(y)i(it)f (is)f(imp)s(ortan)m(t)h(to)i(c)m(hec)m(k)g(that)f(our)f(answ)m(er)h (from)f(the)g(adv)m(ersary)h(is)f Fj(c)-5 b(orr)g(e)g(ct)36 b Fl(b)s(efore)150 2723 y(w)m(e)31 b(use)f(that)h(to)g(compute)g Fh(b)p Fl(.)150 3009 y Fi(Problem)45 b(3:)61 b(O)l(WP)45 b Fb(\))f Fi(PR)l(G)150 3212 y Fl(\(20)35 b(p)s(oin)m(ts\))f(Let)h Fh(P)44 b Fl(:)32 b Fg(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)1169 3179 y Fe(\003)1243 3212 y Fg(!)31 b(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)1585 3179 y Fe(\003)1661 3212 y Fl(b)s(e)34 b(a)g(O)m(WP)-8 b(,)36 b(and)d(let)h Fh(b)h Fl(b)s(e)e(a)i(hard-core)f (predicate)g(of)h Fh(P)13 b Fl(.)150 3325 y(Consider)28 b(the)h(follo)m(wing)f(construction,)i(due)f(to)h(Blum)e(and)h(Micali,) g(of)h(a)g(PR)m(G.)g(T)-8 b(o)30 b(generate)h(a)f(k)m(ey)150 3438 y Fh(P)13 b(K)7 b Fl(,)30 b(just)g(use)g(the)h(k)m(ey)g (generation)g(algorithm)e(for)h Fh(P)13 b Fl(.)41 b(Let)31 b Fh(s)f Fl(denote)g(the)h(input,)e(where)h Fg(j)p Fh(s)p Fg(j)25 b Fl(=)g Fh(k)s Fl(.)291 3551 y(Let)31 b Fh(x)506 3565 y Ff(0)571 3551 y Fl(=)25 b Fh(s)p Fl(,)30 b(and)g(let)g Fh(x)1125 3565 y Fd(i)1178 3551 y Fl(=)25 b Fh(P)13 b Fl(\()p Fh(x)1432 3565 y Fd(i)p Fe(\000)p Ff(1)1551 3551 y Fl(\))31 b(for)f(all)f Fh(i)d(>)e Fl(0.)42 b(De\014ne)1230 3777 y Fh(G)p Fl(\()p Fh(s)p Fl(\))26 b(=)f Fh(b)p Fl(\()p Fh(s)g Fl(=)g Fh(x)1826 3791 y Ff(0)1866 3777 y Fl(\))20 b Fg(\016)h Fh(b)p Fl(\()p Fh(x)2113 3791 y Ff(1)2153 3777 y Fl(\))f Fg(\016)h Fh(:)15 b(:)g(:)21 b Fg(\016)g Fh(b)p Fl(\()p Fh(x)2592 3792 y Fd(k)2635 3777 y Fl(\))291 4043 y Fm(\(a\))30 b Fl(Pro)m(v)m(e)h(that)g Fh(G)g Fl(is)e Fj(next-bit)k(unpr)-5 b(e)g(dictable)p Fl(.)42 b(That)30 b(is,)g(that)150 4256 y Fg(8f)p Fh(A)p Fg(g)359 4270 y Fd(n)437 4256 y Fg(9)p Fh(f)k Fg(2)25 b Fh(neg)34 b Fg(8)p Fh(k)150 4369 y Fl(Pr)o([\()p Fh(\013;)15 b(j)5 b Fl(\))28 b Fg( )d Fh(A)694 4384 y Fd(k)737 4369 y Fl(;)15 b Fh(s)25 b Fg( )h(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)1182 4336 y Fd(k)1227 4369 y Fl(;)273 4482 y Fh(r)28 b Fg( )d Fh(G)p Fl(\()p Fh(s)p Fl(\);)15 b Fh(b)26 b Fg( )f Fh(A)931 4497 y Fd(k)974 4482 y Fl(\()p Fh(\013;)15 b(r)1148 4496 y Ff(1)1209 4482 y Fg(\016)21 b Fh(:)15 b(:)g(:)21 b Fg(\016)g Fh(r)1508 4496 y Fd(j)1544 4482 y Fl(\))26 b(:)f Fh(b)h Fl(=)f Fh(r)1857 4496 y Fd(j)t Ff(+1)1983 4482 y Fl(])h Fg(\024)f Fl(1)p Fh(=)p Fl(2)c(+)f Fh(f)10 b Fl(\()p Fh(k)s Fl(\))291 4694 y(where)30 b Fh(r)e Fl(=)d Fh(r)760 4708 y Ff(1)819 4694 y Fg(\016)c Fh(r)926 4708 y Ff(2)986 4694 y Fg(\016)f Fh(:)15 b(:)g(:)22 b Fg(\016)e Fh(r)1284 4709 y Fd(k)r Ff(+1)1417 4694 y Fl(.)291 4907 y Fm(Solution:)53 b Fl(The)36 b(structure)g(of)g(the)h(argumen)m(t)g(w) m(e)g(will)c(use)j(is)g(that)h(if)e Fh(G)i Fl(is)e(not)i(next-bit)f (un-)150 5020 y(predictable,)31 b(then)g Fh(b)h Fl(is)f(not)h(a)g (hard-core)f(predicate)h(of)g Fh(P)13 b Fl(.)44 b(Giv)m(en)32 b(a)g(v)-5 b(alue)31 b Fh(P)13 b Fl(\()p Fh(x)p Fl(\))32 b(w)m(e)h(can)f(compute)150 5133 y Fh(x)202 5147 y Ff(1)281 5133 y Fl(=)40 b Fh(P)13 b Fl(\()p Fh(x)p Fl(\),)43 b Fh(x)705 5147 y Ff(2)784 5133 y Fl(=)d Fh(P)13 b Fl(\()p Fh(x)1053 5147 y Ff(1)1093 5133 y Fl(\),)42 b(and)c(so)i(on,)i(to)e (compute)f(the)h(bit)e(v)-5 b(alues)39 b Fh(b)p Fl(\()p Fh(x)2883 5147 y Ff(1)2922 5133 y Fl(\))p Fh(;)15 b(b)p Fl(\()p Fh(x)3123 5147 y Ff(2)3164 5133 y Fl(\))p Fh(;)g(:)g(:)g(:)r Fl(.)67 b(Ho)m(w)40 b(do)1867 5416 y(13-4)p eop %%Page: 5 5 5 4 bop 150 159 a Fl(w)m(e)36 b(use)g(this)f(abilit)m(y?)56 b(W)-8 b(ell,)37 b(w)m(e)f(ha)m(v)m(e)i(an)d(adv)m(ersary)i(whic)m(h,)f (giv)m(en)g(the)g(\014rst)f Fh(j)42 b Fl(bits)35 b(of)h Fh(G)p Fl(\()p Fh(s)p Fl(\))h(can)150 271 y(compute)28 b(the)g Fh(j)22 b Fl(+)15 b(1)860 238 y Fd(st)922 271 y Fl(.)40 b(So)28 b(one)g(basic)g(idea)f(is)g(that)i(w)m(e)f(can)h(mak) m(e)g(a)f(string)f Fh(b)15 b Fg(\016)h Fh(b)p Fl(\()p Fh(x)3075 285 y Ff(1)3115 271 y Fl(\))g Fg(\016)g Fh(:)f(:)g(:)h Fg(\016)g Fh(b)p Fl(\()p Fh(x)3535 286 y Fd(k)3578 271 y Fl(\))28 b(for)150 384 y(a)h(random)f Fh(b)p Fl(.)40 b(If)29 b Fh(b)c Fl(=)g Fh(b)p Fl(\()p Fh(x)p Fl(\))k(then)g(this)f (string)f(is)h(the)h(output)g(of)g Fh(G)g Fl(on)g(input)e Fh(x)p Fl(.)40 b(Th)m(us,)28 b(if)g(w)m(e)h(giv)m(e)g(our)150 497 y(adv)m(ersary)i(the)g(\014rst)f Fh(j)36 b Fl(bits)30 b(of)h(this)e(string,)h(w)m(e)i(exp)s(ect)f(it)f(to)i(predict)d(the)i Fh(j)26 b Fl(+)21 b(1)3015 464 y Fd(st)3108 497 y Fl(with)29 b(probabilit)m(y)150 610 y(1)p Fh(=)p Fl(2)22 b(+)d Fh(\017)p Fl(\()p Fh(k)s Fl(\).)291 723 y(Ho)m(w)m(ev)m(er,)29 b(if)c Fh(b)h Fg(6)p Fl(=)e Fh(b)p Fl(\()p Fh(x)p Fl(\))j(w)m(e)g (don't)f(kno)m(w)g(what)f(the)i(adv)m(ersary)f(will)d(do.)39 b(P)m(erhaps)26 b(it)g(has)f(the)i(same)150 836 y(probabilit)m(y)g(1)p Fh(=)p Fl(2)19 b(+)f Fh(\017)p Fl(\()p Fh(k)s Fl(\))30 b(of)f(predicting)e(the)j Fh(j)23 b Fl(+)18 b(1)1924 803 y Fd(st)2016 836 y Fl(bit)28 b({)i(ma)m(yb)s(e)f(it)g(do)s(esn't)g (need)g(the)g(\014rst)g(bit.)39 b(So)150 949 y(w)m(e'll)30 b(ha)m(v)m(e)h(to)g(use)f(a)h(h)m(ybrid)d(argumen)m(t.)291 1062 y(Let)j Fh(r)495 1076 y Fd(i)554 1062 y Fl(for)f(an)m(y)g Fh(i)h Fl(indicate)e(a)i(random)f(bit.)40 b(The)29 b(h)m(ybrids)f(w)m (e)j(will)d(c)m(ho)s(ose)j(are)981 1379 y Fh(H)1057 1393 y Ff(0)1096 1379 y Fl(\()p Fh(x)1183 1393 y Ff(0)1223 1379 y Fl(\))101 b(=)f Fh(b)p Fl(\()p Fh(x)1656 1393 y Ff(0)1696 1379 y Fl(\))20 b Fg(\016)h Fh(:)15 b(:)g(:)21 b Fg(\016)g Fh(b)p Fl(\()p Fh(x)2135 1394 y Fd(k)2178 1379 y Fl(\))981 1517 y Fh(H)1057 1531 y Ff(1)1096 1517 y Fl(\()p Fh(x)1183 1531 y Ff(0)1223 1517 y Fl(\))101 b(=)f Fh(r)1571 1531 y Ff(0)1631 1517 y Fg(\016)20 b Fh(b)p Fl(\()p Fh(x)1822 1531 y Ff(1)1862 1517 y Fl(\))h Fg(\016)f Fh(:)15 b(:)g(:)22 b Fg(\016)e Fh(b)p Fl(\()p Fh(x)2301 1532 y Fd(k)2344 1517 y Fl(\))1341 1655 y Fg(\001)15 b(\001)g(\001)994 1792 y Fh(H)1070 1807 y Fd(l)1096 1792 y Fl(\()p Fh(x)1183 1806 y Ff(0)1223 1792 y Fl(\))101 b(=)f Fh(r)1571 1806 y Ff(0)1631 1792 y Fg(\016)20 b Fh(:)15 b(:)g(:)22 b Fg(\016)e Fh(r)1929 1807 y Fd(l)q Fe(\000)p Ff(1)2066 1792 y Fg(\016)g Fh(b)p Fl(\()p Fh(x)2257 1807 y Fd(l)2284 1792 y Fl(\))15 b Fh(:)g(:)g(:)21 b Fg(\016)g Fh(:)15 b(:)g(:)21 b Fg(\016)g Fh(b)p Fl(\()p Fh(x)2844 1807 y Fd(k)2887 1792 y Fl(\))1341 1930 y Fg(\001)15 b(\001)g(\001)978 2068 y Fh(H)1054 2083 y Fd(k)1096 2068 y Fl(\()p Fh(x)1183 2082 y Ff(0)1223 2068 y Fl(\))101 b(=)f Fh(r)1571 2082 y Ff(0)1626 2068 y Fh(:)15 b(:)g(:)h(r)1788 2083 y Fd(k)291 2410 y Fl(No)m(w,)44 b(since)d Fh(H)846 2425 y Fd(k)928 2410 y Fl(is)f(totally)h(random,)i(the)e(probabilit)m (y)d(that)j(our)g(adv)m(ersary)g Fh(A)3216 2425 y Fd(k)3299 2410 y Fl(succeeds)g(in)150 2523 y(predicting)i(the)i Fh(j)35 b Fl(+)30 b(1)983 2490 y Fd(st)1090 2523 y Fl(bit)44 b(m)m(ust)g(b)s(e)h(1)p Fh(=)p Fl(2.)85 b(Ho)m(w)m(ev)m(er,)50 b(the)45 b(probabilit)m(y)d(that)k Fh(A)3208 2538 y Fd(k)3295 2523 y Fl(succeeds)f(in)150 2636 y(predicting)25 b(the)j Fh(j)19 b Fl(+)14 b(1)916 2603 y Fd(st)1006 2636 y Fl(bit)26 b(in)g Fh(H)1323 2650 y Ff(0)1389 2636 y Fl(m)m(ust)h(b)s(e)g(1)p Fh(=)p Fl(2)14 b(+)g Fh(\017)p Fl(\()p Fh(k)s Fl(\).)41 b(Let)28 b Fh(P)2407 2650 y Fd(i)2461 2636 y Fl(=)d(Pr)o([)p Fh(A)2747 2651 y Fd(k)2818 2636 y Fl(giv)m(es)i(correct)i Fh(j)19 b Fl(+)14 b(1)3524 2603 y Fd(st)3614 2636 y Fl(bit,)150 2749 y(giv)m(en)32 b(input)e(from)i Fh(H)926 2763 y Fd(i)954 2749 y Fl(])p Fh(:)g Fl(Let)h Fh(P)1272 2716 y Fe(0)1259 2774 y Fd(i)1323 2749 y Fl(=)28 b(Pr)o([)p Fh(A)1612 2764 y Fd(k)1688 2749 y Fl(giv)m(es)k(correct)h Fh(j)27 b Fl(+)21 b(1)2418 2716 y Fd(st)2513 2749 y Fl(bit,)32 b(giv)m(en)g(input)e(from)i Fh(H)3458 2763 y Fd(i)3517 2749 y Fl(where)150 2862 y Fh(r)191 2876 y Fd(i)p Fe(\000)p Ff(1)335 2862 y Fg(6)p Fl(=)25 b Fh(b)p Fl(\()p Fh(x)557 2876 y Fd(i)p Fe(\000)p Ff(1)675 2862 y Fl(])p Fh(:)291 2975 y Fl(No)m(w)k(w)m(e)g(kno)m(w)f(that)h Fh(P)1116 2989 y Fd(i)p Ff(+1)1260 2975 y Fl(=)c Fh(P)1414 2989 y Fd(i)1442 2975 y Fh(=)p Fl(2)17 b(+)f Fh(P)1707 2942 y Fe(0)1694 3000 y Fd(i)1730 2975 y Fh(=)p Fl(2)29 b(since)f(if)f(the)i Fh(r)2347 2989 y Fd(i)p Fe(\000)p Ff(1)2493 2975 y Fl(bit)f(is)f (\\correct")j(then)e(w)m(e)h(are)g(just)150 3088 y(in)g(the)i(same)f (case)i(as)e Fh(H)1020 3102 y Fd(i)1048 3088 y Fl(.)291 3200 y(No)m(w,)h(w)m(e)g(use)f(a)h(randomized)e(h)m(ybrid)f(reduction.) 150 3413 y Fh(A)218 3380 y Fe(0)218 3441 y Fd(k)261 3413 y Fl(\()p Fh(z)t Fl(\):)150 3526 y(1.)41 b(Randomly)29 b(c)m(ho)s(ose)j Fh(i)25 b Fg( )g(f)p Fl(0)p Fh(;)15 b(:)g(:)g(:)j(;)d(k)24 b Fg(\000)c Fl(1)p Fg(g)p Fl(.)150 3639 y(2.)41 b(Compute)30 b Fh(x)707 3653 y Fd(i)p Ff(+1)851 3639 y Fl(=)25 b Fh(z)t Fl(,)30 b Fh(x)1100 3653 y Fd(i)p Ff(+2)1244 3639 y Fl(=)25 b Fh(P)13 b Fl(\()p Fh(x)1498 3653 y Fd(i)p Ff(+1)1617 3639 y Fl(\),)30 b(etc.)150 3752 y(3.)41 b(Cho)s(ose)30 b Fh(b)c Fg( )f(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)p Fh(;)g(r)30 b Fg( )25 b(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)1425 3719 y Fd(i)p Fe(\000)p Ff(1)1546 3752 y Fl(.)41 b(Let)30 b Fh(y)e Fl(=)d Fh(r)e Fg(\016)e Fh(b)f Fg(\016)h Fh(b)p Fl(\()p Fh(x)2324 3766 y Fd(i)p Ff(+1)2442 3752 y Fl(\))g Fg(\016)g Fh(:)15 b(:)g(:)21 b Fg(\016)g Fh(b)p Fl(\()p Fh(x)2882 3767 y Fd(k)2924 3752 y Fl(\).)150 3865 y(4.)41 b(Get)32 b(\()p Fh(\013;)15 b(j)5 b Fl(\))27 b Fg( )e Fh(A)860 3880 y Fd(k)903 3865 y Fl(\(1)983 3832 y Fd(k)1026 3865 y Fl(\).)150 3978 y(5.)41 b(Giv)m(e)31 b(the)f(\014rst)g Fh(j)36 b Fl(bits)29 b(of)i Fh(y)i Fl(to)e Fh(A)1429 3993 y Fd(k)1472 3978 y Fl(.)150 4091 y(6.)41 b(If)30 b Fh(A)420 4106 y Fd(k)493 4091 y Fl(returns)f(the)i Fh(j)26 b Fl(+)20 b(1)1163 4058 y Fd(st)1256 4091 y Fl(bit)29 b(of)i Fh(y)s Fl(,)f(return)f Fh(b)p Fl(.)150 4206 y(7.)41 b(Otherwise,)29 b(return)p 995 4132 40 4 v 30 w Fh(b)p Fl(.)291 4418 y(No)m(w,)i(what)g(is)e(the)i (probabilit)m(y)d(that)j(w)m(e)f(are)h(correct?)1103 4708 y(Pr)o([)p Fh(S)1281 4722 y Fd(H)5 b(C)g(B)1460 4708 y Fl(\()p Fh(A)1563 4671 y Fe(0)1587 4708 y Fl(\)])26 b(=)1770 4595 y Fd(k)r Fe(\000)p Ff(1)1769 4622 y Fa(X)1773 4817 y Fd(j)t Ff(=0)1915 4708 y Fl(Pr)o([)p Fh(i)g Fl(=)f Fh(j)5 b Fl(])2267 4647 y Fh(P)2325 4661 y Fd(i)2375 4647 y Fl(+)20 b(\(1)h Fg(\000)f Fh(P)2729 4614 y Fe(0)2716 4672 y Fd(i)2752 4647 y Fl(\))p 2267 4687 520 4 v 2505 4771 a(2)291 4972 y(Solving)k(for)g Fh(P)811 4939 y Fe(0)798 4997 y Fd(i)860 4972 y Fl(giv)m(es)h(that)h Fh(P)1340 4939 y Fe(0)1327 4997 y Fd(i)1388 4972 y Fl(=)f(2)p Fh(P)1587 4986 y Fd(i)p Ff(+1)1716 4972 y Fg(\000)10 b Fh(P)1855 4986 y Fd(i)1908 4972 y Fl(from)25 b(the)g(ab)s(o)m(v)m(e)h(equation.) 39 b(Th)m(us,)25 b Fh(P)3235 4986 y Fd(i)3273 4972 y Fl(+)10 b(\(1)g Fg(\000)g Fh(P)3596 4939 y Fe(0)3583 4997 y Fd(i)3619 4972 y Fl(\))25 b(=)150 5084 y(1)20 b(+)g Fh(P)364 5098 y Fd(i)412 5084 y Fg(\000)g Fh(P)574 5051 y Fe(0)561 5110 y Fd(i)622 5084 y Fl(=)25 b(1)20 b(+)g Fh(P)932 5098 y Fd(i)980 5084 y Fg(\000)f Fl(2)p Fh(P)1173 5098 y Fd(i)p Ff(+1)1312 5084 y Fl(+)h Fh(P)1461 5098 y Fd(i)1515 5084 y Fl(=)25 b(1)20 b(+)f(2)p Fh(P)1869 5098 y Fd(i)1918 5084 y Fg(\000)h Fl(2)p Fh(P)2112 5098 y Fd(i)p Ff(+1)2231 5084 y Fl(.)40 b(Th)m(us,)30 b(the)g(probabilit)m (y)e Fh(A)3241 5051 y Fe(0)3294 5084 y Fl(is)i(correct)h(is)1867 5416 y(13-5)p eop %%Page: 6 6 6 5 bop 1140 427 a Fl(Pr[)p Fh(S)1319 441 y Fd(H)5 b(C)g(B)1497 427 y Fl(\()p Fh(A)1600 389 y Fe(0)1624 427 y Fl(\)])84 b(=)1923 313 y Fd(k)r Fe(\000)p Ff(1)1922 341 y Fa(X)1926 536 y Fd(j)t Ff(=0)2080 365 y Fl(1)p 2078 406 51 4 v 2078 489 a Fh(k)2138 427 y Fl(\()2183 365 y(1)p 2183 406 46 4 v 2183 489 a(2)2259 427 y(+)20 b Fh(P)2408 441 y Fd(i)2457 427 y Fg(\000)g Fh(P)2606 441 y Fd(i)p Ff(+1)2724 427 y Fl(\))1768 760 y(=)1934 699 y(1)p 1932 739 51 4 v 1932 822 a Fh(k)2008 646 y Fd(k)r Fe(\000)p Ff(1)2007 674 y Fa(X)2011 869 y Fd(j)t Ff(=0)2138 760 y Fl(\()2183 699 y(1)p 2183 739 46 4 v 2183 822 a(2)2259 760 y(+)g Fh(P)2408 774 y Fd(i)2457 760 y Fg(\000)g Fh(P)2606 774 y Fd(i)p Ff(+1)2724 760 y Fl(\))1768 1093 y(=)1932 1032 y(1)p 1932 1072 V 1932 1156 a(2)2007 1093 y(+)2110 1032 y(1)p 2108 1072 51 4 v 2108 1156 a Fh(k)2185 980 y Fd(k)r Fe(\000)p Ff(1)2183 1007 y Fa(X)2188 1202 y Fd(j)t Ff(=0)2330 1093 y Fh(P)2388 1107 y Fd(i)2437 1093 y Fg(\000)f Fh(P)2585 1107 y Fd(i)p Ff(+1)1768 1378 y Fl(=)1932 1316 y(1)p 1932 1357 46 4 v 1932 1440 a(2)2007 1378 y(+)2110 1316 y(1)p 2108 1357 51 4 v 2108 1440 a Fh(k)2168 1378 y Fl(\()p Fh(P)2261 1392 y Ff(0)2322 1378 y Fg(\000)g Fh(P)2470 1393 y Fd(k)2514 1378 y Fl(\))1768 1603 y(=)1932 1541 y(1)p 1932 1582 46 4 v 1932 1665 a(2)2007 1603 y(+)2108 1541 y Fh(\017)p Fl(\()p Fh(k)s Fl(\))p 2108 1582 158 4 v 2162 1665 a Fh(k)291 1833 y Fl(whic)m(h)j(is)g(non-negligibly)e(b)s (etter)j(than)g(1)p Fh(=)p Fl(2,)j(since)d(w)m(e)g(assumed)g Fh(\017)p Fl(\()p Fh(k)s Fl(\))h(w)m(as)f(non-negligible.)36 b(Th)m(us,)150 1946 y Fh(A)218 1913 y Fe(0)275 1946 y Fl(reco)m(v)m(ers)g Fh(b)p Fl(\()p Fh(x)p Fl(\))f(from)e Fh(P)13 b Fl(\()p Fh(x)p Fl(\))35 b(with)e(go)s(o)s(d)h(probabilit)m(y) -8 b(,)33 b(whic)m(h)g(con)m(tradicts)h(that)h Fh(b)f Fl(is)f(a)i(hard-core)150 2059 y(predicate)30 b(of)h Fh(P)13 b Fl(.)40 b(Th)m(us,)30 b Fh(G)h Fl(m)m(ust)f(b)s(e)g(next-bit) g(unpredictable.)291 2271 y Fm(\(b\))40 b Fl(Pro)m(v)m(e)j(that)e(for)g Fj(any)h Fh(G)p Fl(,)i(if)c Fh(G)h Fl(is)g(next-bit)f(unpredictable)f (then)h(it)h(is)f(a)h(pseudorandom)150 2384 y(generator)32 b(\(in)d(the)i(sense)f(w)m(e)h(used)e(in)g(lecture)i(-)f(that)h Fh(G)g Fl(is)e(indistinguishable)c(from)30 b(random\).)291 2597 y Fm(Solution:)58 b Fl(Here,)42 b(again,)f(w)m(e)e(will)d(need)j (to)g(use)g(a)g(h)m(ybrid)d(argumen)m(t.)67 b(Supp)s(ose)37 b Fh(G)i Fl(is)f(not)h(a)150 2710 y(pseudorandom)33 b(generator,)k(and) d(that)h Fh(A)1646 2725 y Fd(k)1723 2710 y Fl(is)e(an)i(adv)m(ersary)f (that)i(distinguishes)30 b(it)k(from)g(random,)150 2823 y(and)i(w)m(e)g(w)m(an)m(t)h(to)g(giv)m(e)g(an)f(adv)m(ersary)g(that)h (breaks)f(next-bit)f(unpredictabilit)m(y)-8 b(.)55 b(Supp)s(ose)35 b(that)i Fh(G)150 2936 y Fl(returns)29 b(an)h(output)g(of)h(length)f Fh(l)r Fl(\()p Fh(k)s Fl(\))h(on)f(an)g(input)f(of)h(length)g Fh(k)s Fl(.)291 3048 y(Let)h Fh(p)25 b Fl(=)g Fh(p)667 3062 y Ff(1)726 3048 y Fg(\016)c Fh(:)15 b(:)g(:)21 b Fg(\016)g Fh(p)1030 3067 y Fd(l)q Ff(\()p Fd(k)r Ff(\))1179 3048 y Fl(b)s(e)30 b(a)h(pseudorandom)e(string,)g(and)h(let)g Fh(r)j Fl(b)s(e)d(a)h(truly)e(random)g(string.)1450 3366 y Fh(H)1526 3380 y Ff(0)1666 3366 y Fl(=)100 b Fh(p)1450 3503 y(H)1526 3517 y Ff(1)1666 3503 y Fl(=)g Fh(r)1878 3517 y Ff(1)1938 3503 y Fg(\016)20 b Fh(p)2049 3517 y Ff(2)2109 3503 y Fg(\016)g Fh(:)15 b(:)g(:)i(p)2342 3522 y Fd(l)q Ff(\()p Fd(k)r Ff(\))1648 3641 y Fg(\001)e(\001)g(\001)1280 3779 y Fh(H)1356 3798 y Fd(l)q Ff(\()p Fd(k)r Ff(\))p Fe(\000)p Ff(1)1666 3779 y Fl(=)100 b Fh(r)1878 3793 y Ff(1)1938 3779 y Fg(\016)20 b Fh(:)15 b(:)g(:)22 b Fg(\016)e Fh(r)2236 3798 y Fd(l)q Ff(\()p Fd(k)r Ff(\))p Fe(\000)p Ff(1)2466 3779 y Fg(\016)h Fh(p)2578 3794 y Fd(k)1370 3917 y Fh(H)1446 3935 y Fd(l)q Ff(\()p Fd(k)r Ff(\))1666 3917 y Fl(=)100 b Fh(r)291 4121 y Fl(Note)37 b(that)f Fh(A)786 4136 y Fd(k)865 4121 y Fl(distinguishes)c(b)s(et)m(w) m(een)k Fh(H)1839 4140 y Fd(l)q Ff(\()p Fd(k)r Ff(\))1994 4121 y Fl(and)f Fh(H)2252 4135 y Ff(0)2291 4121 y Fl(.)56 b(No)m(w)37 b(w)m(e)f(giv)m(e)g(a)g(randomized)e(h)m(ybrid)150 4234 y(reduction)29 b(\(assume)i Fh(A)972 4249 y Fd(k)1045 4234 y Fl(returns)e(0)i(to)g(mean)f(\\pseudo-random"\):)150 4447 y Fh(A)218 4414 y Fe(0)218 4474 y Fd(k)261 4447 y Fl(:)150 4560 y(1.)41 b(On)30 b(input)e(1)700 4527 y Fd(k)743 4560 y Fl(,)j(pic)m(k)f(a)g(random)g Fh(j)h Fg( )25 b(f)p Fl(0)p Fh(;)15 b(:)g(:)g(:)j(;)d(l)r Fl(\()p Fh(k)s Fl(\))21 b Fg(\000)f Fl(1)p Fg(g)p Fl(.)150 4672 y(2.)41 b(Return)30 b Fh(\013)25 b Fl(=)g Fh(j;)15 b(j)5 b Fl(.)150 4785 y(3.)41 b(On)30 b(input)e Fh(p)701 4799 y Ff(1)760 4785 y Fg(\016)21 b Fh(:)15 b(:)g(:)21 b Fg(\016)g Fh(p)1064 4799 y Fd(j)1100 4785 y Fl(,)31 b(pic)m(k)f Fh(b)25 b Fg( )g(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)p Fl(.)150 4902 y(4.)41 b(Pic)m(k)30 b Fh(r)e Fg( )d(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)869 4869 y Fd(l)q Ff(\()p Fd(k)r Ff(\))p Fe(\000)p Fd(j)t Fe(\000)p Ff(1)1168 4902 y Fl(.)150 5015 y(5.)41 b(Let)31 b Fh(b)463 4982 y Fe(0)512 5015 y Fg( )25 b Fh(A)696 5030 y Fd(k)739 5015 y Fl(\()p Fh(p)820 5029 y Ff(1)879 5015 y Fg(\016)c Fh(:)15 b(:)g(:)21 b Fg(\016)g Fh(p)1183 5029 y Fd(j)1239 5015 y Fg(\016)g Fh(b)f Fg(\016)h Fh(r)s Fl(\).)150 5128 y(6.)41 b(Return)30 b Fh(b)20 b Fg(\010)g Fh(b)760 5095 y Fe(0)783 5128 y Fl(.)1867 5416 y(13-6)p eop %%Page: 7 7 7 6 bop 291 159 a Fl(This)27 b(time,)h(let)g Fh(P)914 173 y Fd(i)971 159 y Fl(b)s(e)f(the)i(probabilit)m(y)d(that)i Fh(A)1972 174 y Fd(k)2043 159 y Fl(giv)m(es)h(a)f(0)h(giv)m(en)f(that)h (the)f(input)f(w)m(as)h(from)g Fh(H)3697 173 y Fd(i)3725 159 y Fl(.)150 271 y(Let)k Fh(P)385 238 y Fe(0)372 297 y Fd(i)440 271 y Fl(b)s(e)g(the)g(probabilit)m(y)d(that)k Fh(A)1457 286 y Fd(k)1531 271 y Fl(giv)m(es)f(a)h(0)f(giv)m(en)g(that)g (the)g(input)e(w)m(as)i(from)g Fh(H)3218 285 y Fd(i)3277 271 y Fl(and)f Fh(r)3496 285 y Fd(i)3552 271 y Fg(6)p Fl(=)d Fh(p)3697 285 y Fd(i)3725 271 y Fl(.)150 384 y(Then,)i(again,)g (w)m(e)h(ha)m(v)m(e)h Fh(P)1082 398 y Fd(i)p Ff(+1)1226 384 y Fl(=)25 b Fh(P)1380 398 y Fd(i)1408 384 y Fh(=)p Fl(2)c(+)f Fh(P)1681 351 y Fe(0)1668 410 y Fd(i)1705 384 y Fh(=)p Fl(2.)291 497 y(The)30 b(probabilit)m(y)e(the)i(reduction) g(w)m(orks)g(is)1008 873 y(Pr)o([)p Fh(S)1186 887 y Fd(N)7 b(B)s(U)1365 873 y Fl(\()p Fh(A)1468 836 y Fe(0)1492 873 y Fl(\)])83 b(=)1789 751 y Fd(l)q Ff(\()p Fd(k)r Ff(\))p Fe(\000)p Ff(1)1826 787 y Fa(X)1831 982 y Fd(j)t Ff(=0)2010 873 y Fl(Pr)o([)p Fh(i)26 b Fl(=)f Fh(j)5 b Fl(])2362 812 y Fh(P)2420 826 y Fd(i)2470 812 y Fl(+)20 b(\(1)h Fg(\000)e Fh(P)2823 779 y Fe(0)2810 837 y Fd(i)2847 812 y Fl(\))p 2362 852 520 4 v 2600 936 a(2)1635 1219 y(=)1799 1157 y(1)p 1799 1198 46 4 v 1799 1281 a(2)1875 1219 y(+)2028 1157 y(1)p 1976 1198 150 4 v 1976 1281 a Fh(l)r Fl(\()p Fh(k)s Fl(\))2151 1096 y Fd(l)q Ff(\()p Fd(k)r Ff(\))p Fe(\000)p Ff(1)2188 1132 y Fa(X)2192 1328 y Fd(j)t Ff(=0)2371 1219 y Fh(P)2429 1233 y Fd(i)2478 1219 y Fg(\000)h Fh(P)2627 1233 y Fd(i)p Ff(+1)1635 1512 y Fl(=)1799 1451 y(1)p 1799 1492 46 4 v 1799 1575 a(2)1875 1512 y(+)1976 1451 y Fh(\017)p Fl(\()p Fh(k)s Fl(\))p 1976 1492 158 4 v 1980 1575 a Fh(l)r Fl(\()p Fh(k)s Fl(\))291 1837 y(b)m(y)45 b(the)f(same)i(reasoning)e(as)g(in)g(part)g(3a.)85 b(Th)m(us,)47 b(w)m(e)e(succeed)g(in)e(predicting)g(the)i(next)g(bit) 150 1950 y(with)33 b(nonnegligible)e(probabilit)m(y)-8 b(.)51 b(Th)m(us)33 b(if)g Fh(G)i Fl(is)e(next-bit)h(unpredictable,)f Fh(G)i Fl(is)e(a)h(pseudorandom)150 2063 y(generator.)150 2346 y Fi(Problem)45 b(4:)61 b(Pumping)44 b(Pseudorandomness)150 2549 y Fl(\(20)34 b(p)s(oin)m(ts\))d(Supp)s(ose)g(y)m(ou)h(are)h(giv)m (en)g(a)f(PR)m(G)h Fh(G)g Fl(suc)m(h)f(that)h Fg(j)p Fh(G)p Fl(\()p Fh(x)p Fl(\))p Fg(j)d Fl(=)e Fg(j)p Fh(x)p Fg(j)22 b Fl(+)g(1,)33 b(and)f(a)h(p)s(olynomial)150 2662 y Fh(p)p Fl(.)43 b(Construct)31 b(a)h Fh(G)838 2629 y Fe(0)893 2662 y Fl(suc)m(h)f(that)h Fg(j)p Fh(G)1393 2629 y Fe(0)1417 2662 y Fl(\()p Fh(x)p Fl(\))p Fg(j)c Fl(=)e Fh(p)p Fl(\()p Fg(j)p Fh(x)p Fg(j)p Fl(\),)33 b(and)e(pro)m(v)m(e)h(that)g Fh(G)2658 2629 y Fe(0)2713 2662 y Fl(is)e(also)h(a)h(PR)m(G.)g(Y)-8 b(our)32 b(pro)s(of)150 2775 y(should)21 b(b)s(e)i(along)g(similar)e(lines)h(to)i(the)f(pro)s (of)g(y)m(ou)g(use)g(in)f(problem)g(3.)39 b(A)23 b(certain)g(lev)m(el)g (of)h(informalit)m(y)150 2888 y(is)29 b(acceptable.)291 3089 y Fm(Solution:)57 b Fl(Let)39 b Fh(G)996 3056 y Fe(0)1020 3089 y Fl(\()p Fh(x)p Fl(\))g(=)f Fh(G)1361 3056 y Fd(p)p Ff(\()p Fe(j)p Fd(x)p Fe(j)p Ff(\))p Fe(\000j)p Fd(x)p Fe(j)1669 3089 y Fl(\()p Fh(x)p Fl(\))h(where)f Fh(G)2172 3056 y Ff(1)2212 3089 y Fl(\()p Fh(x)p Fl(\))h(=)g Fh(G)p Fl(\()p Fh(x)p Fl(\))g(and)f Fh(G)2971 3056 y Fd(i)2999 3089 y Fl(\()p Fh(x)p Fl(\))i(=)e Fh(G)p Fl(\()p Fh(G)3447 3056 y Fd(i)p Fe(\000)p Ff(1)3567 3089 y Fl(\()p Fh(x)p Fl(\)\).)150 3202 y(That)26 b(is,)h Fh(G)558 3169 y Fe(0)608 3202 y Fl(just)f(rep)s(eatedly)g(applies)f Fh(G)p Fl(.)40 b(No)m(w,)28 b(since)e(eac)m(h)i(application)d(of)i Fh(G)f Fl(increases)h(the)g(length)150 3315 y(b)m(y)j(1,)h(it)f(should) f(b)s(e)g(clear)i(that)g Fh(G)1366 3282 y Fe(0)1420 3315 y Fl(increases)f(the)g(length)g(from)g Fg(j)p Fh(x)p Fg(j)h Fl(to)g Fh(p)p Fl(\()p Fg(j)p Fh(x)p Fg(j)p Fl(\),)g(as)g (required.)291 3427 y(No)m(w,)41 b(supp)s(ose)36 b(there)j(is)e(an)g (adv)m(ersary)i Fh(A)1844 3442 y Fd(k)1924 3427 y Fl(that)g(can)f (distinguish)c(b)s(et)m(w)m(een)39 b Fh(G)3201 3395 y Fe(0)3263 3427 y Fl(=)e Fh(G)3442 3395 y Fd(p)p Ff(\()p Fe(j)p Fd(x)p Fe(j)p Ff(\))p Fe(\000j)p Fd(x)p Fe(j)150 3540 y Fl(and)32 b(a)i(random)e(string)g(with)g(adv)-5 b(an)m(tage)35 b Fh(\017)p Fl(\()p Fh(k)s Fl(\).)49 b(Let)34 b Fh(k)f Fl(=)c Fg(j)p Fh(x)p Fg(j)p Fl(,)34 b(and)f(let)g Fh(x)2751 3554 y Fd(i)2812 3540 y Fl(denote)g(a)h(random)e(string)150 3653 y(of)f(length)e Fh(k)24 b Fl(+)c Fh(i)p Fl(.)41 b(Then,)29 b(w)m(e)i(de\014ne)f(a)g(sequence)h(of)g(h)m(ybrid)d (distributions:)1476 3953 y Fh(D)1551 3967 y Ff(0)1692 3953 y Fl(=)100 b Fh(G)1934 3915 y Fe(0)1958 3953 y Fl(\()p Fh(x)2045 3967 y Ff(0)2084 3953 y Fl(\))26 b(=)f Fh(G)2312 3915 y Fd(p)p Ff(\()p Fd(k)r Ff(\))p Fe(\000)p Fd(k)2539 3953 y Fl(\()p Fh(x)2626 3967 y Ff(0)2666 3953 y Fl(\))1476 4091 y Fh(D)1551 4105 y Ff(1)1692 4091 y Fl(=)100 b Fh(G)1934 4053 y Fd(p)p Ff(\()p Fd(k)r Ff(\))p Fe(\000)p Fd(k)r Fe(\000)p Ff(1)2251 4091 y Fl(\()p Fh(x)2338 4105 y Ff(1)2378 4091 y Fl(\))1674 4228 y Fg(\001)15 b(\001)g(\001)1199 4366 y Fh(D)1274 4385 y Fd(p)p Ff(\()p Fd(k)r Ff(\))p Fe(\000)p Fd(k)r Fe(\000)p Ff(1)1692 4366 y Fl(=)100 b Fh(G)p Fl(\()p Fh(x)2021 4385 y Fd(p)p Ff(\()p Fd(k)r Ff(\))p Fe(\000)p Fd(k)r Fe(\000)p Ff(1)2338 4366 y Fl(\))1289 4504 y Fh(D)1364 4523 y Fd(p)p Ff(\()p Fd(k)r Ff(\))p Fe(\000)p Fd(k)1692 4504 y Fl(=)g Fh(x)1915 4523 y Fd(p)p Ff(\()p Fd(k)r Ff(\))p Fe(\000)p Fd(k)291 4828 y Fl(No)m(w,)37 b(b)m(y)e(the)g(standard)g(h)m(ybrid)d(argumen)m(t,)37 b(there)e(m)m(ust)g(b)s(e)g(an)g Fh(i)g Fl(suc)m(h)g(that)g Fh(A)g Fl(distinguishes)150 4941 y(b)s(et)m(w)m(een)c Fh(D)576 4955 y Fd(i)635 4941 y Fl(and)f Fh(D)887 4955 y Fd(i)p Ff(+1)1005 4941 y Fl(,)h(that)g(is,)827 5167 y(Pr[)p Fh(y)d Fg( )d Fh(D)1214 5181 y Fd(i)1268 5167 y Fl(:)h Fh(A)p Fl(\()p Fh(y)s Fl(\))g(=)f(0])20 b Fg(\000)g Fl(Pr[)p Fh(y)28 b Fg( )d Fh(D)2195 5181 y Fd(i)p Ff(+1)2339 5167 y Fl(:)h Fh(A)p Fl(\()p Fh(y)s Fl(\))g(=)f(0])h Fh(>)f(\017)p Fl(\()p Fh(k)s Fl(\))p Fh(:)1867 5416 y Fl(13-7)p eop %%Page: 8 8 8 7 bop 291 159 a Fl(W)-8 b(e)32 b(can)e(use)g(this)g(to)h(break)f Fh(G)h Fl(as)f(a)h(PR)m(G)g(as)g(follo)m(ws:)150 371 y Fh(A)218 338 y Fe(0)218 399 y Fd(k)261 371 y Fl(\()p Fh(y)s Fl(\))g(where)f Fg(j)p Fh(y)s Fg(j)25 b Fl(=)g Fg(j)p Fh(x)p Fg(j)c Fl(+)f Fh(i)p Fl(:)150 488 y(1.)41 b(Compute)30 b Fh(z)g Fg( )25 b Fh(G)914 455 y Fd(p)p Ff(\()p Fe(j)p Fd(x)p Fe(j)p Ff(\))p Fe(\000)p Fd(i)1167 488 y Fl(\()p Fh(y)s Fl(\).)150 601 y(2.)41 b(Return)30 b Fh(A)639 616 y Fd(k)682 601 y Fl(\()p Fh(z)t Fl(\).)291 813 y(If)36 b Fh(A)456 828 y Fd(k)535 813 y Fl(is)f(correct,)k(then)d Fh(A)1249 780 y Fe(0)1249 841 y Fd(k)1327 813 y Fl(is)g(correct)h(as)f (w)m(ell.)57 b(Note)38 b(that)e(for)g(this)f(construction)h(for)g Fh(G)3587 780 y Fe(0)3646 813 y Fl(w)m(e)150 926 y(cannot)i(do)g(a)g (randomized)e(h)m(ybrid)f(argumen)m(t,)40 b(since,)f(for)e(this)g (construction,)i(whic)m(h)d(h)m(ybrids)f(w)m(e)150 1039 y(distinguish)d(sp)s(ecify)i(the)i(length)f(of)h(the)g(input)d(w)m(e)j (need.)57 b(This)33 b(pro)m(v)m(es)k(that)f(if)f Fh(G)3149 1006 y Fe(0)3208 1039 y Fl(is)g(not)g(a)h(PR)m(G)150 1152 y(then)30 b(neither)f(is)h Fh(G)p Fl(.)41 b(Th)m(us,)30 b Fh(G)1221 1119 y Fe(0)1275 1152 y Fl(is)f(a)i(PR)m(G,)g(whic)m(h)e (concludes)h(the)g(pro)s(of.)150 1438 y Fi(Problem)45 b(5:)61 b(Predictable)46 b(O)l(WFs)150 1641 y Fl(Let)30 b Fh(F)38 b Fl(:)26 b Fg(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)679 1608 y Fe(\003)746 1641 y Fg(!)25 b(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)1082 1608 y Fe(\003)1153 1641 y Fl(b)s(e)29 b(a)h(length-preserving)e(one-w)m(a)m(y)j(function.)39 b(Let)30 b Fh(b)3008 1655 y Fd(i)3036 1641 y Fl(\()p Fh(x)p Fl(\))g(b)s(e)f(a)h(predicate)150 1754 y(whic)m(h)j(returns)g (the)h Fh(i)923 1721 y Fd(th)1027 1754 y Fl(bit)g(of)g Fh(x)p Fl(.)52 b(Construct)33 b(a)i(length)e(preserving)g(function)f Fh(F)3070 1721 y Fe(0)3094 1754 y Fl(,)j(and)e(pro)m(v)m(e)i(that)150 1867 y Fh(F)221 1834 y Fe(0)275 1867 y Fl(is)29 b(\(1\))j(a)f(one-w)m (a)m(y)h(function,)d(and)h(\(2\))h(for)g(an)m(y)f(1)c Fg(\024)f Fh(i)h Fg(\024)f Fh(k)s Fl(,)31 b(it)f(is)f(easy)i(for)f (random)g Fh(x)g Fl(of)h(length)f Fh(k)150 1980 y Fl(to)h(compute)g Fh(b)669 1994 y Fd(i)697 1980 y Fl(\()p Fh(x)p Fl(\),)g(giv)m(en)g (only)e Fh(F)1383 1947 y Fe(0)1406 1980 y Fl(\()p Fh(x)p Fl(\))i(as)g(input.)291 2193 y Fm(Solution:)291 2305 y Fl(Here)g(is)e(a)i(construction)f(due)g(to)h(Y)-8 b(evgeniy)31 b(Do)s(dis,)e(Eric)h(Lehman,)g(and)g(Matthias)h(Ruhl.)291 2418 y(De\014ne)c Fh(c)h Fl(and)e Fh(d)p Fl(,)j(functions)c(from)i (three-bit)g(inputs)e(to)j(one)f(and)g(t)m(w)m(o-bit)g(outputs,)h(resp) s(ectiv)m(ely)-8 b(,)150 2531 y(as)31 b(follo)m(ws:)1717 2848 y Fh(x)83 b(c)p Fl(\()p Fh(x)p Fl(\))h Fh(d)p Fl(\()p Fh(x)p Fl(\))1633 2986 y(000)142 b(0)g(00)1633 3124 y(001)g(0)g(01)1633 3262 y(010)g(0)g(10)1633 3400 y(100)g(0)g(11)1633 3537 y(011)g(1)g(00)1633 3675 y(101)g(1)g(01)1633 3813 y(110)g(1)g(10)1633 3951 y(111)g(1)g(11)291 4155 y(Note)43 b(that)g(kno)m(wing)e Fh(c)p Fl(\()p Fh(x)p Fl(\))i(do)s(es)e(not)h(help)f(in)f(determining)g Fh(d)p Fl(\()p Fh(x)p Fl(\))j(if)d Fh(x)i Fl(is)f(unkno)m(wn.)74 b(De\014ne)150 4268 y Fh(c)189 4282 y Fd(i)218 4268 y Fl(\()p Fh(x)305 4282 y Ff(1)368 4268 y Fg(\016)25 b Fh(:)15 b(:)g(:)25 b Fg(\016)f Fh(x)689 4282 y Ff(3)p Fd(n)771 4268 y Fl(\))35 b(=)f Fh(c)p Fl(\()p Fh(x)1072 4282 y Ff(3)p Fd(i)p Fe(\000)p Ff(2)1251 4268 y Fg(\016)24 b Fh(x)1372 4282 y Ff(3)p Fd(i)p Fe(\000)p Ff(1)1550 4268 y Fg(\016)g Fh(x)1671 4282 y Ff(3)p Fd(i)1734 4268 y Fl(\),)38 b(and)e(de\014ne)f Fh(d)2330 4282 y Fd(i)2394 4268 y Fl(exactly)i(the)f(same)h(w)m(a)m(y)-8 b(,)39 b(but)c(with)g Fh(d)150 4381 y Fl(instead)30 b(of)g Fh(c)p Fl(.)291 4494 y(No)m(w,)h(if)f Fh(F)43 b Fl(is)30 b(a)g (length-preserving)f(one-w)m(a)m(y)j(function,)d(then)h(let)h Fh(F)2733 4461 y Fe(0)2786 4494 y Fl(b)s(e)f(as)h(follo)m(ws:)745 4720 y Fh(F)816 4682 y Fe(0)840 4720 y Fl(\()p Fh(x)927 4734 y Ff(1)986 4720 y Fg(\016)21 b Fh(:)15 b(:)g(:)21 b Fg(\016)g Fh(x)1296 4734 y Ff(3)p Fd(n)1378 4720 y Fl(\))26 b(=)f Fh(c)1574 4734 y Ff(1)1614 4720 y Fl(\()p Fh(x)p Fl(\))20 b Fg(\016)h Fh(:)15 b(:)g(:)21 b Fg(\016)g Fh(c)2053 4734 y Fd(n)2100 4720 y Fl(\()p Fh(x)p Fl(\))g Fg(\016)g Fh(F)13 b Fl(\()p Fh(d)2462 4734 y Ff(1)2502 4720 y Fl(\()p Fh(x)p Fl(\))21 b Fg(\016)g Fh(:)15 b(:)g(:)21 b Fg(\016)g Fh(d)2950 4734 y Fd(n)2997 4720 y Fl(\()p Fh(x)p Fl(\)\))291 4887 y(\(If)32 b Fh(F)490 4854 y Fe(0)545 4887 y Fl(is)f(giv)m(en)h(input)d(of)j(a)h(length)e(not)h(of)g(the)g (form)f(3)p Fh(n)p Fl(,)h(it)g(just)f(passes)h(through)f(a)h(bit)f(or)h (t)m(w)m(o)150 5000 y(at)f(the)g(end)e(of)i(its)f(input,)e(and)i(uses)g (this)f(construction)h(on)h(the)f(rest)h(of)f(the)h(bits.\))291 5112 y(No)m(w,)g(if)f Fh(F)678 5079 y Fe(0)731 5112 y Fl(is)g(not)h(one-w)m(a)m(y)-8 b(,)32 b(and)e Fh(A)1598 5127 y Fd(k)1671 5112 y Fl(in)m(v)m(erts)g(it,)g(then)h(w)m(e)f(can)h (in)m(v)m(ert)f Fh(F)44 b Fl(as)30 b(follo)m(ws:)1867 5416 y(13-8)p eop %%Page: 9 9 9 8 bop 150 159 a Fh(A)218 126 y Fe(0)218 186 y Fd(k)261 159 y Fl(\()p Fh(F)13 b Fl(\()p Fh(x)p Fl(\)\):)150 271 y(1.)41 b(Cho)s(ose)30 b(a)h(random)f Fh(c)g Fl(of)h(length)f Fg(j)p Fh(F)13 b Fl(\()p Fh(x)p Fl(\))p Fg(j)p Fh(=)p Fl(2.)150 384 y(2.)41 b(Let)31 b Fh(x)476 351 y Fe(0)524 384 y Fg( )26 b Fh(A)709 399 y Fd(k)751 384 y Fl(\()p Fh(c)21 b Fg(\016)g Fh(F)13 b Fl(\()p Fh(x)p Fl(\)\).)150 497 y(3.)41 b(Output)29 b Fh(d)631 511 y Ff(1)671 497 y Fl(\()p Fh(x)758 464 y Fe(0)782 497 y Fl(\))20 b Fg(\016)h Fh(:)15 b(:)g(:)21 b Fg(\016)g Fh(d)1142 511 y Fd(n)1189 497 y Fl(\()p Fh(x)1276 464 y Fe(0)1300 497 y Fl(\))p Fh(:)291 710 y Fl(If)31 b Fh(A)451 725 y Fd(k)526 710 y Fl(giv)m(es)h(a)g(correct)h(answ)m(er)e(then)g(certainly)g Fh(A)2088 677 y Fe(0)2088 737 y Fd(k)2163 710 y Fl(will.)42 b(Th)m(us,)31 b Fh(F)2702 677 y Fe(0)2757 710 y Fl(is)f(one-w)m(a)m(y) -8 b(.)47 b(On)31 b(the)h(other)150 823 y(hand,)d Fh(c)441 837 y Fd(i)470 823 y Fl(\()p Fh(x)p Fl(\))i(giv)m(es)f(a)g(predictor)g (for)f(bits)h Fh(x)1678 837 y Ff(3)p Fd(i)p Fe(\000)p Ff(2)1831 823 y Fh(;)15 b(x)1923 837 y Ff(3)p Fd(i)p Fe(\000)p Ff(1)2077 823 y Fh(;)30 b Fl(and)g Fh(x)2361 837 y Ff(3)p Fd(i)2454 823 y Fl(whic)m(h)f(is)g(correct)j(with)c (probabilit)m(y)150 936 y(3/4.)1867 5416 y(13-9)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF handouts/solutions/sol6.ps0100664003325600237220000110625007402753727015433 0ustar 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(di\013eren)m(t)150 384 y(from)e Fl(p)411 398 y Fj(m)477 384 y Fp(,)h(and)f(so)g(w)m(e)h(are)g(done.)291 597 y(\(b\))i(Supp)s (ose)e(that,)k(instead,)e(Alice's)f Fl(P)13 b(K)40 b Fp(is)32 b(a)h(public)d(k)m(ey)k(of)f(a)g(trap)s(do)s(or)f(p)s(erm)m (utation)g Fl(p)3602 611 y Fj(P)10 b(K)3725 597 y Fp(.)150 710 y(In)27 b(order)h(to)h(send)e Fl(m)p Fp(,)i(Bob)g(comes)g(up)e (with)g(a)h(random)g Fl(k)s Fp(-bit)f(seed)i Fl(s)p Fp(,)f(and)g (computes)g Fl(r)g Fp(=)d Fl(G)p Fp(\()p Fl(s)p Fp(\).)40 b(He)150 823 y(then)30 b(sends)g(\()p Fl(m)20 b Fk(\010)g Fl(r)m(;)15 b(p)951 837 y Fj(P)10 b(K)1074 823 y Fp(\()p Fl(s)p Fp(\)\).)42 b(Is)30 b(the)g(resulting)f(encryption)g(sc)m(heme)i (seman)m(tically)f(secure?)291 1035 y Fq(Solution:)48 b Fp(No.)k(W)-8 b(e)36 b(will)31 b(pro)m(v)m(e)k(it)e(b)m(y)h(a)h(coun) m(terexample,)g(i.e.)51 b(w)m(e)35 b(will)c(exhibit)h(an)i(example)150 1148 y(PR)m(G)d Fl(G)g Fp(and)e(TDP)i(family)e Fl(p)1208 1162 y Fj(P)10 b(K)1360 1148 y Fp(suc)m(h)30 b(that)h(the)g(resulting)e (cryptosystem)h(is)g(completely)g(brok)m(en.)291 1261 y(Supp)s(ose)38 b(a)i(PR)m(G)g Fl(G)1050 1228 y Fh(0)1154 1261 y Fp(:)81 b Fk(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)1480 1228 y Fj(k)r(=)p Fi(2)1636 1261 y Fk(7!)40 b(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)1987 1228 y Fj(`)2062 1261 y Fp(and)39 b(a)h(TDP)g Fl(p)2616 1228 y Fh(0)2616 1288 y Fj(P)10 b(K)2778 1261 y Fp(o)m(v)m(er)41 b Fk(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)3202 1228 y Fj(k)r(=)p Fi(2)3357 1261 y Fp(are)40 b(giv)m(en.)150 1374 y(F)-8 b(rom)36 b(them)g(let's)f (construct)h Fl(G)g Fp(and)f Fl(p)1569 1388 y Fj(P)10 b(K)1692 1374 y Fp(:)51 b Fl(G)36 b Fp(will)d(simply)g(ignore)j(the)g (second)f(half)g(of)h(its)f(input:)150 1487 y Fl(G)p Fp(\()p Fl(x)p Fp(\))50 b(=)e Fl(G)p Fp(\()p Fl(x)670 1501 y Fi(1)739 1487 y Fk(\016)30 b Fl(x)866 1501 y Fi(2)906 1487 y Fp(\))48 b(=)h Fl(G)1180 1454 y Fh(0)1203 1487 y Fp(\()p Fl(x)1290 1501 y Fi(1)1330 1487 y Fp(\).)83 b(It)44 b(is)g(easy)g(to)h(see)g(that)g(distinguishing)40 b Fl(G)45 b Fp(from)e(random)h(is)150 1600 y(equiv)-5 b(alen)m(t)37 b(to)h(distinguishing)32 b Fl(G)1366 1567 y Fh(0)1427 1600 y Fp(from)37 b(random)f(and)h(is)f(therefore)i(a)f (con)m(tradiction.)61 b Fl(p)3397 1614 y Fj(P)10 b(K)3520 1600 y Fp(\()p Fl(x)p Fp(\))37 b(=)150 1713 y Fl(p)196 1727 y Fj(P)10 b(K)319 1713 y Fp(\()p Fl(x)406 1727 y Fi(1)467 1713 y Fk(\016)22 b Fl(x)586 1727 y Fi(2)625 1713 y Fp(\))29 b(=)g Fl(x)841 1727 y Fi(1)902 1713 y Fk(\016)22 b Fl(p)1015 1680 y Fh(0)1038 1713 y Fp(\()p Fl(x)1125 1727 y Fi(2)1165 1713 y Fp(\),)33 b(i.e.)47 b Fl(p)1467 1727 y Fj(P)10 b(K)1622 1713 y Fp(lea)m(v)m(es)33 b(the)g(\014rst)f(half)f(of)i(its)e(input)g(unc)m(hanged.)46 b(It)33 b(is)e(easy)150 1826 y(to)g(see)g(that)g Fl(p)651 1840 y Fj(P)10 b(K)804 1826 y Fp(is)29 b(still)g(a)h(p)s(erm)m(utation) g(and)g(that)h(in)m(v)m(erting)e(it)h(is)g(equiv)-5 b(alen)m(t)29 b(to)i(in)m(v)m(erting)f Fl(p)3577 1793 y Fh(0)3577 1853 y Fj(P)10 b(K)3700 1826 y Fp(.)291 1939 y(No)m(w)29 b(what)f(do)s(es)g (Bob's)h(encryption)e(lo)s(ok)g(lik)m(e?)39 b(It's)29 b(\()p Fl(m)16 b Fk(\010)f Fl(G)p Fp(\()p Fl(s)p Fp(\))p Fl(;)g(p)2682 1953 y Fj(P)10 b(K)2806 1939 y Fp(\()p Fl(s)p Fp(\)\))26 b(=)f(\()p Fl(m)16 b Fk(\010)f Fl(G)3364 1906 y Fh(0)3388 1939 y Fp(\()p Fl(s)3466 1953 y Fi(1)3506 1939 y Fp(\))p Fl(;)g(s)3624 1953 y Fi(1)3679 1939 y Fk(\010)150 2051 y Fl(p)196 2065 y Fj(P)10 b(K)319 2051 y Fp(\()p Fl(s)397 2065 y Fi(2)436 2051 y Fp(\)\).)41 b(F)-8 b(rom)31 b(this,)f(it)g(is)f(easy)i(to)g(compute)g Fl(m)p Fp(!)291 2264 y(Common)h(errors:)44 b(A)32 b(common)g(error)g(p) s(eople)f(made)i(w)m(as)f(to)h(try)f(a)h(coun)m(terexample)f(argumen)m (t)150 2377 y(where)f(the)h(construction)g(of)g Fl(G)g Fp(is)f(based)g(on)h Fl(p)p Fp(.)44 b(The)32 b(problem)e(with)g(this)h (is)g(that)h(in)f(this)f(problem,)150 2490 y Fl(G)k Fp(is)f(a)h Fn(single)f Fp(PR)m(G,)i(but)d Fl(p)i Fp(is)e(a)i(mem)m(b)s(er)f(of)h (a)g(trap)s(do)s(or)f(p)s(erm)m(utation)f(family)-8 b(.)49 b(In)33 b(order)g(for)h(the)150 2603 y(argumen)m(t)g(to)g(b)s(e)f(v)-5 b(alid,)33 b Fl(G)h Fp(cannot)g(dep)s(end)e(in)g(an)m(y)i(w)m(a)m(y)g (on)g(the)g(k)m(ey)g Fl(P)13 b(K)40 b Fp(used)33 b(to)h(sp)s(ecify)e (whic)m(h)150 2716 y(mem)m(b)s(er)h(of)g(the)h(TDP)f(family)e Fl(p)i Fp(is)g(to)h(b)s(e)e(used.)49 b(The)33 b(kinds)e(of)i(solutions) f(I)h(sa)m(w)h(that)g(fell)d(prey)i(to)150 2828 y(this)c(trap)h(w)m (ere)h(not)g(easily)e(\014xable.)291 3141 y Fq(\(c\))h Fp(Consider)f(the)h(follo)m(wing)f(cryptosystem:)291 3254 y(Key)45 b(generation:)69 b(generate)46 b Fl(n)j Fp(=)g Fl(pq)e Fp(where)d Fl(p)g Fp(and)g Fl(q)k Fp(are)d(primes)e(of)h (length)g Fl(k)k Fp(suc)m(h)c(that)150 3366 y Fl(p)25 b Fk(\021)g Fl(q)j Fk(\021)d Fp(7)h(mo)s(d)e(8.)41 b(Public)28 b(k)m(ey)k(is)d Fl(n)p Fp(,)h(secret)h(k)m(ey)h(is)d Fl(p;)15 b(q)s Fp(.)291 3479 y(Encryption:)42 b(to)33 b(encrypt)e(an)h Fl(`)p Fp(-bit)f(message)i Fl(m)p Fp(,)f(c)m(ho)s(ose) h(a)f(random)f Fl(s)c Fk(2)g Fg(Z)3000 3446 y Fh(\003)3000 3502 y Fj(n)3043 3479 y Fp(,)32 b(create)i(a)e(random)150 3592 y(pad)37 b Fl(r)j Fp(as)e(follo)m(ws:)54 b Fl(x)934 3606 y Fi(1)1011 3592 y Fp(=)36 b Fl(s)1161 3559 y Fi(2)1226 3592 y Fp(mo)s(d)24 b Fl(n)p Fp(;)41 b Fl(x)1598 3606 y Fj(i)1663 3592 y Fp(=)c Fl(x)1823 3559 y Fi(2)1823 3618 y Fj(i)p Fh(\000)p Fi(1)1967 3592 y Fp(mo)s(d)24 b Fl(n)37 b Fp(for)h(2)f Fk(\024)g Fl(i)h Fk(\024)f Fl(`)25 b Fp(+)f(1;)42 b Fl(r)3083 3606 y Fj(i)3149 3592 y Fp(=)36 b Fl(LS)5 b(B)g Fp(\()p Fl(x)3540 3606 y Fj(i)3568 3592 y Fp(\))38 b(for)150 3705 y(1)26 b Fk(\024)f Fl(i)g Fk(\024)g Fl(`)p Fp(.)41 b(Output)29 b(\()p Fl(m)20 b Fk(\010)g Fl(r)m(;)15 b(x)1252 3720 y Fj(`)p Fi(+1)1376 3705 y Fp(\).)291 3818 y(First,)40 b(describ)s(e)c(ho)m(w)i(to)h(decrypt)e(in) g(this)g(sc)m(heme)h(with)f(t)m(w)m(o)i(mo)s(dular)d(exp)s(onen)m (tiations)i(and)150 3931 y Fl(O)s Fp(\()p Fl(`)p Fp(\))d(mo)s(dular)d (m)m(ultiplications.)50 b Fn(Hint:)e(The)37 b(fact)f(that)i Fl(p)31 b Fk(\021)h Fp(7)26 b(mo)s(d)e(8)37 b Fn(c)-5 b(omes)37 b(in)f(handy.)54 b Fp(Second,)150 4044 y(form)m(ulate)34 b(a)g(hardness)f(assumption)f(that)j(corresp)s(onds)d(to)j(factoring)f Fl(n)g Fp(of)g(the)g(form)f(used)g(in)g(this)150 4157 y(system,)e(and)f(pro)m(v)m(e)h(that)g(this)e(cryptosystem)i(is)e (GM-secure)j(under)d(y)m(our)h(assumption.)291 4369 y Fq(Solution:)64 b Fp(This)41 b(cryptosystem)h(is)f(due)h(to)h(Blum)d (and)i(Goldw)m(asser.)76 b(It)42 b(is)f(notable)h(for)g(its)150 4482 y(e\016ciency)-8 b(,)24 b(b)s(oth)c(in)g(terms)h(of)h(comm)m (unication)e(and)h(in)f(terms)h(of)g(computation,)i(and)e(for)g(the)g (fact)h(that)150 4595 y(its)29 b(securit)m(y)g(rests)h(on)f(the)h (di\016cult)m(y)d(of)j(factoring)g(Blum)e(in)m(tegers,)i(whic)m(h)e(is) h(a)h(w)m(eak)m(er)g(assumption)150 4708 y(than)e(the)g(RSA)g (assumption)f(or)h(the)g(QRA.)g(It)h(is)e(also)h(co)s(ol)g(b)s(ecause,) h(although)f(w)m(e)g(kno)m(w,)h(o)m(wing)f(to)150 4821 y(part)34 b(\(b\))f(of)h(this)f(problem,)g(that)i(in)d(general)i(w)m(e) g(cannot)g(use)g(a)g(TDP)f(in)g(com)m(bination)g(with)f(PR)m(G)150 4934 y(to)f(securely)f(encrypt,)g(in)f(this)g(case,)j(this)d(is)h (\014ne!)291 5047 y(First,)g(let)g(us)g(sho)m(w)g(ho)m(w)h(to)g (compute)g(2)1726 5014 y Fj(`)1759 5047 y Fp(th)f(ro)s(ot)h(of)f Fl(v)k Fp(mo)s(dulo)29 b Fl(p)p Fp(.)1867 5416 y(15-2)p eop %%Page: 3 3 3 2 bop 291 159 a Fp(Note)42 b(that,)i(since)c Fl(v)k Fp(is)39 b(a)i(quadratic)f(residue)f(mo)s(dulo)g Fl(p)j Fp(=)g(8)p Fl(t)27 b Fp(+)g(7,)44 b(and)c(b)m(y)g(F)-8 b(ermat's)42 b(little)150 271 y(theorem,)1515 489 y Fl(v)87 b Fp(=)82 b Fl(v)24 b Fk(\016)c Fl(v)1989 421 y Ff(p)p Fe(\000)p Fd(1)p 1990 436 111 3 v 2030 478 a(2)2140 489 y Fp(mo)s(d)k Fl(p)1646 627 y Fp(=)82 b Fl(v)1846 589 y Fi(1+4)p Fj(t)p Fi(+3)2117 627 y Fp(mo)s(d)25 b Fl(p)1646 765 y Fp(=)82 b Fl(v)1846 727 y Fi(2\(2)p Fj(t)p Fi(+2\))2117 765 y Fp(mo)s(d)25 b Fl(p)1646 903 y Fp(=)82 b(\()p Fl(v)1881 865 y Fi(2)p Fj(t)p Fi(+2)2037 903 y Fp(\))2072 865 y Fi(2)2137 903 y Fp(mo)s(d)25 b Fl(p)291 1107 y Fp(Therefore,)30 b(raising)d Fl(v)33 b Fp(to)c(the)h(p)s(o)m(w)m(er)f(\(2)p Fl(t)19 b Fp(+)e(2\))30 b(giv)m(es)g(a)f(square)g(ro)s(ot.)41 b(What)30 b(is)e(crucial)g(is)g(that)i(is)150 1220 y(do)s(es)k(not)g (just)f(giv)m(e)i(some)f(square)g(ro)s(ot,)h(it)f(giv)m(es)g Fn(the)j(squar)-5 b(e)36 b(r)-5 b(o)g(ot)38 b(that)g(is)d(itself)h(a)h (squar)-5 b(e!)52 b Fp(Wh)m(y?)150 1333 y(Because)29 b(2)p Fl(t)15 b Fp(+)g(2)27 b(is)g(ev)m(en,)i(so)f Fl(v)1221 1300 y Fj(t)p Fi(+1)1369 1333 y Fp(m)m(ust)f(b)s(e)g(its)g(square)g(ro) s(ot.)40 b(So,)29 b(that)f(means)f(that)i(exp)s(onen)m(tiating)150 1446 y(to)42 b(2)p Fl(t)27 b Fp(+)g(2)41 b(in)m(v)m(erts)g(the)g (squaring)f(p)s(erm)m(utation)g(o)m(v)m(er)i(quadratic)e(residues)f(mo) s(dulo)h Fl(p)p Fp(.)71 b(Th)m(us,)43 b(to)150 1558 y(compute)35 b(the)h(2)730 1525 y Fj(`)763 1558 y Fp(th)f(ro)s(ot)g(that)h(it)e (itself)g(a)i(square,)g(w)m(e)f(m)m(ust)g(compute)g Fl(v)2791 1525 y Fi(\(2)p Fj(t)p Fi(+2\))2995 1502 y Ff(`)3057 1558 y Fp(mo)s(d)24 b Fl(p)p Fp(.)55 b(That)35 b(can)150 1671 y(b)s(e)e(accomplished)f(in)g(t)m(w)m(o)j(mo)s(dular)d(exp)s(onen) m(tiations.)49 b(What's)34 b(the)g(running)d(time?)50 b(T)-8 b(o)34 b(compute)150 1784 y(\(2)p Fl(t)8 b Fp(+)g(2\))430 1751 y Fj(`)490 1784 y Fp(mo)s(d)24 b Fl(p)8 b Fk(\000)g Fp(1,)26 b(w)m(e)f(need)f(to)h(carry)g(out)f Fl(O)s Fp(\()p Fl(`)p Fp(\))h(mo)s(dular)d(m)m(ultiplications)g(of)i Fl(k)s Fp(-bit)g(v)-5 b(alues,)25 b(whic)m(h)150 1897 y(tak)m(es)30 b Fl(O)s Fp(\()p Fl(`k)576 1864 y Fi(2)616 1897 y Fp(\))e(time.)40 b(T)-8 b(o)29 b(carry)f(out)h(the)f(second)h (exp)s(onen)m(tiation,)f(w)m(e)h(need)f Fl(O)s Fp(\()p Fl(k)3033 1864 y Fi(3)3073 1897 y Fp(\))h(time.)39 b(The)28 b(total)150 2010 y(time)i(is)f(therefore)i Fl(O)s Fp(\()p Fl(k)987 1977 y Fi(3)1047 2010 y Fp(+)20 b Fl(k)1188 1977 y Fi(2)1228 2010 y Fl(`)p Fp(\).)291 2123 y(Let)28 b(us)f(analyze)h(the)g(e\016ciency)f(of)h(the)g(sc)m(heme.)40 b(The)27 b(length)g(of)h(a)g(ciphertext)f(is)g(exactly)h Fl(`)15 b Fp(+)g(2)p Fl(k)s Fp(.)150 2236 y(The)37 b(cost)i(of)f (encrypting:)54 b(w)m(e)39 b(need)e(to)i(do)e(a)i(m)m(ultiplication)34 b(for)k(eac)m(h)h(encrypted)e(bit,)i(so)f(that's)150 2349 y Fl(O)s Fp(\()p Fl(k)307 2316 y Fi(2)347 2349 y Fl(`)p Fp(\).)i(T)-8 b(o)29 b(decrypt:)40 b(compute)29 b(2)1395 2316 y Fj(`)1428 2349 y Fp(th)f(ro)s(ot)h(of)g Fl(v)j Fp(mo)s(dulo)27 b Fl(p)h Fp(and)g Fl(q)k Fp(\(that)d(tak)m(es)h Fl(O)s Fp(\()p Fl(k)3177 2316 y Fi(3)3234 2349 y Fp(+)16 b Fl(k)3371 2316 y Fi(2)3411 2349 y Fl(`)p Fp(\)\),)29 b(then)150 2462 y(com)m(bine)23 b(using)f(the)i(CR)-8 b(T)23 b(\(that)h(is)f(dominated)f(b)m(y)i(computing)e(in)m(v)m(erses,) j(whic)m(h)d(is)h(extended)g(gcd,)j(so)150 2575 y(it's)j Fl(O)s Fp(\()p Fl(k)458 2542 y Fi(3)498 2575 y Fp(\)\),)i(then)f(one)g (m)m(ultiplication)c(p)s(er)j(bit)g(whic)m(h)g(is)f Fl(O)s Fp(\()p Fl(k)2386 2542 y Fi(2)2426 2575 y Fl(`)p Fp(\),)j(so)f(the)g (total)g(cost)h(of)f(decryption)150 2688 y(is)f Fl(O)s Fp(\()p Fl(k)398 2655 y Fi(3)456 2688 y Fp(+)18 b Fl(k)595 2655 y Fi(2)635 2688 y Fl(`)p Fp(\),)30 b(whic)m(h,)f(for)g(large)h Fl(`)p Fp(,)g(is)f(dominated)f(b)m(y)i Fl(O)s Fp(\()p Fl(k)2323 2655 y Fi(2)2363 2688 y Fl(`)p Fp(\),)g(whic)m(h)e(is)h(the)h (same)g(as)f(the)h(cost)h(of)150 2800 y(encryption.)291 2913 y(Note)f(just)e(ho)m(w)h(e\016cien)m(t)g(this)f(decryption)f(pro)s (cedure)h(is:)39 b(taking)28 b(ro)s(ots)h(is)f(a)h(m)m(uc)m(h)g(more)g (costly)150 3026 y(op)s(eration)g(than)h(squaring.)39 b(Ho)m(w)m(ev)m(er,)32 b(this)d(op)s(eration)g(is)g(only)g(carried)g (out)h(once.)41 b(Ev)m(erything)29 b(else)150 3139 y(is)g(just)h (squaring.)291 3252 y(Before)j(w)m(e)f(can)g(pro)m(v)m(e)g(the)g (securit)m(y)g(of)f(this)g(cryptosystem,)i(w)m(e)f(m)m(ust)f(giv)m(e)h (a)g(formal)f(hardness)150 3365 y(assumption)e(under)f(whic)m(h)h(w)m (e)i(will)c(pro)m(v)m(e)k(securit)m(y)-8 b(.)41 b(Supp)s(ose)29 b Fl(P)13 b(R)q(I)7 b(M)j(E)5 b Fp(\()p Fl(a;)15 b Fp(1)2948 3332 y Fj(k)2992 3365 y Fp(\))30 b(returns)f(a)i(random)150 3478 y(prime)e(of)h(length)g Fl(k)k Fp(equiv)-5 b(alen)m(t)29 b(to)j Fl(a)e Fp(mo)s(dulo)f(8,)i(or)f Fl(")h Fp(if)e(\()p Fl(a;)15 b Fp(8\))27 b Fk(6)p Fp(=)e(1.)291 3591 y(W)-8 b(e)32 b(mak)m(e)f(the)g(follo)m(wing)d(assumption:)150 3803 y Fk(8f)p Fl(A)314 3818 y Fj(k)357 3803 y Fk(g)p Fl(P)13 b(P)g(T)g(F)43 b Fk(9)p Fl(\027)31 b Fk(2)25 b Fl(neg)s(l)32 b Fk(8)p Fl(k)150 3916 y Fp(Pr)o([)p Fl(p)26 b Fk( )f Fl(P)13 b(R)q(I)7 b(M)j(E)5 b Fp(\(7)p Fl(;)15 b Fp(1)983 3883 y Fj(k)1027 3916 y Fp(\);)g Fl(q)29 b Fk( )c Fl(P)13 b(R)q(I)7 b(M)j(E)5 b Fp(\(7)p Fl(;)15 b Fp(1)1811 3883 y Fj(k)1856 3916 y Fp(\);)g(\()p Fl(a;)g(b)p Fp(\))27 b Fk( )e Fl(A)2339 3931 y Fj(k)2382 3916 y Fp(\()p Fl(pq)s Fp(\))h(:)273 4029 y Fk(f)p Fl(a;)15 b(b)p Fk(g)26 b Fp(=)f Fk(f)p Fl(p;)15 b(q)s Fk(g)p Fp(])27 b Fl(<)e(\027)6 b Fp(\()p Fl(k)s Fp(\))291 4242 y(No)m(w)31 b(let's)f(pro)m(v)m(e)h (that)g(the)f(resulting)e(cryptosystem)j(is)e(seman)m(tically)g (secure.)41 b(W)-8 b(e)31 b(will)d(use)i(the)150 4355 y(follo)m(wing)37 b(facts)j(from)e(lecture:)57 b(\(1\))40 b(if)d Fl(n)i Fp(is)e(a)i(Blum)f(in)m(teger,)j(then)e(computing)e(the)i (square)g(ro)s(ot)150 4468 y(of)32 b(a)g(giv)m(en)g(v)-5 b(alue)32 b(in)e Fl(QR)q Fp(\()p Fl(n)p Fp(\))i(is)f(equiv)-5 b(alen)m(t)31 b(to)i(factoring)f Fl(n)g Fp(\(in)f(other)h(w)m(ords,)g (in)m(v)m(erting)f(the)h(Blum)150 4581 y(p)s(erm)m(utation)f(is)g (equivlen)m(t)g(to)h(factoring\);)i(and)d(\(2\))i(LSB)f(is)f(hardcore)g (for)h(the)g(Blum)f(p)s(erm)m(utation)150 4693 y(unless)e(factoring)h (Blum)f(in)m(tegers)i(is)e(easy)-8 b(.)291 4806 y(Our)41 b(reduction)h(will)e(receiv)m(e)j(as)g(input)e(a)i(v)-5 b(alue)42 b Fl(y)48 b Fk(2)e Fl(QR)q Fp(\()p Fl(n)p Fp(\),)f(and)d(its) g(goal)i(is)d(to)j(compute)150 4919 y Fl(LS)5 b(B)g Fp(\()p Fl(B)g(l)r(um)617 4886 y Fh(\000)p Fi(1)617 4942 y Fj(n)710 4919 y Fp(\()p Fl(y)s Fp(\)\).)47 b(Supp)s(ose)31 b(an)h(adv)m(ersary)g Fl(A)h Fp(claims)e(that)i(he)f(can)g(distinguish)d(the)j(encryption)150 5032 y(of)f(message)g Fl(m)f Fp(from)g(that)h(of)g(a)g(random)e (message.)42 b(Then)30 b(b)m(y)g(the)h(h)m(ybrid)d(argumen)m(t)j(there) g(is)e(a)i(bit)150 5145 y Fl(i)36 b Fp(suc)m(h)g(that)g(if)f(the)i (\014rst)e Fl(i)h Fp(bits)f(of)h(the)g(message)h(are)g(random,)g(and)e (the)h(rest)g(come)h(from)f Fl(m)p Fp(,)h(this)1867 5416 y(15-3)p eop %%Page: 4 4 4 3 bop 150 159 a Fp(is)37 b(distinguishable)c(from)38 b(the)g(case)g(when)f(the)h(\014rst)f Fl(i)25 b Fp(+)g(1)38 b(bits)f(are)h(random,)h(and)e(the)h(rest)g(come)150 271 y(from)27 b Fl(m)p Fp(.)40 b(That)27 b(is)g(to)h(sa)m(y)-8 b(,)29 b(if)e Fl(p)1227 285 y Fj(j)1290 271 y Fp(is)g(the)h(probabilit) m(y)d(that)j Fl(A)f Fp(outputs)g(0)h(when)f(giv)m(en)g(the)h Fl(j)5 b Fp(th)28 b(h)m(ybrid,)150 384 y(then)f Fk(j)p Fl(p)425 398 y Fj(i)468 384 y Fk(\000)15 b Fl(p)600 398 y Fj(i)p Fi(+1)717 384 y Fk(j)26 b(\025)f Fl(\017)p Fp(\()p Fl(k)s Fp(\))j(for)f(a)h(non-negligible)d Fl(\017)p Fp(\()p Fl(k)s Fp(\).)41 b(Here's)28 b(what)g(the)f(reduction)g(will)e(do)i(to) h(create)150 497 y(a)38 b(ciphertext)g(\()p Fl(u;)15 b(v)s Fp(\):)57 b(on)37 b(input)f Fl(y)s Fp(,)k(it)d(will)e(mak)m(e)k (the)f(\014rst)f Fl(i)26 b Fp(+)f(1)38 b(bits)e(of)i Fl(u)g Fp(random.)62 b(It)38 b(will)d(set)150 610 y Fl(v)194 624 y Fj(i)p Fi(+2)338 610 y Fp(=)25 b Fl(y)s Fp(,)30 b Fl(v)581 624 y Fj(j)643 610 y Fp(=)25 b Fl(v)786 577 y Fi(2)783 636 y Fj(j)t Fh(\000)p Fi(1)940 610 y Fp(for)30 b Fl(i)21 b Fp(+)e(2)26 b Fl(<)f(j)31 b Fk(\024)25 b Fl(`)20 b Fp(+)g(1,)31 b Fl(u)1854 624 y Fj(j)1916 610 y Fp(=)25 b Fl(m)2092 624 y Fj(j)2148 610 y Fk(\010)20 b Fl(LS)5 b(B)g Fp(\()p Fl(v)2515 624 y Fj(j)2552 610 y Fp(\),)30 b Fl(i)21 b Fp(+)f(2)26 b Fk(\024)f Fl(j)30 b Fk(\024)25 b Fl(`)p Fp(.)41 b(The)30 b(resulting)150 723 y(ciphertext)i(will)e(b)s(e)h(\()p Fl(u;)15 b(v)1054 738 y Fj(`)p Fi(+1)1178 723 y Fp(\).)47 b(It)32 b(will)e(then)i(use)g (the)g(adv)m(ersary)h(to)g(predict)e(the)h Fl(i)22 b Fp(+)f(1st)33 b(bit)e(of)i(the)150 836 y(encrypted)g(message)h(the)f (same)h(w)m(a)m(y)g(as)f(in)f(problem)f(3)j(of)f(problem)e(set)j(5.)49 b(Denote)35 b(this)c(bit)i Fl(m)3607 803 y Fh(0)3607 861 y Fj(i)p Fi(+1)3725 836 y Fp(.)150 949 y(Output)c Fl(b)d Fp(=)e Fl(m)713 916 y Fh(0)713 974 y Fj(i)p Fi(+1)852 949 y Fk(\010)c Fl(u)995 963 y Fj(i)p Fi(+1)1143 949 y Fp(as)31 b(the)g(guess)f(for)g Fl(LS)5 b(B)g Fp(\()p Fl(B)g(l)r(um)2256 916 y Fh(\000)p Fi(1)2256 971 y Fj(n)2349 949 y Fp(\()p Fl(y)s Fp(\)\).)150 1232 y Fm(Problem)45 b(2)150 1435 y Fp(A)c(message)i(authen)m(tication)e(co)s(de)h(\(MA)m (C\))h(is)d(a)i(metho)s(d)f(for)g(ensuring)e(that)j(the)g(data)g (receiv)m(ed)150 1547 y(o)m(v)m(er)34 b(the)f(net)m(w)m(ork)h(came)g (from)e(the)h(righ)m(t)f(p)s(erson.)47 b(More)34 b(precisely)-8 b(,)32 b(it)h(is)f(an)g(ob)5 b(ject)34 b(that)f(satis\014es)150 1660 y(the)h(follo)m(wing)f(prop)s(erties:)46 b(Supp)s(ose)32 b(Alice)h(and)h(Bob)g(share)g(a)g(secret)h Fl(s)p Fp(,)g(and)e(Bob)h(w) m(an)m(ts)h(to)g(send)150 1773 y(message)27 b Fl(m)f Fp(to)h(Alice.)39 b(Then)25 b Fl(M)10 b(AC)1436 1787 y Fj(s)1473 1773 y Fp(\()p Fl(m)p Fp(\))26 b(is)f(an)h(e\016cien)m(tly) g(computable)g(function)f(suc)m(h)g(that)i(ev)m(en)g(if)150 1886 y(an)22 b(activ)m(e)i(attac)m(k)m(er)h(Ev)m(e)f(queries)d(it)h(on) g(a)h(set)g(of)g(messages)g(of)g(its)f(c)m(hoice,)j(it)d(still)e (cannot)k(authen)m(ticate)150 1999 y(an)m(y)k(message)h(not)f (explicitly)e(queried.)38 b(T)-8 b(o)29 b(giv)m(e)f(a)h(more)f(formal)f (de\014nition,)f(let)i(us)f(\014rst)g(extend)h(our)150 2112 y(notation.)40 b(If)27 b Fl(A)704 2079 y Fh(\016)743 2112 y Fp(\()p Fk(\016)p Fp(\))i(is)d(an)h(oracle)h(T)-8 b(uring)25 b(mac)m(hine,)j(then)f(b)m(y)g Fl(Q)e Fk( )h Fl(A)2638 2079 y Fj(O)2697 2112 y Fp(\()p Fl(x)p Fp(\))i(w)m(e)g (denote)g(the)f(con)m(ten)m(ts)150 2225 y(of)36 b Fl(A)p Fp('s)g(query)g(tap)s(e)g(up)s(on)e(termination)h(with)g(oracle)h Fl(O)j Fp(and)d(input)e Fl(x)p Fp(.)57 b(No)m(w:)c(A)36 b(function)f(family)150 2338 y Fk(f)p Fl(M)283 2352 y Fj(s)320 2338 y Fp(\()p Fk(\016)p Fp(\))p Fk(g)29 b Fp(is)e(a)h(MA)m(C) g(if)e(for)h(all)g(PPTF)g Fk(f)p Fl(E)5 b(v)s(e)1735 2353 y Fj(k)1779 2338 y Fk(g)p Fp(,)29 b(there)e(exists)g(a)h (negligible)d(function)h Fl(\027)6 b Fp(\()p Fl(k)s Fp(\))28 b(suc)m(h)f(that)519 2518 y(Pr)o([)p Fl(s)e Fk( )h(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)1046 2481 y Fj(k)1090 2518 y Fp(;)g(\()p Fl(m;)g(x)p Fp(\))p Fl(;)g(Q)27 b Fk( )e Fl(E)5 b(v)s(e)1788 2476 y Fj(M)1856 2487 y Ff(S)1788 2548 y Fj(k)1931 2518 y Fp(:)26 b Fl(x)f Fp(=)g Fl(M)2243 2532 y Fj(s)2280 2518 y Fp(\()p Fl(m)p Fp(\))31 b(and)e(\()p Fl(m;)15 b(x)p Fp(\))37 b Fl(=)-56 b Fk(2)25 b Fl(Q)p Fp(])h Fk(\024)f Fl(\027)6 b Fp(\()p Fl(k)s Fp(\))291 2782 y Fq(\(a\))30 b Fp(Let)g Fk(f)p Fl(F)719 2796 y Fj(s)813 2782 y Fp(:)56 b Fk(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)1114 2749 y Fh(j)p Fj(s)p Fh(j)1217 2782 y Fk(7!)25 b(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)1553 2749 y Fh(j)p Fj(s)p Fh(j)1631 2782 y Fk(g)31 b Fp(b)s(e)f(a)h(PRF)f(family)-8 b(.)40 b(Sho)m(w)29 b(that)i(it)f(is)g(a)h(MA)m(C.)291 2979 y Fq(Solution:)40 b Fp(Supp)s(ose)25 b(it)j(is)e(not)i(a)g(MA)m (C.)g(Then)f(let)g(us)g(use)h(Ev)m(e)g(to)g(tell)f(us)g(whether)g(an)g (oracle)h Fl(O)150 3092 y Fp(is)k(a)h(PRF)g(or)g(a)h(random)e (function.)47 b(The)32 b(reduction)g Fl(R)i Fp(is)e(as)h(follo)m(ws:)45 b(in)m(v)m(ok)m(e)34 b Fl(E)5 b(v)s(e)3161 3107 y Fj(k)3204 3092 y Fp(.)49 b(When)32 b Fl(E)5 b(v)s(e)3706 3107 y Fj(k)150 3205 y Fp(puts)38 b(a)g(query)g Fl(m)785 3219 y Fj(i)852 3205 y Fp(on)m(to)h(its)f(query)g(tap)s(e,)j(query)d Fl(O)s Fp(,)i(and)e(giv)m(e)h Fl(E)5 b(v)s(e)2639 3220 y Fj(k)2721 3205 y Fl(O)s Fp('s)39 b(answ)m(er)f Fl(O)s Fp(\()p Fl(m)3389 3219 y Fj(i)3417 3205 y Fp(\).)66 b(After)150 3318 y(p)s(olynomially)34 b(man)m(y)k(suc)m(h)g(queries,)h Fl(E)5 b(v)s(e)1670 3333 y Fj(k)1751 3318 y Fp(will)35 b(pro)s(duce)i(an)h(output)f(\()p Fl(m;)15 b(x)p Fp(\))49 b Fl(=)-55 b Fk(2)37 b Fl(Q)p Fp(.)63 b(\(If)38 b(it)f(breaks)150 3431 y(do)m(wn)31 b(without)g(an)h(output,)g(or)g(runs)e(longer)h(than) h(it)f(is)g(supp)s(osed)f(to,)j(or)f(if)f(\()p Fl(m;)15 b(x)p Fp(\))28 b Fk(2)g Fl(Q)p Fp(,)k(then)g(just)150 3544 y(set)38 b(\()p Fl(m;)15 b(x)p Fp(\))38 b(at)g(random)f(from)g (the)g(set)h(of)g Fl(k)s Fp(-bit)f(strings.\))61 b(No)m(w,)40 b(query)d Fl(O)j Fp(on)d Fl(m)p Fp(.)62 b(If)37 b Fl(x)g Fp(=)f Fl(O)s Fp(\()p Fl(m)p Fp(\),)150 3657 y(ouput)30 b(\\pseudorandom,")g(else)g(output)g(\\random.")291 3769 y(Note)36 b(that,)g(in)d(case)j Fl(O)h Fp(is)c(actually)h Fl(F)1658 3783 y Fj(s)1730 3769 y Fp(for)g(some)h Fl(s)p Fp(,)g(w)m(e)g(will)c(b)s(e)j(righ)m(t)g(whenev)m(er)g Fl(E)5 b(v)s(e)3437 3784 y Fj(k)3515 3769 y Fp(w)m(ould)150 3882 y(ha)m(v)m(e)32 b(succeeded)e(to)h(output)f(a)h(forgery)-8 b(,)31 b(i.e.)41 b(with)29 b(probabilit)m(y)f Fl(\017)p Fp(\()p Fl(k)s Fp(\).)291 3995 y(In)j(case)h Fl(O)i Fp(is)c(a)i(random) e(function,)h(w)m(e)g(can)h(mak)m(e)g(a)g(mistak)m(e)f(only)g(if)f Fl(E)5 b(v)s(e)2985 4010 y Fj(k)3060 3995 y Fp(manages)32 b(to)g(guess)150 4108 y Fl(R)q Fp(\()p Fl(m)p Fp(\))d(correctly)-8 b(.)40 b(But)29 b(this)e(happ)s(ens)g(with)g(probabilit)m(y)f(exactly)j (2)2535 4075 y Fh(\000)p Fj(n)2637 4108 y Fp(,)g(since)f Fl(R)q Fp(\()p Fl(m)p Fp(\))g(is)g(a)g(completely)150 4221 y(random)33 b(string)g(indep)s(enden)m(t)f(on)i(an)m(y)h(observ)m (ed)f(v)-5 b(alues.)52 b(So)34 b(if)f Fl(O)k Fp(is)c(a)i(random)e (function,)h(w)m(e)h(are)150 4334 y(correct)d(with)d(probabilit)m(y)f (1)20 b Fk(\000)g Fp(2)1327 4301 y Fh(\000)p Fj(k)1425 4334 y Fp(.)41 b(Observing)28 b(that)j(for)f(su\016cien)m(tly)f(large)i Fl(k)s Fp(,)g(2)3088 4301 y Fh(\000)p Fj(k)3211 4334 y Fk(\024)25 b Fl(\017)p Fp(\()p Fl(k)s Fp(\))p Fl(=)p Fp(6:)150 4627 y(Pr)o([)p Fl(R)32 b Fp(is)d(correct)q(])84 b(=)e(Pr[)p Fl(R)31 b Fp(is)f(correct)q Fk(j)p Fl(O)j Fp(is)d(random)o(])20 b Fk(\003)h Fp(1)p Fl(=)p Fp(2)h(+)e(Pr)o([)p Fl(R)31 b Fp(is)f(correct)q Fk(j)p Fl(O)j Fp(is)d(a)h(PRF])20 b Fk(\003)h Fp(1)p Fl(=)p Fp(2)846 4765 y Fk(\025)82 b Fl(\017)p Fp(\()p Fl(k)s Fp(\)1)p Fl(=)p Fp(2)23 b(+)d(1)p Fl(=)p Fp(2\(1)i Fk(\000)e Fp(2)1778 4728 y Fh(\000)p Fj(k)1876 4765 y Fp(\))846 4903 y Fk(\025)82 b Fp(1)p Fl(=)p Fp(2)22 b(+)e Fl(\017)p Fp(\()p Fl(k)s Fp(\))p Fl(=)p Fp(3)291 5167 y(\(b\))31 b(Is)f(it)g(the)g(case)i(that)f(a)f(MA) m(C)h(is)f(a)g(PRF?)1867 5416 y(15-4)p eop %%Page: 5 5 5 4 bop 291 159 a Fq(Solution:)39 b Fp(No.)h(Let)26 b Fl(M)1160 173 y Fj(s)1223 159 y Fp(b)s(e)f(a)i(MA)m(C.)g(Let)f Fl(M)1936 126 y Fh(0)1926 181 y Fj(s)1963 159 y Fp(\()p Fl(m)p Fp(\))g(=)f(0)11 b Fk(\016)g Fl(M)2435 173 y Fj(s)2473 159 y Fp(\()p Fl(m)p Fp(\))27 b(for)f(all)e Fl(m)i Fp(and)g(for)f(all)g Fl(s)p Fp(.)39 b(No)m(w,)150 271 y Fl(M)248 238 y Fh(0)238 294 y Fj(s)308 271 y Fp(is)32 b(still)e(a)k(MA)m(C)f(\(if)f(an)h(adv)m (ersary)g(could)f(forge)i(an)e Fl(m;)15 b(M)2391 238 y Fh(0)2381 294 y Fj(s)2418 271 y Fp(\()p Fl(m)p Fp(\))34 b(pair,)e(this)g(is)g(an)h Fl(m;)15 b Fp(0)22 b Fk(\016)g Fl(M)3562 285 y Fj(s)3599 271 y Fp(\()p Fl(m)p Fp(\))150 384 y(pair,)31 b(so)i(w)m(e)f(can)g(compute)h Fl(m;)15 b(M)1360 398 y Fj(s)1397 384 y Fp(\()p Fl(m)p Fp(\).)46 b(If)31 b(the)i(adv)m(ersary)f(nev)m(er)g(queried)f Fl(M)2948 351 y Fh(0)2938 407 y Fj(s)3007 384 y Fp(on)h(input)e Fl(m)p Fp(,)i(it)g(will)150 497 y(succeed)f(and)f(so)g(will)e(w)m (e.\),)k(but)d Fl(M)1423 464 y Fh(0)1413 520 y Fj(s)1481 497 y Fp(is)g(not)i(a)f(PRF)h(since)e(it)h(alw)m(a)m(ys)h(starts)g (with)e(0.)291 610 y(T)-8 b(o)32 b(distinguish)27 b Fl(M)985 577 y Fh(0)1039 610 y Fp(from)k(random,)g(query)f(one)h(random)g(v)-5 b(alue.)42 b(If)30 b(the)i(\014rst)e(bit)g(of)h(the)g(output)150 723 y(is)g(0,)h(guess)g(pseduorandom.)43 b(Oth)m(w)m(erise,)32 b(guess)g(random.)43 b(This)30 b(will)f(b)s(e)i(correct)i(with)d (probabilit)m(y)150 836 y(3/4.)291 949 y(\(c\))i(Is)e(it)f(true)i(that) g(if)e(MA)m(Cs)i(exist)f(then)g(PRFs)g(exist?)291 1161 y Fq(Solution:)52 b Fp(Y)-8 b(es.)57 b(Recall)36 b(that,)i(b)m(y)d (HILL,)h(existence)g(of)g(O)m(WFs)g(implies)d(existence)j(of)g(PR)m (Gs.)150 1274 y(Then)h(b)m(y)i(GGM,)g(existence)g(of)g(PR)m(Gs)g (implies)c(existence)k(of)g(PRFs.)65 b(So)38 b(w)m(e)h(need)f(to)h(sho) m(w)f(that)150 1387 y(MA)m(C)31 b Fk(\))f Fp(O)m(WF)h(and)f(then)g(w)m (e)h(will)d(b)s(e)i(all)f(set.)291 1500 y(Let)40 b Fk(f)p Fl(M)596 1514 y Fj(s)674 1500 y Fp(:)h Fk(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)960 1467 y Fh(j)p Fj(s)p Fh(j)1078 1500 y Fk(7!)41 b(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)1430 1467 y Fj(p)p Fi(\()p Fh(j)p Fj(s)p Fh(j)p Fi(\))1598 1500 y Fk(g)40 b Fp(b)s(e)f(a)h(MA)m(C.)g(Consider)e(the)i(follo)m (wing)d(function,)k(whic)m(h)150 1613 y(w)m(e)e(shall)d(pro)m(v)m(e)j (is)f(one-w)m(a)m(y)1184 1580 y Fi(1)1225 1613 y Fp(:)56 b Fl(f)86 b Fp(:)77 b Fk(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)1759 1580 y Fj(k)r Fi(+2)p Fj(k)1927 1557 y Fd(2)2006 1613 y Fk(7!)38 b(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)2355 1580 y Fi(2)p Fj(k)r(p)p Fi(\()p Fj(k)r Fi(\))2564 1613 y Fp(,)40 b Fl(f)10 b Fp(\()p Fl(x)p Fp(\))39 b(=)f Fl(f)10 b Fp(\()p Fl(s;)15 b(m)3207 1627 y Fi(1)3246 1613 y Fl(;)g(:)g(:)g(:)h (;)f(m)3527 1628 y Fi(2)p Fj(k)3606 1613 y Fp(\))38 b(=)150 1726 y Fl(m)230 1740 y Fi(1)292 1726 y Fk(\016)23 b Fl(:)15 b(:)g(:)23 b Fk(\016)g Fl(m)636 1741 y Fi(2)p Fj(k)736 1726 y Fk(\016)g Fl(M)892 1740 y Fj(s)929 1726 y Fp(\()p Fl(m)1044 1740 y Fi(1)1084 1726 y Fp(\))g Fk(\016)g Fl(:)15 b(:)g(:)23 b Fk(\016)g Fl(M)1494 1740 y Fj(s)1531 1726 y Fp(\()p Fl(m)1646 1741 y Fi(2)p Fj(k)1724 1726 y Fp(\).)52 b(W)-8 b(e)35 b(will)c(sho)m(w)j(that)g(in)m(v)m(erting)f(this)g Fl(f)43 b Fp(is)32 b(equiv)-5 b(alen)m(t)150 1839 y(to)32 b(forging)e(authen)m(tication,)h(i.e.)42 b(that,)32 b(from)e(an)h(adv)m (ersary)g Fl(A)2401 1854 y Fj(k)2475 1839 y Fp(that)g(in)m(v)m(erts)g Fl(f)10 b Fp(,)30 b(w)m(e)h(can)h(construct)150 1952 y Fl(E)5 b(v)s(e)311 1912 y Fj(M)379 1920 y Ff(s)311 1981 y Fj(k)447 1952 y Fp(that)31 b(can)g(output)f(\()p Fl(m;)15 b(M)1352 1966 y Fj(s)1390 1952 y Fp(\()p Fl(m)p Fp(\)\))31 b(with)e(non-negligible)f(probabilit)m(y)-8 b(.)291 2065 y(Note)33 b(that)f(this)e(problem)g(is)h(harder)f(than)h (y)m(ou)h(migh)m(t)g(ha)m(v)m(e)g(though)m(t.)45 b(The)31 b(reason)g(it)g(is)g(tric)m(ky)150 2178 y(is)d(that)i(just)f(b)s (ecause)g(w)m(e)g(ha)m(v)m(e)i(in)m(v)m(erted)e(the)g(function)f(and)h (obtained)f(a)i(p)s(ossible)c(shared)j(secret)h Fl(s)3702 2145 y Fh(0)3725 2178 y Fp(,)150 2291 y(do)s(es)i(not)h(mean)f(that)h (the)f(secret)i Fl(s)e Fp(that)g(Alice)g(and)g(Bob)h(ha)m(v)m(e)g(got)h (is)d(equal)h(to)h(it!)46 b(Care)32 b(m)m(ust)h(b)s(e)150 2403 y(tak)m(en)e(to)h(mak)m(e)f(sure)f(that)h Fl(s)1179 2370 y Fh(0)1232 2403 y Fp(is)e(helpful)f(to)j(Ev)m(e)g(ev)m(en)g(if)e (it)h(isn't)g(equal)g(to)h Fl(s)p Fp(.)291 2516 y(Construct)c Fl(E)5 b(v)s(e)874 2531 y Fj(k)944 2516 y Fp(as)28 b(follo)m(ws:)38 b(it)26 b(queries)g Fl(M)1863 2530 y Fj(s)1927 2516 y Fp(on)h(2)p Fl(k)k Fp(randomly)25 b(c)m(hosen)j(messages)g Fl(m)3326 2530 y Fi(1)3365 2516 y Fl(;)15 b(:)g(:)g(:)i(;)e(m)3647 2531 y Fi(2)p Fj(k)3725 2516 y Fp(,)150 2629 y(and)25 b(obtains)h Fl(x)689 2643 y Fi(1)728 2629 y Fl(;)15 b(:)g(:)g(:)i(;)e (x)982 2644 y Fi(2)p Fj(k)1060 2629 y Fp(.)39 b(It)26 b(then)g(runs)e(in)m(v)m(erter)i Fl(A)2019 2644 y Fj(k)2062 2629 y Fp(\()p Fl(x)2149 2643 y Fi(1)2189 2629 y Fl(;)15 b(:)g(:)g(:)h(;)f(x)2442 2644 y Fi(2)p Fj(k)2521 2629 y Fp(\))26 b(and)f(obtains)h(\()p Fl(s)3147 2596 y Fh(0)3170 2629 y Fl(;)15 b(m)3290 2643 y Fi(1)3330 2629 y Fl(;)g(:)g(:)g(:)h(;)f (m)3611 2644 y Fi(2)p Fj(k)3689 2629 y Fp(\).)150 2742 y(It)30 b(then)h(c)m(ho)s(oses)g(a)g(random)e Fl(m)h Fp(and)g(outputs)g(outputs)g(\()p Fl(m;)15 b(M)2387 2757 y Fj(s)2420 2739 y Fe(0)2447 2742 y Fp(\()p Fl(m)p Fp(\)\).)291 2855 y(W)-8 b(e)32 b(wish)c(to)j(sho)m(w)g(that)g(with)e (non-negligible)e(probabilit)m(y)-8 b(,)29 b Fl(M)2547 2869 y Fj(s)2584 2855 y Fp(\()p Fl(m)p Fp(\))d(=)f Fl(M)2944 2870 y Fj(s)2977 2851 y Fe(0)3003 2855 y Fp(\()p Fl(m)p Fp(\).)291 2968 y(Supp)s(ose)31 b Fl(s)i Fp(is)f(\014xed.)47 b(Let)34 b Fl(bad)p Fp(\()p Fl(s)p Fp(\))f(denote)h(strings)e Fl(s)2164 2935 y Fh(0)2216 2968 y Fk(2)d(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)2526 2935 y Fh(j)p Fj(s)p Fh(j)2637 2968 y Fp(suc)m(h)33 b(that)h(Pr)o([)p Fl(m)c Fk( )f(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)3617 2935 y Fh(j)p Fj(s)p Fh(j)3725 2968 y Fp(:)150 3081 y Fl(M)238 3095 y Fj(s)275 3081 y Fp(\()p Fl(m)p Fp(\))26 b(=)f Fl(M)635 3096 y Fj(s)668 3077 y Fe(0)694 3081 y Fp(\()p Fl(m)p Fp(\)])h Fl(<)f Fp(1)p Fl(=)p Fp(2.)42 b(Clearly)-8 b(,)28 b(if)g Fl(s)1650 3048 y Fh(0)1701 3081 y Fp(giv)m(en)h(to)h(us)e(b)m(y)h Fl(A)2355 3096 y Fj(k)2426 3081 y Fp(is)f Fn(not)i Fp(bad,)e(then)h (our)f(Ev)m(e)i(succeeds)150 3194 y(with)c(probabilit)m(y)e(at)k(least) g(1)p Fl(=)p Fp(2.)41 b(So)27 b(w)m(e)g(w)m(an)m(t)h(to)g(mak)m(e)g (sure)e(that)i(there)f(are)h(no)f(bad)f Fl(s)3248 3161 y Fh(0)3271 3194 y Fp(s)h(among)h(the)150 3307 y(preimages)i(of)h Fl(f)10 b Fp(\()p Fl(s;)15 b(m)932 3321 y Fi(1)971 3307 y Fl(;)g(:)g(:)g(:)i(;)e(m)1253 3322 y Fi(2)p Fj(k)1331 3307 y Fp(\).)41 b(Here,)32 b(the)f(union)d(b)s(ound)h(is)g(helpful.)39 b(Recall)30 b(the)h(union)e(b)s(ound:)150 3420 y(Pr)o([)p Fl(A)i Fp(or)g Fl(B)5 b Fp(])25 b Fk(\024)g Fp(Pr)o([)p Fl(A)p Fp(])c(+)f(Pr)o([)p Fl(B)5 b Fp(].)41 b(So)469 3737 y(Pr)o([)p Fl(s)25 b Fk( )h(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)996 3699 y Fj(k)1040 3737 y Fp(;)g Fk(f)p Fl(m)1205 3751 y Fj(i)1260 3737 y Fk( )25 b(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)1596 3699 y Fj(k)1666 3737 y Fp(:)25 b(1)h Fk(\024)f Fl(i)h Fk(\024)f Fp(2)p Fl(k)s Fk(g)106 b Fp(:)g(Bad)31 b Fl(s)2647 3704 y Fh(0)2700 3737 y Fp(exists)f(among)h Fl(f)3299 3704 y Fh(\000)p Fi(1)3393 3737 y Fp(\()p Fl(f)10 b Fp(\()p Fl(s;)15 b(m)3681 3751 y Fi(1)3720 3737 y Fl(;)g(:)g(:)g(:)i (;)e(m)4002 3752 y Fi(2)p Fj(k)4080 3737 y Fp(\)\))q(])25 b Fk(\024)532 3802 y Fc(X)459 4007 y Fj(s)p Fh(2f)p Fi(0)p Fj(;)p Fi(1)p Fh(g)699 3988 y 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Fp(\()p Fl(f)10 b Fp(\()p Fl(s;)15 b(m)3027 4175 y Fi(1)3066 4161 y Fl(;)g(:)g(:)g(:)i (;)e(m)3348 4176 y Fi(2)p Fj(k)3426 4161 y Fp(\)\)])26 b(=)224 4348 y Fc(X)150 4553 y Fj(s)p Fh(2f)p Fi(0)p Fj(;)p Fi(1)p Fh(g)390 4534 y Ff(k)444 4434 y Fp(Pr)o([)p Fl(s)p Fp(])731 4348 y Fc(X)649 4549 y Fj(s)682 4530 y Fe(0)704 4549 y Fh(2)p Fj(bad)p Fi(\()p Fj(s)p Fi(\))958 4434 y Fp(Pr[)p Fk(f)p Fl(m)1206 4448 y Fj(i)1260 4434 y Fk( )f(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)1596 4397 y Fj(k)1666 4434 y Fp(:)25 b(1)h Fk(\024)f Fl(i)h Fk(\024)f Fp(2)p Fl(k)s Fk(g)106 b Fp(:)g Fl(M)2501 4449 y Fj(s)2534 4431 y Fe(0)2561 4434 y Fp(\()p Fl(m)2676 4448 y Fj(i)2704 4434 y Fp(\))26 b(=)f Fl(M)2949 4448 y Fj(s)2986 4434 y Fp(\()p Fl(m)3101 4448 y Fj(i)3129 4434 y Fp(\))p Fk(8)p Fp(1)h Fk(\024)f Fl(i)g Fk(\024)g Fp(2)p Fl(k)s Fp(])h(=)1264 4621 y Fc(X)1190 4826 y Fj(s)p Fh(2f)p Fi(0)p Fj(;)p Fi(1)p Fh(g)1430 4807 y Ff(k)1484 4707 y Fp(Pr)o([)p Fl(s)p Fp(])1771 4621 y Fc(X)1689 4822 y Fj(s)1722 4803 y Fe(0)1745 4822 y Fh(2)p Fj(bad)p Fi(\()p Fj(s)p Fi(\))1998 4707 y Fp(2)2043 4670 y Fh(\000)p Fi(2)p Fj(k)2260 4707 y Fl(<)2487 4621 y Fc(X)2413 4826 y Fj(s)p Fh(2f)p Fi(0)p Fj(;)p Fi(1)p Fh(g)2653 4807 y Ff(k)2707 4707 y Fp(Pr[)p Fl(s)p Fp(]2)2943 4670 y Fj(k)2986 4707 y Fp(2)3031 4670 y Fh(\000)p Fi(2)p Fj(k)3189 4707 y Fp(=)f(2)3330 4670 y Fj(k)3373 4707 y Fp(2)3418 4670 y Fh(\000)p Fj(k)3516 4707 y Fp(2)3561 4670 y Fh(\000)p Fj(k)3685 4707 y Fp(=)g(2)3826 4670 y Fh(\000)p Fj(k)p 150 4867 1440 4 v 254 4921 a Fd(1)288 4953 y Fb(Th)n(us)c(function)g(do)r(esn't)h(w)n(ork)f(on)g(ev) n(ery)f(input)h(size,)i(but)d(using)h(the)g(preliminary)f(result)i(in)f (problem)f(3a,)j(w)n(e)e(pro)n(v)n(e)150 5044 y(this)26 b(do)r(esn't)g(matter.)1867 5416 y Fp(15-5)p eop %%Page: 6 6 6 5 bop 291 159 a Fp(Therefore,)36 b(the)g(probabilit)m(y)c(that)k (there)f(exists)g(a)h(bad)e Fl(s)2365 126 y Fh(0)2423 159 y Fp(is)g(negligible.)53 b(So)35 b(with)e(probabilit)m(y)150 271 y(1)24 b Fk(\000)e Fp(2)357 238 y Fh(\000)p Fj(k)455 271 y Fp(,)36 b(the)f Fl(s)720 238 y Fh(0)778 271 y Fp(giv)m(en)g(to)g (us)f(b)m(y)h(the)g(in)m(v)m(erter)f(will)e(yield)h(probabilit)m(y)g(1) p Fl(=)p Fp(2)j(of)f(successful)e(forgery)-8 b(.)150 384 y(The)28 b(in)m(v)m(erter)g(will)d(succeed)k(with)e(probabilit)m(y) f Fl(\017)p Fp(\()p Fl(k)s Fp(\),)j(so)g(the)f(forgery)g(will)e (succeed)j(with)d(probabilit)m(y)150 497 y Fl(\017)p Fp(\()p Fl(k)s Fp(\))p Fl(=)p Fp(2\(1)d Fk(\000)d Fp(2)636 464 y Fj(k)679 497 y Fp(\).)291 610 y(P)m(oten)m(tial)31 b(erroneous)g(solutions:)39 b(trying)29 b(to)i(go)h(directly)d(from)h (MA)m(C)h(to)g(PRF)f(is)g(v)m(ery)h(di\016cult.)150 723 y(The)37 b(problem)f(is)h(that)h(GL)g(bit,)h(i.e.,)g(taking)f(a)g (random)f Fl(r)j Fp(and)d(setting)h Fl(F)2871 737 y Fj(s)2908 723 y Fp(\()p Fl(x)p Fp(\))g(=)f Fl(M)3264 737 y Fj(s)3301 723 y Fp(\()p Fl(x)p Fp(\))p Fl(:r)k Fp(w)m(on't)150 836 y(w)m(ork.)51 b(This)32 b(is)h(b)s(ecause)h(a)g(MA)m(C)h(can)f(b)s (e)f(designed)g(suc)m(h)g(that)i Fl(M)2559 850 y Fj(s)2596 836 y Fp(\()p Fl(r)s Fp(\))p Fl(:r)h Fp(is)d(alw)m(a)m(ys)i(0!)51 b(So)34 b(there)g(is)150 949 y(hop)s(e)c(only)f(if)h Fl(r)i Fp(is)e(k)m(ept)h(secret.)41 b(The)30 b(actual)h(pro)s(of)f(is)f (tric)m(ky)-8 b(.)150 1235 y Fm(Problem)45 b(3)150 1438 y Fp(In)27 b(this)g(problem,)g(w)m(e)i(will)c(try)j(to)g(relate)h(v)-5 b(arious)27 b(kinds)f(of)i(assumptions)e(ab)s(out)i(one-w)m(a)m(y)h (functions)150 1551 y(to)i(eac)m(h)h(other.)291 1664 y(\(a\))23 b(Pro)m(v)m(e)g(that)f(if)f(one-w)m(a)m(y)i(function)e (families)e(exist,)24 b(then)d(one-w)m(a)m(y)j(functions)c(exist.)38 b(Do)s(es)22 b(y)m(our)150 1777 y(pro)s(of)k(go)i(through)e(for)h (one-w)m(a)m(y)i(p)s(erm)m(utations)d(from)g(one-w)m(a)m(y)j(p)s(erm)m (utation)d(families?)37 b(Informally)150 1890 y(argue)31 b(wh)m(y)f(or)g(wh)m(y)g(not.)291 2101 y Fq(Solution:)44 b Fp(First)31 b(of)h(all,)f(w)m(e)h(pro)m(v)m(e)h(a)f(related)g (result.)44 b(Supp)s(ose)30 b(w)m(e)i(ha)m(v)m(e)h(a)f(function)e Fl(F)45 b Fp(whic)m(h)150 2214 y(is)28 b(one-w)m(a)m(y)i(but)e(whic)m (h)f(w)m(orks)i(on)f(only)g(in)m(tegers)h(of)f(the)h(form)f Fl(p)p Fp(\()p Fl(k)s Fp(\))h(for)g(some)g(\014xed)f(p)s(olynomial)e Fl(p)3686 2181 y Fi(2)3725 2214 y Fp(.)150 2327 y(W)-8 b(e)32 b(can)e(create)i(a)f(function)e Fl(F)1244 2294 y Fh(0)1293 2327 y Fp(:)c Fk(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)1563 2294 y Fh(\003)1630 2327 y Fk(!)25 b(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)1966 2294 y Fh(\003)2038 2327 y Fp(whic)m(h)29 b(is)h(one)g(w)m(a)m(y)i(as)e(follo)m(ws:)150 2539 y(1.)41 b(On)30 b(input)e Fl(x)i Fp(where)g Fk(j)p Fl(x)p Fk(j)c Fp(=)f Fl(y)s Fp(,)30 b(\014nd)f Fl(y)1557 2506 y Fh(0)1610 2539 y Fp(suc)m(h)i(that)g Fl(y)2061 2506 y Fh(0)2109 2539 y Fp(=)25 b Fl(p)p Fp(\()p Fl(k)s Fp(\))31 b(but)e Fl(y)g(<)c(p)p Fp(\()p Fl(k)e Fp(+)d(1\).)41 b(2.)g(Sa)m(y)31 b Fl(x)25 b Fp(=)g Fl(x)3634 2553 y Fi(0)3694 2539 y Fk(\016)c Fl(x)3812 2553 y Fi(1)3881 2539 y Fp(where)30 b Fk(j)p Fl(x)4221 2553 y Fi(0)4261 2539 y Fk(j)25 b Fp(=)g Fl(y)4455 2506 y Fh(0)4478 2539 y Fp(.)150 2652 y(3.)41 b(Output)29 b Fl(F)13 b Fp(\()p Fl(x)742 2666 y Fi(0)782 2652 y Fp(\).)291 2863 y(No)m(w,)38 b(the)e(only)f(hitc)m(h)g(in)f(pro)m(ving)h(that)h Fl(F)1850 2830 y Fh(0)1909 2863 y Fp(is)e(a)i(O)m(WF)h(is)e(that)h(p)s(erhaps)e (w)m(e)i(use)f(to)s(o)h(little)f(of)150 2976 y(the)i(input.)57 b(Supp)s(ose)34 b(that)k Fl(y)1223 2943 y Fh(0)1281 2976 y Fk(\030)d Fp(log)17 b Fl(y)s Fp(;)39 b(then,)f(an)f(adv)m(ersary)f (could)g(\014nd)f(an)h(appropriate)g Fl(x)3541 2943 y Fh(0)3601 2976 y Fp(just)150 3089 y(b)m(y)30 b(brute)f(force.)41 b(Ho)m(w)m(ev)m(er,)32 b(this)d(will)e(not)k(b)s(e)e(the)h(case)h(for)f (p)s(olynomials.)37 b(W)-8 b(e)31 b(kno)m(w)f(that)h(for)f(some)150 3202 y(p)s(olynomial)e Fl(q)s Fp(,)j Fl(y)768 3169 y Fh(0)817 3202 y Fk(\025)25 b Fp(\(1)p Fl(=q)s Fp(\)\()p Fl(y)s Fp(\).)44 b(The)30 b(probabilit)m(y)e(that)j(an)g(adv)m(ersary)g (reco)m(v)m(ers)h(an)m(y)f Fl(x)3264 3169 y Fh(0)3318 3202 y Fp(from)f Fl(F)3604 3169 y Fh(0)3627 3202 y Fp(\()p Fl(x)p Fp(\))150 3314 y(is)35 b(negligible)e(in)h Fl(y)815 3282 y Fh(0)838 3314 y Fp(,)j(since)e Fl(F)49 b Fp(is)35 b(a)h(O)m(WF.)h(This,)e(in)f(turn,)i(is)f(negligible)e(in)i Fl(y)j Fp(since)d Fl(\027)6 b Fp(\(\(1)p Fl(=q)s Fp(\)\()p Fl(y)s Fp(\)\))38 b(is)150 3427 y(negligible)28 b(in)h Fl(y)s Fp(.)291 3540 y(Th)m(us,)35 b(w)m(e)g(need)g(only)e(pro)m(v)m(e) j(that)f(the)g(existence)g(of)g(O)m(WF)g(families)e(implies)f(the)j (existence)g(of)150 3653 y(O)m(WF's)c(de\014ned)f(on)g(suc)m(h)g (inputs.)291 3865 y(No)m(w,)42 b(the)d(trouble)e(going)i(from)f(a)h(O)m (WFF)h(to)f(a)g(O)m(WF)h(is)e(that)h(in)e(a)i(O)m(WFF,)h Fl(f)3252 3879 y Fj(P)10 b(K)3413 3865 y Fp(ma)m(y)40 b(not)150 3978 y(b)s(e)32 b(one-w)m(a)m(y)i(for)f(an)m(y)g(particular)e Fl(P)13 b(K)7 b Fp(.)47 b(There)32 b(could)g(actually)g(b)s(e)g(a)i (class)e(of)h(adv)m(ersaries,)g(one)g(for)150 4091 y(ev)m(ery)i Fl(P)13 b(K)7 b Fp(,)35 b(suc)m(h)f(that)h Fl(A)1087 4105 y Fj(P)10 b(K)1245 4091 y Fp(can)34 b(in)m(v)m(ert)h Fl(f)1725 4105 y Fj(P)10 b(K)1881 4091 y Fp(and)34 b(y)m(et)i(this)d(w) m(ould)g(still)f(b)s(e)i(a)h(O)m(WFF:)h(no)e Fn(single)150 4204 y Fp(adv)m(ersary)d(could)e(encompass)i(all)e(these)i(and)e(still) g(b)s(e)g(p)s(olynomial)f(size.)2729 4171 y Fi(3)291 4317 y Fp(Ho)m(w)m(ev)m(er,)34 b(for)e(randomly)e(c)m(hosen)i Fl(P)13 b(K)38 b Fp(the)32 b(function)e(should)g(b)s(e)h(hard)f(to)j (in)m(v)m(ert.)44 b(Supp)s(ose)30 b(the)150 4430 y(generation)c (algorithm)e Fl(G)i Fp(tak)m(es)h Fl(p)p Fp(\()p Fl(k)s Fp(\))f(bits)e(of)i(randomness)e(for)i(some)f(p)s(olynomial)e Fl(p)p Fp(.)39 b(No)m(w,)27 b(supp)s(ose)150 4542 y Fk(j)p Fl(x)p Fk(j)44 b Fp(=)f Fl(p)p Fp(\()p Fl(k)s Fp(\))28 b(+)f Fl(k)44 b Fp(\(w)m(e)e(presume)e(here)h(that)h Fl(f)1804 4556 y Fj(P)10 b(K)1967 4542 y Fp(simply)39 b(op)s(erates)i(on)g Fk(f)p Fp(0)p Fl(;)15 b Fp(1)p Fk(g)2998 4509 y Fj(k)3043 4542 y Fp(,)44 b(but)d(actually)f(w)m(e)150 4655 y(could)33 b(assume)i(nothing)e(and)h(let)g Fl(q)s Fp(\()p Fl(k)s Fp(\))h(b)s(e)f(the)g(amoun)m(t)h(of)g(randomness)e (needed)h(b)m(y)g(the)h(sampling)p 150 4716 1440 4 v 254 4770 a Fd(2)288 4802 y Fb(Actually)-6 b(,)29 b(w)n(e)g(assume)g Fa(p)f Fb(is)h(monotonically)g(increasing,)i(but)d(this)h(can)g(b)r(e)f (relaxed.)44 b(Using)29 b(only)g(monotonically)150 4893 y(increasing)23 b Fa(p)e Fb(will)h(not)g(b)r(e)f(problematic)g(as)h Fa(p)p Fb(\()p Fa(k)r Fb(\))f(=)g Fa(O)r Fb(\()p Fa(k)1821 4861 y Ff(c)1854 4893 y Fb(\))g(for)h(some)g(constan)n(t)f Fa(c)p Fb(,)i(and)e Fa(k)2776 4861 y Ff(c)2829 4893 y Fb(is)h(monotonically)g(increasing.)254 4953 y Fd(3)288 4985 y Fb(W)-6 b(e)22 b(actually)i(do)e(require)g(that)h(our)f(adv)n (ersarial)i(TM's)g(b)r(e)f(p)r(olynomial)f(size)i(\(in)e(the)g(securit) n(y)g(parameter\))h(alw)n(a)n(ys.)150 5076 y(F)-6 b(or)25 b(instance,)i(if)f(w)n(e)g(relax)f(this)h(requiremen)n(t,)e(a)i(TM)g (with)g(a)f(lo)r(okup)h(table)g(of)g(factorizations)i(for)e(all)g(n)n (um)n(b)r(ers)e(up)h(to)150 5167 y(2)188 5135 y Ff(k)252 5167 y Fb(could)h(factor.)1867 5416 y Fp(15-6)p eop %%Page: 7 7 7 6 bop 150 159 a Fp(algorithm)32 b Fl(S)38 b Fp(and)32 b(let)g Fk(j)p Fl(m)p Fk(j)e Fp(=)f Fl(p)p Fp(\()p Fl(k)s Fp(\))23 b(+)e Fl(q)s Fp(\()p Fl(k)s Fp(\)\),)35 b(and)d(let)h Fl(x)c Fp(=)g Fl(x)2319 173 y Fi(0)2380 159 y Fk(\016)23 b Fl(x)2500 173 y Fi(1)2572 159 y Fp(where)32 b Fk(j)p Fl(x)2914 173 y Fi(0)2954 159 y Fk(j)d Fp(=)g Fl(p)p Fp(\()p Fk(j)p Fl(x)3266 173 y Fi(1)3306 159 y Fk(j)p Fp(\).)48 b(No)m(w)34 b(let)150 271 y Fl(F)13 b Fp(\()p Fl(x)p Fp(\))33 b(=)f Fl(G)p Fp(\(1)630 238 y Fh(j)p Fj(x)690 247 y Fd(1)725 238 y Fh(j)749 271 y Fp(;)15 b Fl(x)841 285 y Fi(0)880 271 y Fp(\))24 b Fk(\016)f Fl(f)1052 300 y Fj(G)p Fi(\(1)1169 277 y Fe(j)p Ff(x)1223 292 y Fd(1)1258 277 y Fe(j)1281 300 y Fi(;)p Fj(x)1341 309 y Fd(0)1375 300 y Fi(\))1406 271 y Fp(\()p Fl(x)1493 285 y Fi(1)1533 271 y Fp(\).)53 b(\(Here,)37 b Fl(G)p Fp(\(1)2078 238 y Fj(k)2122 271 y Fp(;)15 b Fl(r)s Fp(\))35 b(denotes)g(the)f(output)g(of)h Fl(G)g Fp(on)f(input)f(1)3707 238 y Fj(k)150 393 y Fp(with)c(randomness)g Fl(r)s Fp(.\))291 506 y(This)c(de\014nes)h(a)h(function)e(that)i(w)m(orks)g(on)f(all)g (inputs)e(of)j(size)f Fl(p)p Fp(\()p Fl(k)s Fp(\))13 b(+)g Fl(k)29 b Fp(for)e(some)g(in)m(teger)g Fl(k)s Fp(.)39 b(If)26 b(w)m(e)150 618 y(can)h(pro)m(v)m(e)g(this)e(function)g(is)h (one-w)m(a)m(y)i(w)m(e'll)d(b)s(e)h(done.)39 b(So,)27 b(assume)f(not.)40 b(Assume)26 b(there)h(exists)f(some)150 731 y(adv)m(ersary)35 b Fl(A)635 746 y Fj(k)713 731 y Fp(whic)m(h)f(\014nds)f(preimages)h(of)i Fl(F)47 b Fp(with)34 b(probabilit)m(y)f(1)p Fl(=q)s Fp(\()p Fl(p)p Fp(\()p Fl(k)s Fp(\))25 b(+)e Fl(k)s Fp(\))35 b(when)f(the)h(inputs)150 844 y(ha)m(v)m(e)d(size)e Fl(p)p Fp(\()p Fl(k)s Fp(\))21 b(+)f Fl(k)s Fp(,)30 b(where)g Fl(q)j Fp(is)d(some)h(p)s(olynomial.)38 b(T)-8 b(o)30 b(break)h Fl(f)2506 858 y Fj(P)10 b(K)2628 844 y Fp(,)31 b(w)m(e)f(do)h(the)f(follo)m(wing:)150 1057 y(On)g(input)e Fl(P)13 b(K)q(;)i(y)s Fp(,)150 1170 y(1.)41 b(Let)31 b Fl(P)13 b(K)579 1137 y Fh(0)622 1170 y Fk(\016)21 b Fl(x)740 1137 y Fh(0)788 1170 y Fk( )26 b Fl(A)973 1185 y Fj(k)1015 1170 y Fp(\()p Fl(P)13 b(K)28 b Fk(\016)20 b Fl(y)s Fp(\).)150 1283 y(2.)41 b(Output)29 b Fl(x)636 1250 y Fh(0)660 1283 y Fp(.)291 1495 y(If)c Fl(A)445 1510 y Fj(k)512 1495 y Fp(is)f(successful)g(then)g Fl(P)13 b(K)32 b Fp(=)25 b Fl(P)13 b(K)1638 1462 y Fh(0)1686 1495 y Fp(and)24 b Fl(f)1902 1509 y Fj(P)10 b(K)2025 1495 y Fp(\()p Fl(x)2112 1462 y Fh(0)2135 1495 y Fp(\))26 b(=)f Fl(y)s Fp(,)h(so)f(w)m(e)g(will)d(b)s(e)j(successful.)37 b(The)25 b(prob-)150 1608 y(abilit)m(y)j(of)h(this)f(is)g(at)i(least)f (1)p Fl(=q)s Fp(\()p Fl(p)p Fp(\()p Fl(k)s Fp(\))20 b(+)d Fl(k)s Fp(\),)30 b(whic)m(h)e(is)g(equal)h(to)h(1)p Fl(=q)2536 1575 y Fh(0)2560 1608 y Fp(\()p Fl(k)s Fp(\))g(for)f(some)g(other)h(p)s (olynomial)150 1721 y Fl(q)194 1688 y Fh(0)217 1721 y Fp(,)h(so)f(w)m(e)h(are)g(successful)e(with)g(nonnegligible)f (probabilit)m(y)-8 b(.)291 1834 y(As)34 b(for)h(whether)e(this)g (generalizes)i(to)g(O)m(WP's,)h(the)e(quic)m(k)g(answ)m(er)h(is)e(no.) 52 b(The)34 b(main)g(problem)150 1947 y(is)h(that)h(w)m(e)g(ha)m(v)m(e) h(no)e(guaran)m(tee)i(that)g Fl(G)f Fp(can)f(b)s(e)g(in)m(v)m(erted.)57 b(But)35 b(also,)j Fl(P)13 b(K)42 b Fp(ma)m(y)36 b(b)s(e)f(longer)g (than)150 2060 y(the)28 b(randomness)f(used)g(to)i(create)g(it,)g(so)f (this)f(construction)g(ma)m(y)i(not)f(b)s(e)g(length)f(preserving.)38 b(There)150 2173 y(really)32 b(is)h(no)g(easy)h(\014x)f(to)h(either)f (of)h(these)g(problems.)48 b(One)33 b(idea)g(that)h(do)s(esn't)f(quite) g(w)m(ork)h(for)f(the)150 2286 y(second)24 b(problem)f(is)g(to)h (output)g(the)g(randomness)f(instead)h(of)g(the)g(PK,)g(but)f(this)g (has)h(a)g(problem)f(if)g(the)150 2398 y(O)m(WPF)k(is)e(a)i(TDPF)f (\(for)g(instance\),)h(b)s(ecause)g(giving)e(the)h(randomness)f(w)m (ould)g(allo)m(w)g(the)i(adv)m(ersary)150 2511 y(to)k(reco)m(v)m(er)h (the)f(trap)s(do)s(or.)291 2724 y Fq(Common)f(errors:)40 b Fp(There)27 b(w)m(e)i(t)m(w)m(o)g(ma)5 b(jor)28 b(common)g(errors)f (here.)40 b(One)27 b(\(whic)m(h)g(also)h(o)s(ccurred)150 2837 y(with)20 b(ab)s(out)h(95)h(p)s(ercen)m(t)g(frequency)f(in)f(part) h(\(b\)\))h(w)m(as)f(the)h(mistak)m(e)g(of)f(not)h(dealing)e(with)g (the)i(problem)150 2950 y(that)33 b(the)f(construction)g(only)g(pro)s (duces)f(a)h(function)f(that)i(w)m(orks)g(on)f(certain)g(input)e (lengths.)46 b(It's)33 b(a)150 3063 y(messy)d(detail,)g(but)g (necessary)-8 b(.)291 3176 y(The)25 b(other)h(error)f(w)m(as)h(to)h (only)d(output)h Fl(f)1733 3190 y Fj(P)10 b(K)1856 3176 y Fp(\()p Fl(x)1943 3190 y Fi(1)1983 3176 y Fp(\))26 b(instead)e(of)i Fl(P)13 b(K)k Fk(\016)11 b Fl(f)2717 3190 y Fj(P)f(K)2840 3176 y Fp(\()p Fl(x)2927 3190 y Fi(1)2967 3176 y Fp(\).)39 b(This)24 b(is)h(a)h(problem)150 3288 y(b)s(ecause)34 b(in)f(the)h(reduction,)g(if)f(the)h(adv)m(ersary) h(returns)e(a)h(preimage)g(whic)m(h)f(do)s(esn't)h(use)f(the)i(same)150 3401 y Fl(P)13 b(K)7 b Fp(,)44 b(it's)d(useless)f(to)i(us.)73 b(This)40 b Fn(may)i Fp(not)g(b)s(e)f(a)g(problem)f(in)g(general,)k (but)d(it)g(can)h(b)s(e)e(a)i(serious)150 3514 y(problem.)60 b(Here's)37 b(an)g(example.)61 b(Supp)s(ose)36 b Fl(f)1796 3528 y Fj(P)10 b(K)1955 3514 y Fp(is)36 b(a)i(O)m(WFF,)h(and)d(let)h Fl(f)2855 3481 y Fh(0)2845 3541 y Fj(P)10 b(K)3004 3514 y Fp(=)37 b Fl(f)3157 3528 y Fj(P)10 b(K)3316 3514 y Fp(for)37 b(all)f Fl(P)13 b(K)150 3627 y Fp(except)34 b(one,)g Fl(P)13 b(K)781 3641 y Fi(0)851 3627 y Fk( )29 b Fl(G)p Fp(\(1)1122 3594 y Fj(k)1166 3627 y Fl(;)15 b Fp(0)1251 3594 y Fj(p)p Fi(\()p Fj(k)r Fi(\))1385 3627 y Fp(\).)50 b Fl(f)1550 3594 y Fh(0)1540 3654 y Fj(P)10 b(K)1655 3663 y Fd(0)1692 3627 y Fp(\()p Fl(x)p Fp(\))30 b(=)g Fl(x)j Fp(for)g(all)f Fl(x)p Fp(.)48 b(No)m(w,)35 b Fl(P)13 b(K)2810 3641 y Fi(0)2883 3627 y Fp(o)s(ccurs)32 b(with)g(negligible)150 3740 y(probabilit)m(y)-8 b(,)27 b(so)i(this)e(is)h(still)e(a)j(O)m(WFF.)h(Ho)m(w)m(ev)m(er,)h(the)d (construction)g(without)g(giving)f Fl(P)13 b(K)35 b Fp(is)28 b(easily)150 3853 y(defeated:)41 b(return)30 b Fl(x)877 3820 y Fh(0)877 3877 y Fi(0)941 3853 y Fp(=)25 b(0)1082 3820 y Fj(k)1156 3853 y Fp(and)30 b Fl(x)1385 3820 y Fh(0)1385 3877 y Fi(1)1449 3853 y Fp(=)25 b Fl(f)1590 3867 y Fj(P)10 b(K)1713 3853 y Fp(\()p Fl(x)1800 3867 y Fi(1)1839 3853 y Fp(\).)291 4066 y(\(b\))40 b(A)g(function)e(is)h (length-preserving)f(if)h(the)h(length)f(of)h(its)f(output)g(is)g(alw)m (a)m(ys)h(equal)f(to)i(the)150 4178 y(length)33 b(of)h(its)g(input.)49 b(Pro)m(v)m(e)35 b(that)g(if)e(one-w)m(a)m(y)i(functions)e(exist,)h (then)g(length-preserving)e(one-w)m(a)m(y)150 4291 y(functions)d (exist.)291 4504 y Fq(Solution:)40 b Fp(Assume)29 b(the)g(existence)g (of)g Fl(F)13 b Fp(,)30 b(a)f(O)m(WF.)h(FIrst)f(of)g(all,)g(w)m(e'll)f (sho)m(w)h(that)g(there)g(exists)150 4617 y(an)h Fl(F)347 4584 y Fh(0)401 4617 y Fp(suc)m(h)g(that)h Fk(j)p Fl(x)p Fk(j)26 b Fp(=)f Fk(j)p Fl(y)s Fk(j)g(\))g(j)p Fl(F)1362 4584 y Fh(0)1386 4617 y Fp(\()p Fl(x)p Fp(\))p Fk(j)h Fp(=)f Fk(j)p Fl(F)1751 4584 y Fh(0)1775 4617 y Fp(\()p Fl(y)s Fp(\))p Fk(j)p Fp(.)291 4730 y(Since)42 b Fl(F)56 b Fp(is)43 b(computable)f(in)g(p)s(olynomial)e(time,)47 b(let)c Fl(p)f Fp(b)s(e)h(a)g(p)s(olynomial)e(whic)m(h)h(b)s(ounds)f (its)150 4843 y(running)29 b(time.)44 b(Let)32 b Fl(F)971 4810 y Fh(0)995 4843 y Fp(\()p Fl(x)p Fp(\))g(simply)d(output)j Fl(F)13 b Fp(\()p Fl(x)p Fp(\))32 b(padded)e(with)h(enough)g(prep)s (ended)f(zeros)i(to)g(giv)m(e)150 4956 y(an)37 b(output)f(of)i(length)e Fl(p)p Fp(\()p Fk(j)p Fl(x)p Fk(j)p Fp(\).)61 b(No)m(w,)40 b(since)c Fk(j)p Fl(F)13 b Fp(\()p Fl(x)p Fp(\))p Fk(j)38 b(\024)e Fl(p)p Fp(\()p Fk(j)p Fl(x)p Fk(j)p Fp(\))i(this)e(mak)m(es)h (the)h(output)e(a)i(constan)m(t)150 5068 y(size.)45 b(F)-8 b(urthermore,)32 b(if)e(this)h(can)h(b)s(e)f(easily)g(in)m(v)m(erted,)h (w)m(e)g(could)f(just)g(tak)m(e)i(an)f(input)e Fl(y)s Fp(,)i(prep)s(end)d(it)1867 5416 y(15-7)p eop %%Page: 8 8 8 7 bop 150 159 a Fp(with)31 b(enough)h(zeros)h(to)g(ha)m(v)m(e)h (length)d Fl(p)p Fp(\()p Fl(k)s Fp(\),)j(and)e(then)f(get)j(the)e (preimage)g(of)h(this)e(under)g Fl(F)3439 126 y Fh(0)3462 159 y Fp(,)i(whic)m(h)150 271 y(will)28 b(b)s(e)h(the)i(preimage)f(of)h Fl(y)i Fp(under)c Fl(F)43 b Fp(if)29 b(the)i(adv)m(ersary)g(correctly)f (in)m(v)m(erts)h Fl(F)2929 238 y Fh(0)2952 271 y Fp(.)291 384 y(No)m(w,)25 b(supp)s(ose)20 b(w)m(e)j(ha)m(v)m(e)g(a)f(O)m(WF)h Fl(F)35 b Fp(suc)m(h)22 b(that)g Fk(j)p Fl(F)13 b Fp(\()p Fl(x)p Fp(\))p Fk(j)27 b Fp(=)e Fl(p)p Fp(\()p Fk(j)p Fl(x)p Fk(j)p Fp(\))d(for)g(some)g(p)s(olynomial.)36 b(Supp)s(ose)150 497 y Fl(p)p Fp(\()p Fl(k)s Fp(\))26 b Fl(<)f(k)30 b Fp(for)c(some)h Fl(k)s Fp(.)39 b(No)m(w,)29 b(w)m(e)e(can)g(construct)g(an)f Fl(F)2100 464 y Fh(0)2150 497 y Fp(whic)m(h)f(\014xes)i(length)f(as)g(follo)m(ws.)39 b(If)26 b Fl(x)h Fp(is)e(suc)m(h)150 610 y(that)35 b Fl(p)p Fp(\()p Fk(j)p Fl(x)p Fk(j)p Fp(\))d Fl(>)e Fk(j)p Fl(x)p Fk(j)p Fp(,)36 b(just)d(ha)m(v)m(e)i Fl(F)1332 577 y Fh(0)1356 610 y Fp(\()p Fl(x)p Fp(\))d(=)e Fl(F)13 b Fp(\()p Fl(x)p Fp(\).)52 b(Otherwise,)34 b(let)g Fl(F)2547 577 y Fh(0)2570 610 y Fp(\()p Fl(x)p Fp(\))e(=)f(0)2871 577 y Fh(j)p Fj(x)p Fh(j\000j)p Fj(F)10 b Fi(\()p Fj(x)p Fi(\))p Fh(j)3220 610 y Fk(\016)23 b Fl(F)13 b Fp(\()p Fl(x)p Fp(\).)52 b(By)35 b(a)150 723 y(practically)29 b(iden)m(tical)g(argumen)m(t,)i Fl(F)1466 690 y Fh(0)1520 723 y Fp(is)f(still)e(a)j(O)m(WF.)291 836 y(Finally)-8 b(,)36 b(supp)s(ose)e(w)m(e)i(ha)m(v)m(e)h(a)f(O)m(WF)h Fl(F)49 b Fp(suc)m(h)35 b(that)i(for)e(all)g Fl(x)p Fp(,)i Fk(j)p Fl(F)13 b Fp(\()p Fl(x)p Fp(\))p Fk(j)35 b Fp(=)f Fl(p)p Fp(\()p Fk(j)p Fl(x)p Fk(j)p Fp(\))j(and)e Fl(p)p Fp(\()p Fl(k)s Fp(\))g Fk(\025)f Fl(k)150 949 y Fp(for)e(all)e Fl(k)s Fp(.)45 b(Supp)s(ose)30 b Fl(x)i Fp(is)f(suc)m(h)g(that)i Fk(j)p Fl(x)p Fk(j)28 b Fp(=)f Fl(p)p Fp(\()p Fl(k)s Fp(\))32 b(for)g(some)g(in)m(teger)g Fl(k)s Fp(.)45 b(No)m(w)33 b(let)e Fl(x)d Fp(=)f Fl(x)3267 963 y Fi(0)3328 949 y Fk(\016)21 b Fl(x)3446 963 y Fi(1)3517 949 y Fp(where)150 1062 y Fk(j)p Fl(x)227 1076 y Fi(0)267 1062 y Fk(j)29 b Fp(=)g Fl(k)s Fp(,)k(and)f(let)h Fl(F)913 1029 y Fh(0)936 1062 y Fp(\()p Fl(x)p Fp(\))d(=)e Fl(F)13 b Fp(\()p Fl(x)1345 1076 y Fi(0)1385 1062 y Fp(\).)48 b(No)m(w)33 b Fk(j)p Fl(F)1798 1029 y Fh(0)1822 1062 y Fp(\()p Fl(x)p Fp(\))p Fk(j)d Fp(=)e Fl(p)p Fp(\()p Fl(k)s Fp(\))i(=)f Fk(j)p Fl(x)p Fk(j)p Fp(,)k(so)g Fl(F)2739 1029 y Fh(0)2795 1062 y Fp(is)f(length-preserving.)45 b(T)-8 b(o)150 1175 y(sho)m(w)30 b(that)h Fl(F)643 1142 y Fh(0)697 1175 y Fp(is)e(a)i(O)m(WF,)150 1460 y Fm(Problem)45 b(4)150 1663 y Fp(Supp)s(ose)25 b(that)i Fl(F)754 1677 y Fj(s)791 1663 y Fp(\()p Fl(x)p Fp(\))g(is)f(a)h(pseudorandom)e(function)g(from)h Fl(k)k Fp(bit)25 b(input)g(to)i Fl(k)j Fp(bit)c(output,)h(where)f Fl(s)g Fp(is)150 1776 y(the)j(seed)g(and)f Fl(x)h Fp(is)f(the)h(input,) e(decide)h(whether)h(eac)m(h)h(of)f(the)g(follo)m(wing)e(are)i(alw)m(a) m(ys)g(pseudorandom)150 1889 y(functions.)43 b(If)31 b(so,)h(pro)m(v)m(e)h(it.)43 b(If)31 b(not,)i(pro)m(vide)d(a)i(coun)m (terexample)g(and)f(explain)f(ho)m(w)h(to)i(distinguish)150 2002 y(the)h(giv)m(en)f(function)g(from)g(random.)50 b(It)33 b(is)g(ok)-5 b(a)m(y)35 b(to)f(lea)m(v)m(e)h(out)f(the)g (details)e(of)i(y)m(our)g(pro)s(ofs)e(so)i(long)150 2115 y(as)d(they)f(can)h(b)s(e)f(\015eshed)f(out)i(correctly)f(without)g(to) s(o)h(m)m(uc)m(h)f(trouble.)291 2322 y Fq(\(a\))g Fl(F)525 2289 y Fi(1)512 2345 y Fj(s)564 2322 y Fp(\()p Fl(x)p Fp(\))c(=)f Fl(F)866 2344 y Fi(0)901 2326 y Ff(k)944 2322 y Fp(\()p Fl(x)p Fp(\))c Fk(\016)f Fl(F)1210 2336 y Fj(s)1248 2322 y Fp(\(0)1328 2289 y Fj(k)1371 2322 y Fp(\))291 2530 y Fq(Solution:)46 b Fp(This)31 b(is)h(not)h(a)g(PRF.)g (\(Actually)f(none)h(of)g(these)g(are)g(PRFs,)g(so)g(I'll)f(just)g(lea) m(v)m(e)i(this)150 2643 y(part)k(out)g(of)g(the)g(solution)f(from)g(no) m(w)h(on.\))64 b(This)36 b(one)i(has)g(t)m(w)m(o)h(big)e(problems)f (with)h(it.)63 b(First)37 b(of)150 2756 y(all,)g(the)f(second)h(part)f (of)g(the)h(output)e(do)s(esn't)i(dep)s(end)d(on)i Fl(x)p Fp(,)i(so)f(if)e(w)m(e)i(query)e(the)i(oracle)f(on)h(t)m(w)m(o)150 2869 y(di\013eren)m(t)30 b(inputs)e(and)i(the)h(second)g(part)f(is)g (the)g(same)h(in)f(b)s(oth,)g(w)m(e)h(can)g(b)s(e)f(prett)m(y)h(sure)f (the)g(output)150 2982 y(is)e(pseudorandom.)39 b(Also,)30 b(the)f(\014rst)g(part)g(do)s(esn't)h(dep)s(end)d(on)j Fl(s)p Fp(,)f(so)h(w)m(e)g(can)f(compute)h(it)f(ourselv)m(es.)150 3094 y(So,)h(w)m(e)h(can)f(see)h(if)e Fl(O)s Fp(\(0)983 3061 y Fj(k)1026 3094 y Fp(\))d(=)f Fl(F)1241 3117 y Fi(0)1276 3098 y Ff(k)1319 3094 y Fp(\(0)1399 3061 y Fj(k)1442 3094 y Fp(\))c Fk(\016)f Fl(r)32 b Fp(for)e(some)h Fl(r)s Fp(,)f(and)f(if)g(so)h(w)m(e)h(can)f(b)s(e)g(prett)m(y)h(sure)e (the)h(oracle)150 3207 y(is)f(pseudorandom.)291 3320 y(The)d(probabilit)m(y)d(of)k(success)f(for)g(either)g(of)g(these)h(t)m (w)m(o)g(strategies)g(is)e(1)12 b Fk(\000)g Fl(neg)s(l)29 b Fp(since)c(with)g(a)h(small)150 3433 y(probabilit)m(y)i(the)i(prop)s (erties)f(w)m(e)i(are)g(c)m(hec)m(king)g(will)d(also)i(hold)f(for)h(a)h (random)f(function.)291 3641 y Fq(\(b\))g Fl(F)532 3608 y Fi(2)519 3663 y Fj(s)571 3641 y Fp(\()p Fl(x)p Fp(\))c(=)f Fl(F)873 3663 y Fi(0)908 3644 y Ff(k)951 3641 y Fp(\()p Fl(s)p Fp(\))c Fk(\016)f Fl(F)1208 3655 y Fj(s)1246 3641 y Fp(\()p Fl(x)p Fp(\))291 3849 y Fq(Solution:)38 b Fp(By)25 b(the)f(same)h(argumen)m(t)g(as)g(part)f(\(a\),)j(the)d(\014rst)g(half) f(of)i(the)f(output)g(do)s(esn't)h(dep)s(end)150 3961 y(on)37 b Fl(x)p Fp(,)i(so)e(w)m(e)g(can)g(just)g(try)g(t)m(w)m(o)h (di\013eren)m(t)e Fl(x)h Fp(v)-5 b(alues)37 b(and)f(if)g(the)h(\014rst) f(half)g(is)g(the)h(same)h(for)e(b)s(oth,)150 4074 y(guess)30 b(pseudorandom.)291 4282 y Fq(\(c\))g Fl(F)520 4249 y Fi(3)507 4305 y Fj(s)560 4282 y Fp(\()p Fl(x)p Fp(\))c(=)f Fl(F)862 4296 y Fj(s)899 4282 y Fp(\()p Fl(x)p Fp(\))c Fk(\016)f Fl(F)1165 4296 y Fj(x)1210 4282 y Fp(\()p Fl(s)p Fp(\))291 4490 y Fq(Solution:)41 b Fp(This)27 b(do)s(esn't)j (necessarily)e(end)h(up)g(b)s(eing)f(pseudorandom)g(b)s(ecause)h Fl(F)43 b Fp(ma)m(y)30 b(ha)m(v)m(e)h(a)150 4603 y(w)m(eak)h(seed.)42 b(F)-8 b(or)31 b(example,)g(let)g Fl(F)1360 4570 y Fh(0)1347 4625 y Fj(s)1415 4603 y Fp(b)s(e)f(a)h(PRF,)g(and)f(let)h Fl(F)2226 4617 y Fj(s)2263 4603 y Fp(\()p Fl(x)p Fp(\))c(=)e Fl(F)2579 4570 y Fh(0)2566 4625 y Fj(s)2603 4603 y Fp(\()p Fl(x)p Fp(\))32 b(if)e Fl(s)25 b Fk(6)p Fp(=)h(0)3051 4570 y Fj(k)3094 4603 y Fp(,)31 b(but)f Fl(F)3375 4625 y Fi(0)3410 4606 y Ff(k)3453 4603 y Fp(\()p Fl(x)p Fp(\))c(=)g Fl(x)150 4716 y Fp(for)40 b(all)e Fl(x)p Fp(.)70 b(No)m(w,)43 b(with)38 b(probabilit)m(y)g(1)27 b Fk(\000)f Fp(2)1730 4683 y Fh(\000)p Fj(k)1828 4716 y Fp(,)43 b(an)c Fl(F)2089 4730 y Fj(s)2166 4716 y Fp(oracle)i(is)e(an)h Fl(F)2747 4683 y Fh(0)2734 4738 y Fj(s)2811 4716 y Fp(oracle,)j(so)d(no)g(adv)m (ersary)150 4828 y(can)29 b(gain)f(a)h(nonnegligible)c(adv)-5 b(an)m(tage)31 b(distinguishing)24 b Fl(F)41 b Fp(from)28 b(random.)39 b(Ho)m(w)m(ev)m(er,)31 b(to)f(distinguish)150 4941 y Fl(F)221 4908 y Fi(3)208 4964 y Fj(s)294 4941 y Fp(from)k(random,)g(simply)e(try)i Fl(x)d Fp(=)g(0)1554 4908 y Fj(k)1597 4941 y Fp(,)k(and)e(when)g(w)m(e)i(get)g Fl(r)25 b Fk(\016)e Fl(s)33 b Fp(c)m(hec)m(k)j(whether)d Fl(r)h Fp(=)d Fl(F)3417 4955 y Fj(s)3454 4941 y Fp(\(0)3534 4908 y Fj(k)3578 4941 y Fp(\).)51 b(If)150 5054 y(so,)37 b(output)e(pseudorandom,)g(otherwise)f(output)h(random.)54 b(W)-8 b(e)36 b(will)d(b)s(e)h(correct)j(with)c(probabilit)m(y)150 5167 y(1)21 b Fk(\000)f Fl(neg)s Fp(.)1867 5416 y(15-8)p eop %%Page: 9 9 9 8 bop 291 159 a Fq(\(d\))30 b Fl(F)532 126 y Fi(4)519 181 y Fj(s)571 159 y Fp(\()p Fl(x)p Fp(\))c(=)f Fl(F)873 173 y Fj(s)906 182 y Fd(1)945 159 y Fp(\()p Fl(x)p Fp(\))c Fk(\016)g Fl(F)1212 173 y Fj(s)1245 182 y Fd(2)1283 159 y Fp(\()p Fl(x)p Fp(\))31 b(where)f Fl(s)1742 173 y Fi(1)1807 159 y Fp(=)25 b Fl(F)1961 173 y Fj(s)1998 159 y Fp(\()p Fl(x)p Fp(\))31 b(and)f Fl(s)2371 173 y Fi(2)2435 159 y Fp(=)25 b Fl(F)p 2589 135 37 3 v 13 x Fj(s)2626 159 y Fp(\()p Fl(x)p Fp(\))291 364 y Fq(Solution:)55 b Fp(Supp)s(ose)35 b(w)m(e)j(can)f(construct)h(an)f Fl(F)50 b Fp(suc)m(h)37 b(that)h Fl(F)2551 378 y Fj(s)2588 364 y Fp(\()p Fl(x)p Fp(\))f(=)g Fl(F)p 2913 340 V 14 x Fj(s)2950 364 y Fp(\()p Fl(x)p Fp(\))h(for)f(all)f Fl(s)p Fp(.)61 b(Then,)150 477 y Fl(s)193 491 y Fi(1)266 477 y Fp(=)34 b Fl(s)414 491 y Fi(2)489 477 y Fp(so)i(the)g(\014rst)f(and)g(second)g(half)g(of)h (the)g(output)f(w)m(ould)g(b)s(e)g(the)h(same,)h(whic)m(h)e(o)s(ccurs)g (with)150 590 y(only)29 b(negligible)g(probabilit)m(y)e(in)i(a)i (random)f(function.)291 703 y(No)m(w,)40 b(assume)d Fl(F)924 670 y Fh(0)911 725 y Fj(s)985 703 y Fp(is)g(a)g(PRF)g(that)h(tak)m(es)h Fk(j)p Fl(s)p Fk(j)25 b Fp(+)f(1)38 b(bit)e(inputs)f(and)i(giv)m(es)g Fk(j)p Fl(s)p Fk(j)25 b Fp(+)f(1)38 b(bit)e(outputs.)150 816 y(\(W)-8 b(e)28 b(can)e(assume)g(this)f(safely)h(b)s(ecause)g(w)m (e)h(kno)m(w)f(that)h(if)e(PRFs)h(exist,)h(then)f(O)m(WFs)h(exist,)g (so)g(PR)m(Gs)150 929 y(exist,)j(and)e(the)i(GGM)g(construction)f(can)h (clearly)e(b)s(e)h(set)h(up)e(to)i(mo)s(di\014ed)d(to)j(set)g(up)f(the) g(input)f(and)150 1042 y(output)33 b(and)g(seed)h(lengths)f(as)h(w)m(e) g(require.\))49 b(Let)34 b Fl(F)2011 1057 y Fj(b)p Fh(\016)p Fj(s)2114 1042 y Fp(\()p Fl(x)p Fp(\))d(=)g Fl(F)2440 1009 y Fh(0)2427 1064 y Fj(s)2464 1042 y Fp(\()p Fl(x)p Fp(\))23 b Fk(\010)f Fl(b)2741 1009 y Fj(k)2784 1042 y Fp(.)50 b(That)34 b(is,)g(if)e Fl(b)f Fp(=)f(1,)35 b(then)150 1154 y Fl(F)208 1169 y Fj(b)p Fh(\016)p Fj(s)311 1154 y Fp(\()p Fl(x)p Fp(\))40 b(=)p 583 1075 218 4 v 39 w Fl(F)654 1128 y Fh(0)641 1177 y Fj(s)678 1154 y Fp(\()p Fl(x)p Fp(\))q(,)h(and)d(if)g Fl(b)i Fp(=)f(0)g(then)g Fl(F)1691 1169 y Fj(b)p Fh(\016)p Fj(s)1794 1154 y Fp(\()p Fl(x)p Fp(\))h(=)f Fl(F)2137 1121 y Fh(0)2124 1177 y Fj(s)2161 1154 y Fp(\()p Fl(x)p Fp(\).)67 b(No)m(w,)42 b Fl(F)52 b Fp(is)38 b(clearly)g(a)i(PRF)f(if)e Fl(F)3601 1121 y Fh(0)3664 1154 y Fp(is,)150 1267 y(since)d(w)m(e)h(cannot)h (distinguish)31 b(one)k(from)f(the)h(other.)54 b(Ho)m(w)m(ev)m(er,)38 b Fl(F)48 b Fp(do)s(es)34 b(ha)m(v)m(e)i(the)f(prop)s(ert)m(y)f(that) 150 1380 y Fl(F)208 1394 y Fj(s)245 1380 y Fp(\()p Fl(x)p Fp(\))26 b(=)f Fl(F)p 547 1356 37 3 v 14 x Fj(s)584 1380 y Fp(\()p Fl(x)p Fp(\),)32 b(so)e Fl(F)945 1347 y Fi(4)932 1403 y Fj(s)1015 1380 y Fp(w)m(ould)f(not)i(b)s(e)f(a)g(PRF.)291 1586 y Fq(\(e\))g Fl(F)522 1553 y Fi(5)509 1608 y Fj(s)561 1586 y Fp(\()p Fl(x)p Fp(\))c(=)f Fl(F)863 1600 y Fj(s)900 1586 y Fp(\()p Fl(F)993 1600 y Fj(x)1038 1586 y Fp(\()p Fl(x)p Fp(\)\))291 1791 y Fq(Solution:)48 b Fp(Let)34 b Fl(F)982 1758 y Fh(0)969 1814 y Fj(s)1040 1791 y Fp(b)s(e)f(a)h(PRF,) g(and)f(let)h Fl(F)1866 1805 y Fj(s)1903 1791 y Fp(\()p Fl(x)p Fp(\))d(=)g Fl(F)2229 1758 y Fh(0)2216 1814 y Fj(s)2253 1791 y Fp(\()p Fl(x)p Fp(\))j(as)g(long)f(as)h Fl(x)d Fk(6)p Fp(=)g Fl(s)p Fp(,)j(but)f Fl(F)3354 1805 y Fj(s)3391 1791 y Fp(\()p Fl(s)p Fp(\))e(=)g(0)3682 1758 y Fj(k)3725 1791 y Fp(.)150 1904 y(First)g(of)h(all,)g(let)f(us)h (pro)m(v)m(e)g(that)h Fl(F)1399 1871 y Fi(5)1386 1927 y Fj(s)1470 1904 y Fp(is)e(not)h(a)h(PRF.)f(F)-8 b(or)32 b(an)m(y)h(input)c Fl(x)p Fp(,)k(the)f(output)f(will)f(b)s(e)h Fl(F)3554 1918 y Fj(s)3591 1904 y Fp(\(0)3671 1871 y Fj(k)3715 1904 y Fp(\))150 2017 y({)i(it)g(will)d(b)s(e)i(the)h(same)g (for)g(ev)m(ery)g(input,)f(so)h(w)m(e)g(can)g(just)f(try)h(t)m(w)m(o)h (di\013eren)m(t)e(v)-5 b(alues)33 b(and)f(if)g(w)m(e)h(get)150 2130 y(the)e(same)f(answ)m(er)h(eac)m(h)g(time,)f(guess)h (pseudorandom.)291 2243 y(Ho)m(w)m(ev)m(er,)i Fl(F)44 b Fp(is)29 b(a)i(PRF.)g(Supp)s(ose)e(not.)42 b(W)-8 b(ell,)31 b(if)e(w)m(e)i(can)g(distinguish)c Fl(F)44 b Fp(from)30 b(random)g(but)f(w)m(e)150 2356 y(cannot)34 b(distinguish)c Fl(F)986 2323 y Fh(0)1043 2356 y Fp(from)j(random)g(\(b)m(y)g (assumption\),)h(then)f(w)m(e)h(m)m(ust)f(b)s(e)g(able)g(to)i (distinguish)150 2469 y Fl(F)54 b Fp(from)40 b Fl(F)558 2436 y Fh(0)581 2469 y Fp(.)72 b(Ho)m(w)m(ev)m(er,)45 b(the)c(only)f(w)m(a)m(y)h(w)m(e)g(could)f(do)h(this)e(w)m(ould)g(b)s (e)h(to)i(query)e Fl(F)53 b Fp(on)41 b(the)g(seed)150 2582 y(with)28 b(nonnegligible)e(probabilit)m(y)-8 b(.)38 b(No)m(w,)30 b(a)f(priori,)e(w)m(e)j(cannot)f(sa)m(y)h(this)e(has)g (negligible)f(probabilit)m(y)-8 b(.)150 2695 y(Ho)m(w)m(ev)m(er,)30 b(supp)s(ose)c Fl(A)i Fp(w)m(as)g(an)f(adv)m(ersary)g(that)h (distinguished)c Fl(F)40 b Fp(from)27 b Fl(F)2802 2662 y Fh(0)2826 2695 y Fp(.)39 b(Then,)28 b Fl(A)3219 2662 y Fh(0)3269 2695 y Fp(as)g(describ)s(ed)150 2808 y(b)s(elo)m(w)i(can)g (distinguish)d Fl(F)1106 2775 y Fh(0)1160 2808 y Fp(from)i(random)h(as) h(follo)m(ws:)150 3010 y(1.)41 b(Run)29 b Fl(A)i Fp(with)e(the)h (oracle)h(giv)m(en)g(to)g Fl(A)1601 2977 y Fh(0)1624 3010 y Fp(,)g(remem)m(b)s(ering)e(all)g(the)i(inputs)d Fl(x)2839 3024 y Fj(i)2897 3010 y Fp(and)i(outputs)g Fl(y)3453 3024 y Fj(i)3481 3010 y Fp(.)150 3122 y(2.)41 b(Once)30 b Fl(A)h Fp(is)e(\014nished,)f(try)j(ev)m(ery)g Fl(x)1484 3136 y Fj(i)1542 3122 y Fp(to)g(see)g(if)f(the)g(oracle)h(w)m (as)g Fl(F)2549 3090 y Fh(0)2536 3145 y Fj(x)2576 3155 y Ff(i)2607 3122 y Fp(.)261 3235 y(Do)g(this)f(b)m(y)g(testing)g (whether)g Fl(F)1415 3249 y Fj(x)1455 3259 y Ff(i)1486 3235 y Fp(\()p Fl(x)1573 3249 y Fj(j)1610 3235 y Fp(\))25 b(=)g Fl(y)1811 3249 y Fj(j)1878 3235 y Fp(for)30 b(all)f Fl(j)5 b Fp(.)150 3348 y(3.)41 b(If)30 b(an)m(y)h Fl(i)f Fp(passes)h(this)e(test,)i(output)f(\\pseudorandom,")g(otherwise)g (output)g(\\random.")291 3550 y(No)m(w)40 b(if)f(w)m(e)h(are)g(giv)m (en)g(a)g(random)f(oracle,)j(w)m(e)f(ha)m(v)m(e)f(a)g(negligible)e (probabilit)m(y)f(of)j(outputting)150 3663 y(\\pseudorandom.")e(Ho)m(w) m(ev)m(er,)29 b(if)c(w)m(e)h(are)f(giv)m(en)h(an)f(oracle)h(to)g Fl(F)2399 3677 y Fj(s)2437 3663 y Fp(,)g Fl(A)g Fp(m)m(ust)f(query)g Fl(F)38 b Fp(on)26 b(input)d Fl(s)i Fp(with)150 3776 y(nonnegligible)30 b(probabilit)m(y)g(\(or)j(it)f(could)f(not)i (distinguish)c Fl(F)45 b Fp(from)32 b Fl(F)2674 3743 y Fh(0)2730 3776 y Fp(with)g(nonnegligible)d(proba-)150 3889 y(bilit)m(y\),)i(so)h(w)m(e)h(output)e(\\pseudorandom")h(with)e (nonnegligible)f(probabilit)m(y)-8 b(.)44 b(Th)m(us)31 b(w)m(e)h(are)h(correct)150 4002 y(with)f(probabilit)m(y)e(1)p Fl(=)p Fp(2)23 b Fk(\000)f Fl(neg)s(l)35 b Fp(when)d(the)h(oracle)h(is) e(a)h(random)f(function,)h(and)f(with)g(nonnegligible)150 4115 y(probabilit)m(y)i(when)h(the)h(oracle)h(is)e Fl(F)1450 4129 y Fj(s)1487 4115 y Fp(.)58 b(Th)m(us,)37 b(w)m(e)f(are)h(correct)g (with)e(probabilit)m(y)f(at)j(least)f(1)p Fl(=)p Fp(2)25 b(+)f Fl(\017)150 4228 y Fp(where)30 b Fl(\017)g Fp(is)f (nonnegligible.)150 4513 y Fm(Problem)45 b(5)150 4716 y Fp(\(10)34 b(p)s(oin)m(ts\))e(Summarize)g(what)h(w)m(e)g(kno)m(w)g (so)h(far.)48 b(W)-8 b(e)34 b(ha)m(v)m(e)g(discussed)d(sev)m(eral)j (primitiv)m(es)c(in)i(this)150 4828 y(class:)k(secure)22 b(public-k)m(ey)f(cryptosystems)h(\(PK)m(C\),)h(one-w)m(a)m(y)g (functions)e(and)g(one-w)m(a)m(y)j(function)d(fam-)150 4941 y(ilies)32 b(\(O)m(WF)j(and)f(O)m(WFF\),)i(one-w)m(a)m(y)g(p)s (erm)m(utations)d(and)g(one-w)m(a)m(y)j(p)s(erm)m(utation)d(families)f (\(O)m(WP)150 5054 y(and)g(O)m(WPF\),)j(pseudorandom)c(generators)j (\(PR)m(G\),)h(pseudorandom)c(functions)h(\(PRF\),)i(hardcore)150 5167 y(bits)25 b(\(HB\),)i(and)e(trap)s(do)s(or)g(predicates)h(\(TP\).) g(F)-8 b(or)27 b(eac)m(h)g(pair)d(of)i(these,)i(sp)s(ecify)c(whether)i (w)m(e)g(kno)m(w)g(if)1867 5416 y(15-9)p eop %%Page: 10 10 10 9 bop 150 159 a Fp(the)28 b(existence)f(of)h(one)g(implies)c(the)k (existence)g(of)f(the)h(other.)40 b(Giv)m(e)27 b(as)h(man)m(y)g(suc)m (h)f(inferences)f(as)i(y)m(ou)150 271 y(can,)f(and)f(for)f(eac)m(h)i (inference)e(giv)m(e)i(either)e(a)h(reference)h(to)f(a)g(pap)s(er,)g(b) s(o)s(ok,)h(or)f(problem)e(set)i(problem,)150 384 y(or)33 b(giv)m(e)h(a)g(brief)e(pro)s(of)h(idea.)49 b Fn(Sinc)-5 b(e)35 b(implic)-5 b(ations)38 b(ar)-5 b(e)36 b(tr)-5 b(ansitive,)37 b(you)f(do)g(not)g(ne)-5 b(e)g(d)36 b(to)g(give)f(every) 150 497 y(arr)-5 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Ff(P)13 b(P)g(T)43 b Fh(9)p Ff(\027)31 b Fh(2)25 b Ff(neg)s(l)32 b Fh(8)p Ff(k)150 2310 y Fl(Pr)o([)p Ff(S)f Fh( )25 b(f)p Fl(0)p Ff(;)15 b Fl(1)p Fh(g)695 2277 y Fe(k)740 2310 y Fl(;)g Fh(O)852 2324 y Fg(0)917 2310 y Fh( )26 b Ff(P)1092 2324 y Fe(S)1143 2310 y Fl(;)273 2423 y Fh(O)345 2437 y Fg(1)410 2423 y Fh( )f Fl(\005\()p Ff(l)r Fl(\()p Ff(k)s Fl(\)\);)273 2545 y Ff(b)312 2512 y Fd(0)360 2545 y Fh( )g Ff(A)544 2502 y Fd(O)600 2514 y Fc(b)544 2575 y Fe(k)661 2545 y Fl(:)h Ff(b)751 2512 y Fd(0)799 2545 y Fl(=)f Ff(b)p Fl(])h Ff(<)f Fl(1)p Ff(=)p Fl(2)c(+)f Ff(\027)6 b Fl(\()p Ff(k)s Fl(\))291 2758 y Fm(\(a\))30 b Fl(Pro)m(v)m(e)h(that)g Ff(LR)21 b Fh(\000)f Fl(2)31 b(is)e(not)i(a)f(pseudorandom)f(p)s(erm)m(utation)h (family)-8 b(.)291 2970 y Fm(Solution:)40 b Fl(Note)29 b(that)g(if)d(w)m(e)j(only)e(ha)m(v)m(e)i(t)m(w)m(o)g(rounds,)e(then)h Ff(LR)16 b Fh(\000)f Fl(2\()p Ff(L;)g(R)q Fl(\))26 b(=)f Ff(L)15 b Fh(\010)g Ff(F)3324 2984 y Fg(1)3364 2970 y Fl(\()p Ff(R)q Fl(\))p Ff(;)g(Z)35 b Fl(for)150 3083 y(some)g Ff(Z)7 b Fl(.)55 b(No)m(w,)37 b(w)m(e)e(can)h(test)g(whether)e (an)h(oracle)g(is)g Ff(LR)23 b Fh(\000)g Fl(2)36 b(b)m(y)f(querying)e (it)i(on)g(t)m(w)m(o)h(pairs)e Ff(L;)15 b(R)150 3196 y Fl(and)30 b Ff(L)20 b Fh(\010)g Fl(1)p Ff(;)15 b(R)q Fl(.)41 b(If)30 b(the)h(oracle)g(is)e Ff(LR)21 b Fh(\000)f Fl(2,)31 b(then)f(the)h(\014rst)f(half)f(of)h(the)h(outputs)f(will)e (di\013er)h(in)g(only)h(1)150 3309 y(bit.)39 b(If)26 b(the)h(oracle)g(w)m(as)g(a)h(random)e(p)s(erm)m(utation,)g(this)g(w)m (ould)g(happ)s(en)f(with)g(negligible)g(probabilit)m(y)-8 b(.)150 3422 y(Th)m(us,)30 b(our)f(algorithm)h(is:)150 3635 y Ff(A)218 3602 y Fd(O)280 3635 y Fl(\(1)360 3602 y Fg(2)p Fe(k)439 3635 y Fl(\):)150 3747 y(1.)41 b(Randomly)29 b(generate)j Ff(L)25 b Fh( )g(f)p Fl(0)p Ff(;)15 b Fl(1)p Fh(g)1486 3714 y Fe(k)1531 3747 y Fl(,)31 b Ff(R)26 b Fh( )f(f)p Fl(0)p Ff(;)15 b Fl(1)p Fh(g)2018 3714 y Fe(k)2063 3747 y Fl(.)150 3860 y(2.)41 b(Get)32 b Ff(X)514 3874 y Fg(1)573 3860 y Fh(\016)21 b Ff(X)714 3874 y Fg(2)779 3860 y Fl(=)k Fh(O)s Fl(\()p Ff(L)20 b Fh(\016)h Ff(R)q Fl(\))30 b(and)g Ff(X)1527 3827 y Fd(0)1520 3885 y Fg(1)1580 3860 y Fh(\016)21 b Ff(X)1728 3827 y Fd(0)1721 3885 y Fg(2)1786 3860 y Fl(=)k Fh(O)s Fl(\()p Ff(L)20 b Fh(\010)g Fl(1)h Fh(\016)f Ff(R)q Fl(\).)150 3973 y(3.)41 b(If)30 b Ff(X)427 3987 y Fg(1)487 3973 y Fh(\016)21 b Ff(X)635 3940 y Fd(0)628 3998 y Fg(1)693 3973 y Fl(=)k(1,)31 b(output)f (\\pseudorandom,")g(else)g(output)g(\\random.")291 4285 y(W)-8 b(e)32 b(sa)m(y)f(a)g(p)s(erm)m(utation)e(family)g Ff(P)1537 4299 y Fe(S)1618 4285 y Fl(is)h(strongly)f(pseudorandom)g (if:)150 4498 y Fh(8f)p Ff(A)314 4454 y Fg(?)p Fe(;)p Fg(?)314 4510 y Fe(n)405 4498 y Fh(g)c(2)g Ff(P)13 b(P)g(T)43 b Fh(9)p Ff(\027)31 b Fh(2)24 b Ff(neg)s(l)33 b Fh(8)p Ff(k)150 4613 y Fl(Pr)o([)p Ff(S)e Fh( )25 b(f)p Fl(0)p Ff(;)15 b Fl(1)p Fh(g)695 4580 y Fe(k)740 4613 y Fl(;)g Fh(O)852 4627 y Fg(0)917 4613 y Fh( )26 b Ff(P)1092 4627 y Fe(S)1143 4613 y Fl(;)15 b Fh(O)1258 4580 y Fd(0)1281 4627 y Fg(0)1346 4613 y Fh( )25 b Ff(P)1533 4575 y Fd(\000)p Fg(1)1520 4642 y Fe(S)273 4726 y Fh(O)345 4740 y Fg(1)410 4726 y Fh( )g Fl(\005;)15 b Fh(O)709 4693 y Fd(0)733 4740 y Fg(1)797 4726 y Fh( )26 b Fl(\005)982 4693 y Fd(\000)p Fg(1)1076 4726 y Fl(;)273 4844 y Ff(b)312 4811 y Fd(0)360 4844 y Fh( )f Ff(A)544 4811 y Fd(O)600 4823 y Fc(b)632 4811 y Fe(;)p Fd(O)710 4787 y Fb(0)732 4823 y Fc(b)793 4844 y Fl(:)g Ff(b)882 4811 y Fd(0)931 4844 y Fl(=)g Ff(b)p Fl(])g Ff(<)g Fl(1)p Ff(=)p Fl(2)d(+)e Ff(\027)6 b Fl(\()p Ff(k)s Fl(\))1867 5416 y(17-2)p eop %%Page: 3 3 3 2 bop 291 159 a Fl(where)34 b(\005)626 126 y Fd(\000)p Fg(1)755 159 y Fl(in)m(v)m(erts)h(\005.)54 b(Think)32 b(of)j(\005)g(as)g(b)s(eing)e(a)i(\014xed)f(but)g(randomly)f(c)m(hosen) j(p)s(erum)m(tation,)150 271 y(and)30 b(\005)395 238 y Fd(\000)p Fg(1)520 271 y Fl(as)g(its)g(in)m(v)m(erse.)291 384 y Fm(\(b\))g Fl(Pro)m(v)m(e)h(that)g Ff(LR)21 b Fh(\000)f Fl(3)31 b(is)e(not)i(a)g(strong)f(pseudorandom)f(p)s(erm)m(utation)g (family)-8 b(.)291 597 y Fm(Solution:)39 b Fl(This)25 b(time)i(w)m(e)g(will)d(ha)m(v)m(e)k(to)g(b)s(e)e(a)h(little)f(more)h (careful)f(with)f(our)i(notation.)39 b(What)28 b(is)150 710 y(going)k(on)f(in)g(this)f(case)j(is)e(that)h(w)m(e)g(can)g(w)m (ork)g(forw)m(ards)f(and)g(bac)m(kw)m(ards)h(o)m(v)m(er)h(a)f(couple)f (of)h(queries)150 823 y(to)c(pro)s(duce)f(some)h(collision)d(that)j (allo)m(ws)f(us)g(to)h(distinguish)c Ff(LR)15 b Fh(\000)g Fl(3)27 b(from)g(a)h(random)f(p)s(erm)m(utation.)291 936 y(This)20 b(time)i(w)m(e)g(will)e(sho)m(w)i(the)g(algorithm)f (\014rst,)i(and)e(then)h(afterw)m(ards)g(describ)s(e)e(wh)m(y)i(it)f (is)h(correct.)150 1148 y Ff(A)218 1115 y Fd(O)280 1148 y Fl(\(1)360 1115 y Fg(2)p Fe(k)439 1148 y Fl(\):)150 1261 y(1.)41 b(Get)32 b Ff(Y)40 b Fh(\016)20 b Ff(Z)32 b Fl(=)25 b Fh(O)s Fl(\(0)942 1228 y Fg(2)p Fe(k)1021 1261 y Fl(\))150 1374 y(2.)41 b(Get)32 b Ff(Y)511 1341 y Fd(0)555 1374 y Fh(\016)21 b Ff(Z)690 1341 y Fd(0)738 1374 y Fl(=)k Fh(O)s Fl(\(1)989 1341 y Fe(k)1052 1374 y Fh(\016)c Fl(0)1163 1341 y Fe(k)1206 1374 y Fl(\))150 1487 y(3.)41 b(Get)32 b Ff(L)20 b Fh(\016)g Ff(R)26 b Fl(=)f Fh(O)852 1454 y Fd(\000)p Fg(1)946 1487 y Fl(\()p Ff(Y)41 b Fh(\016)21 b Fl(1)1186 1454 y Fe(k)1249 1487 y Fh(\010)f Ff(Z)7 b Fl(\))150 1600 y(4.)41 b(If)30 b Ff(R)c Fl(=)f Ff(Y)40 b Fh(\010)20 b Ff(Y)800 1567 y Fd(0)854 1600 y Fl(output)30 b(\\pseudorandom,")g(else)g(output)g (\\random.")291 1813 y(That)e(this)f(algorithm)g(only)g(outputs)g (\\pseudrandom")g(when)g(the)h(oracle)h(is)e(a)h(random)f(function)150 1925 y(with)22 b(negligible)g(probabilit)m(y)f(is)h(clear.)39 b(W)-8 b(e)25 b(will)c(pro)m(v)m(e)j(if)f(the)h(oracle)g(is)f(LR-3,)i (this)e(algorithm)f(alw)m(a)m(ys)150 2038 y(outputs)30 b(\\pseudorandom,")g(whic)m(h)f(will)f(pro)m(v)m(e)j(that)g Ff(LR)21 b Fh(\000)f Fl(3)30 b(is)g(not)g(a)h(strong)g(PRP)-8 b(.)291 2151 y(In)30 b(the)g(\014rst)g(query)-8 b(,)30 b(the)h(v)-5 b(alues)30 b(after)h(eac)m(h)g(round)e(go:)1468 2468 y(0)1513 2431 y Fe(k)1639 2468 y Fh(\016)83 b Fl(0)1812 2431 y Fe(k)1468 2606 y Fl(0)1513 2569 y Fe(k)1639 2606 y Fh(\016)g Ff(F)1825 2620 y Fg(1)1865 2606 y Fl(\(0)1945 2569 y Fe(k)1989 2606 y Fl(\))20 b Fh(\010)g Fl(0)2180 2569 y Fe(k)2248 2606 y Fl(=)25 b Ff(F)2402 2620 y Fg(1)2442 2606 y Fl(\(0)2522 2569 y Fe(k)2566 2606 y Fl(\))1299 2744 y Ff(F)1357 2758 y Fg(1)1397 2744 y Fl(\(0)1477 2707 y Fe(k)1520 2744 y Fl(\))84 b Fh(\016)f Ff(Y)1483 2882 y(Y)103 b Fh(\016)83 b Ff(Z)291 3086 y Fl(Th)m(us,)30 b(w)m(e)h(kno)m(w)f(that)h Ff(F)1173 3100 y Fg(1)1213 3086 y Fl(\(0)1293 3053 y Fe(k)1336 3086 y Fl(\))21 b Fh(\010)f Ff(F)1541 3100 y Fg(3)1581 3086 y Fl(\()p Ff(Y)g Fl(\))26 b(=)f Ff(Z)7 b Fl(,)30 b(so)g Ff(F)2139 3100 y Fg(1)2179 3086 y Fl(\(0)2259 3053 y Fe(k)2303 3086 y Fl(\))25 b(=)g Ff(F)2517 3100 y Fg(3)2557 3086 y Fl(\()p Ff(Y)20 b Fl(\))h Fh(\010)f Ff(Z)37 b Fl(\(1\).)291 3199 y(In)30 b(the)g(second)h(query)-8 b(,)30 b(the)h(v)-5 b(alues)30 b(after)g(eac)m(h)i(round)d(go:)1756 3516 y(1)1801 3479 y Fe(k)1927 3516 y Fh(\016)84 b Fl(0)2101 3479 y Fe(k)1756 3654 y Fl(0)1801 3616 y Fe(k)1927 3654 y Fh(\016)g Ff(F)2114 3668 y Fg(1)2154 3654 y Fl(\(0)2234 3616 y Fe(k)2277 3654 y Fl(\))21 b Fh(\010)f Fl(1)2469 3616 y Fe(k)1388 3792 y Ff(F)1446 3806 y Fg(1)1486 3792 y Fl(\(0)1566 3754 y Fe(k)1610 3792 y Fl(\))g Fh(\010)g Fl(1)1801 3754 y Fe(k)1927 3792 y Fh(\016)84 b Ff(Y)2129 3754 y Fd(0)1748 3930 y Ff(Y)1821 3892 y Fd(0)1927 3930 y Fh(\016)g Ff(Z)2125 3892 y Fd(0)291 4134 y Fl(So)30 b(w)m(e)h(kno)m(w)g(that)g Ff(F)1044 4148 y Fg(2)1083 4134 y Fl(\()p Ff(F)1176 4148 y Fg(1)1217 4134 y Fl(\(0)1297 4101 y Fe(k)1340 4134 y Fl(\))21 b Fh(\010)f Fl(1)1532 4101 y Fe(k)1575 4134 y Fl(\))25 b(=)g Ff(Y)1804 4101 y Fd(0)1858 4134 y Fl(\(2\).)291 4247 y(No)m(w)35 b(in)d(the)i(third)e (query)-8 b(,)35 b(w)m(e)f(m)m(ust)g(solv)m(e)g(for)f(the)h(in)m (termediate)g(v)-5 b(alues.)50 b(The)34 b(v)-5 b(alues)33 b(are)h(\(in)150 4360 y(forw)m(ard)c(order\):)1690 4677 y Ff(L)83 b Fh(\016)g Ff(R)1682 4815 y(R)h Fh(\016)f Ff(X)1669 4952 y(X)91 b Fh(\016)83 b Ff(Y)1679 5090 y(Y)103 b Fh(\016)83 b Ff(Z)27 b Fh(\010)20 b Fl(1)2188 5053 y Fe(k)1867 5416 y Fl(17-3)p eop %%Page: 4 4 4 3 bop 291 159 a Fl(The)30 b(only)g(v)-5 b(alue)30 b(w)m(e)i(don't)e (get)i(to)g(see)f(is)f Ff(X)7 b Fl(.)42 b(Ho)m(w)m(ev)m(er,)33 b(w)m(e)e(kno)m(w)g(that)g Ff(X)j Fl(=)25 b Ff(Z)i Fh(\010)20 b Fl(1)3328 126 y Fe(k)3392 159 y Fh(\010)g Ff(F)3541 173 y Fg(3)3581 159 y Fl(\()p Ff(Y)g Fl(\),)150 271 y(and)30 b(plugging)e(in)i(from)f(\(1\))j(for)e Ff(F)1360 285 y Fg(3)1400 271 y Fl(\()p Ff(Y)20 b Fl(\))h Fh(\010)f Ff(Z)37 b Fl(giv)m(es)30 b Ff(X)j Fl(=)25 b Ff(F)2238 285 y Fg(1)2278 271 y Fl(\(0)2358 238 y Fe(k)2401 271 y Fl(\))c Fh(\010)f Fl(1)2593 238 y Fe(k)2636 271 y Fl(.)291 384 y(W)-8 b(e)34 b(also)f(kno)m(w,)g(b)m(y)g(de\014nition,)e(that)i Ff(Y)49 b Fl(=)29 b Ff(F)1920 398 y Fg(2)1960 384 y Fl(\()p Ff(X)7 b Fl(\))22 b Fh(\010)g Ff(R)30 b Fl(=)e Ff(F)2483 398 y Fg(2)2523 384 y Fl(\()p Ff(F)2616 398 y Fg(1)2657 384 y Fl(\(0)2737 351 y Fe(k)2780 384 y Fl(\))22 b Fh(\010)g Fl(1)2975 351 y Fe(k)3018 384 y Fl(\))g Fh(\010)f Ff(R)q Fl(.)47 b(By)33 b(\(2\))h(this)150 497 y(giv)m(es)e Ff(Y)47 b Fl(=)26 b Ff(Y)644 464 y Fd(0)688 497 y Fh(\010)21 b Ff(R)q Fl(,)31 b(whic)m(h)g(yields)e Ff(R)f Fl(=)e Ff(Y)41 b Fh(\010)21 b Ff(Y)1877 464 y Fd(0)1900 497 y Fl(.)44 b(This)29 b(is)i(the)g(equation)g(w)m(e)h(test,)h(and)e(th)m (us)g(it)g(will)150 610 y(alw)m(a)m(ys)g(b)s(e)f(true)g(if)f Fh(O)f Fl(=)d Ff(LR)c Fh(\000)f Fl(3.)150 897 y Fi(Problem)45 b(3:)61 b(F)-11 b(our-Round)43 b(Lub)l(y-Rac)l(k)l(o\013)150 1100 y Fl(In)29 b(class)g(w)m(e)i(pro)m(v)m(ed)f(that)g Ff(LR)19 b Fh(\000)g Fl(3)30 b(is)f(a)h(pseudorandom)e(p)s(erm)m (utation)h(family)-8 b(.)39 b(Pro)m(v)m(e)31 b(that)f Ff(LR)20 b Fh(\000)f Fl(4)150 1212 y(is)29 b(a)i(strong)g(pseudorandom) e(p)s(erm)m(utation)g(family)-8 b(.)291 1425 y Fm(Solution:)51 b Fl(As)35 b(y)m(ou)g(migh)m(t)g(exp)s(ect,)i(this)d(is)h(fairly)e (long)i(and)f(complicated.)55 b(W)-8 b(e)37 b(will)32 b(pro)s(ceed)150 1538 y(step)40 b(b)m(y)f(step.)68 b(First)39 b(of)h(all,)h(assume)e(not.)69 b(Then)38 b(there)i(m)m(ust)f(b)s(e)g (an)h(adv)m(ersary)f Ff(A)3277 1505 y Fg(?)p Fe(;)p Fg(?)3408 1538 y Fl(that)h(can)150 1651 y(distinguish)27 b Ff(LR)20 b Fh(\000)g Fl(4)31 b(from)f(random)g(with)f(probabilit)m(y)f (non-negligibly)f(b)s(etter)j(than)g(1/2.)291 1764 y Fm(Step)42 b(1:)53 b Fl(W)-8 b(e)38 b(can)f(sa)m(y)-8 b(,)40 b(without)c(loss)g(of)h(generalit)m(y)g(that)g Ff(A)g Fl(nev)m(er)g(queries)f(either)g(oracle)h(on)150 1877 y(the)h(same)f(input)f(t)m(wice,)k(nor)c(queries)h(one)g(oracle)h (on)g(the)f(output)g(of)h(the)f(other.)62 b(Wh)m(y?)g(W)-8 b(ell,)39 b(if)150 1990 y Ff(A)e Fl(is)f(supp)s(osed)f(to)j (distinguish)c(b)s(et)m(w)m(een)j(ha)m(ving)g(oracle)g(access)i(to)f Ff(LR)25 b Fh(\000)f Fl(4)38 b(and)e Ff(LR)25 b Fh(\000)g Fl(4)3472 1957 y Fd(\000)p Fg(1)3604 1990 y Fl(and)150 2102 y(ha)m(ving)33 b(\005)h(and)f(\005)796 2069 y Fd(\000)p Fg(1)925 2102 y Fl(as)h(its)f(oracles,)i(then)f(if)e Ff(A)i Fl(ev)m(er)h(mak)m(es)f(these)h(rep)s(etitiv)m(e)e(queries,)h Ff(A)g Fl(will)d(get)150 2215 y(bac)m(k)k(exactly)h(what)f(is)e(exp)s (ected.)54 b(Th)m(us,)35 b(if)f Ff(A)h Fl(simply)d(k)m(ept)j(trac)m(k)h (itself)e(of)h(whether)f(it)g(w)m(as)h(ev)m(er)150 2328 y(asking)d(a)h(rep)s(etitiv)m(e)f(query)-8 b(,)33 b(and)f(a)m(v)m (oided)h(asking)f(them)h(but)e(instead)h(ga)m(v)m(e)j(the)e(exp)s (ected)g(answ)m(er,)150 2441 y(the)e(p)s(erformance)e(w)m(ould)g(b)s(e) h(exactly)h(the)g(same.)291 2554 y Fm(Step)f(2:)39 b Fl(No)m(w,)29 b(w)m(e)e(construct)g(an)f Ff(LR)14 b Fh(\000)f Fl(4)p Ff(R)27 b Fl(oracle,)h(whic)m(h)e(op)s(erates)h(with)e (randomness,)i(and)f(is)150 2667 y(to)d(exactly)f(sim)m(ulate)g(the)g (execution)g(of)g Ff(LR)t Fh(\000)s Fl(4)h(but)e(using)g(truly)f (random)h(functions)g(instead)g(of)h(pseu-)150 2780 y(dorandom)e (functions.)36 b(The)20 b(construction,)j(whic)m(h)c(w)m(e)j(call)e Ff(B)25 b Fl(is)20 b(as)h(follo)m(ws.)37 b(Supp)s(ose)19 b(w)m(e)i(ha)m(v)m(e)h(a)f(ran-)150 2893 y(dom)g(tap)s(e)h(of)g(length) e(4)p Ff(k)s(m)i Fl(where)f Ff(m)h Fl(is)e(the)i(maxim)m(um)e(of)i(the) g(total)g(n)m(um)m(b)s(er)e(of)i(queries)e Ff(A)i Fl(will)d(mak)m(e,) 150 3006 y(and)30 b(w)m(e)h(divide)e(this)h(tap)s(e)h(in)m(to)f(4)p Ff(m)h Fl(parts)g(of)g Ff(k)j Fl(bits)29 b(eac)m(h:)43 b Ff(W)2378 3020 y Fg(1)2417 3006 y Ff(;)15 b(:)g(:)g(:)i(;)e(W)2705 3020 y Fe(m)2772 3006 y Ff(;)g(X)2887 3020 y Fg(1)2927 3006 y Ff(;)g(:)g(:)g(:)h(;)f(X)3203 3020 y Fe(m)3271 3006 y Ff(;)g(Y)3364 3020 y Fg(1)3403 3006 y Ff(;)g(:)g(:)g(:)i(;)e(Y) 3658 3020 y Fe(m)3725 3006 y Fl(,)150 3119 y(and)30 b Ff(Z)389 3133 y Fg(1)428 3119 y Ff(;)15 b(:)g(:)g(:)i(;)e(Z)692 3133 y Fe(m)759 3119 y Fl(.)291 3232 y(On)41 b(query)g Ff(L)780 3246 y Fe(i)835 3232 y Fh(\016)28 b Ff(R)977 3246 y Fe(i)1006 3232 y Fl(,)44 b Ff(LR)28 b Fh(\000)g Fl(4)p Ff(R)42 b Fl(will)d(b)s(eha)m(v)m(e)j(as)g(follo)m(ws.)73 b(First,)44 b(if)d(there)g(is)g(an)g Ff(l)46 b(<)d(i)f Fl(suc)m(h)150 3344 y(that)g Ff(R)427 3359 y Fe(l)495 3344 y Fl(=)h Ff(R)678 3358 y Fe(i)747 3344 y Fl(then)e(let)g Ff(l)1134 3358 y Fe(R)1232 3344 y Fl(b)s(e)g(the)g(largest)g(suc)m(h)g Ff(l)i Fl(and)d(set)i Ff(\013)2522 3358 y Fe(i)2593 3344 y Fl(=)g Ff(L)2768 3358 y Fe(i)2824 3344 y Fh(\010)27 b Ff(W)3008 3359 y Fe(l)3029 3370 y Fc(R)3083 3344 y Fl(.)72 b(Otherwise,)43 b(let)150 3457 y Ff(\013)208 3471 y Fe(i)267 3457 y Fl(=)31 b Ff(L)431 3471 y Fe(i)481 3457 y Fh(\010)23 b Ff(W)661 3471 y Fe(i)689 3457 y Fl(.)51 b(Next,)35 b(if)e(there)h(is)f(an)g Ff(l)g(<)e(i)j Fl(suc)m(h)f(that)i Ff(\013)2257 3472 y Fe(l)2314 3457 y Fl(=)30 b Ff(\013)2473 3471 y Fe(i)2536 3457 y Fl(then)j(let)h Ff(l)2908 3471 y Fe(\013)2991 3457 y Fl(b)s(e)f(the)h(largest)g(suc)m(h)150 3570 y Ff(l)h Fl(and)d(set)h Ff(\014)586 3584 y Fe(i)644 3570 y Fl(=)c Ff(R)813 3584 y Fe(i)863 3570 y Fh(\010)21 b Ff(X)1030 3585 y Fe(l)1051 3593 y Fc(\013)1100 3570 y Fl(.)47 b(Otherwise,)33 b(let)f Ff(\014)1815 3584 y Fe(i)1873 3570 y Fl(=)d Ff(R)2042 3584 y Fe(i)2092 3570 y Fh(\010)21 b Ff(X)2259 3584 y Fe(i)2288 3570 y Fl(.)48 b(Next,)34 b(if)e(there)h(is)e(an)i Ff(l)e(<)e(i)k Fl(suc)m(h)f(that) 150 3683 y Ff(\014)201 3698 y Fe(l)253 3683 y Fl(=)25 b Ff(\014)400 3697 y Fe(i)428 3683 y Fl(,)31 b(then)f(let)g Ff(l)849 3698 y Fe(\014)927 3683 y Fl(b)s(e)g(the)g(largest)h(suc)m(h)f Ff(l)i Fl(and)e(set)h Ff(\015)2131 3697 y Fe(i)2185 3683 y Fl(=)24 b Ff(\013)2338 3697 y Fe(i)2387 3683 y Fh(\010)c Ff(Y)2531 3698 y Fe(l)2552 3710 y Fc(\014)2597 3683 y Fl(.)41 b(Otherwise,)29 b(let)i Ff(\015)3297 3697 y Fe(i)3350 3683 y Fl(=)25 b Ff(\013)3504 3697 y Fe(i)3553 3683 y Fh(\010)20 b Ff(Y)3697 3697 y Fe(i)3725 3683 y Fl(.)150 3796 y(Finally)-8 b(,)41 b(if)d(there)i(is)f(an)g Ff(l)k(<)d(i)g Fl(suc)m(h)g(that)g Ff(\015)1779 3811 y Fe(l)1846 3796 y Fl(=)g Ff(\015)2004 3810 y Fe(i)2032 3796 y Fl(,)j(then)c(let)h Ff(l)2484 3810 y Fe(\015)2568 3796 y Fl(b)s(e)f(the)h(largest)f(suc)m (h)h Ff(l)h Fl(and)e(set)150 3909 y Ff(\016)190 3923 y Fe(i)244 3909 y Fl(=)25 b Ff(\014)391 3923 y Fe(i)440 3909 y Fh(\010)20 b Ff(Z)593 3924 y Fe(l)614 3932 y Fc(\015)658 3909 y Fl(.)40 b(Otherwise,)30 b(let)g Ff(\016)1350 3923 y Fe(i)1404 3909 y Fl(=)25 b Ff(\014)1551 3923 y Fe(i)1599 3909 y Fh(\010)20 b Ff(Z)1752 3923 y Fe(i)1781 3909 y Fl(.)40 b(Output)30 b Ff(\015)2217 3923 y Fe(i)2245 3909 y Ff(;)15 b(\016)2325 3923 y Fe(i)2354 3909 y Fl(.)291 4022 y(What)29 b(is)f(essen)m(tially)g(going)g(on)h(here)f(is)g(just)g (the)h(same)g(as)g(the)g(construction)f Ff(B)33 b Fl(from)28 b(class:)40 b(W)-8 b(e)150 4135 y(use)31 b(the)h(random)e(tap)s(e)i Ff(W)1092 4149 y Fg(1)1131 4135 y Ff(;)15 b(:)g(:)g(:)h(;)f(W)1418 4149 y Fe(m)1517 4135 y Fl(to)32 b(sim)m(ulate)e(the)i(outputs)e(of)i Ff(F)2646 4149 y Fg(1)2686 4135 y Fl(,)g(w)m(e)f(use)g Ff(X)39 b Fl(to)32 b(sim)m(ulate)f(the)150 4248 y(outputs)36 b(of)g Ff(F)657 4262 y Fg(2)697 4248 y Fl(,)i(and)e(so)h(on.)58 b(This)35 b(is)g(exactly)j(lik)m(e)d Ff(LR)25 b Fh(\000)f Fl(4)37 b(using)e(random)g(functions)g(instead)h(of)150 4361 y(PRFs.)291 4474 y(No)m(w,)42 b(in)37 b(order)i(to)g(argue)g(that) g(this)f(is)g(indistinguishable)33 b(from)38 b(random,)j(it)d(will)e (help)h(us)h(to)150 4586 y(mo)s(dify)32 b(the)i(construction)g(to)g (create)i Ff(B)1604 4553 y Fd(0)1626 4586 y Fl(.)51 b(In)34 b Ff(B)1894 4553 y Fd(0)1916 4586 y Fl(,)h(instead)e(of)h(sp)s (ecifying)e(the)i(outputs)g(of)g(the)g(four)150 4699 y(random)28 b(functions,)h(the)g(four)f(random)h(tap)s(es)g(will)e (represen)m(t)i(the)g(righ)m(t)g(side)f(of)h(the)h(outputs)e(of)i(the) 150 4812 y(\014rst)25 b(through)h(fourth)f Fj(r)-5 b(ounds.)41 b Fl(T)-8 b(o)26 b(simplify)d(things,)j(w)m(e)h(will)d(sa)m(y)i(to)h (\\determine)f Ff(l)3089 4826 y Fe(R)3147 4812 y Fl(")h(\(for)f (example\))150 4925 y(will)i(mean)i(to)h(let)g Ff(l)834 4939 y Fe(R)922 4925 y Fl(b)s(e)e(the)i(largest)g Ff(l)c(<)e(i)30 b Fl(suc)m(h)g(that)h Ff(R)2178 4940 y Fe(l)2230 4925 y Fl(=)25 b Ff(R)2395 4939 y Fe(i)2423 4925 y Fl(,)30 b(or)h(let)f Ff(l)2748 4939 y Fe(R)2831 4925 y Fl(=)25 b Ff(i)31 b Fl(if)e(no)h(suc)m(h)g Ff(l)j Fl(exists.)291 5038 y(On)f(input)e Ff(L)751 5052 y Fe(i)801 5038 y Fh(\016)22 b Ff(R)937 5052 y Fe(i)966 5038 y Fl(,)33 b Ff(LR)22 b Fh(\000)g Fl(4)p Ff(R)q Fl(\()p Ff(B)1494 5005 y Fd(0)1517 5038 y Fl(\))33 b(b)s(eha)m(v)m(es)g(as)g(follo)m(ws.)46 b(Determine)33 b Ff(l)2853 5052 y Fe(R)2911 5038 y Fl(,)g(and)f(let)g Ff(\013)3339 5052 y Fe(i)3397 5038 y Fl(=)c Ff(W)3582 5053 y Fe(l)3603 5064 y Fc(R)3679 5038 y Fh(\010)150 5151 y Ff(L)212 5166 y Fe(l)233 5177 y Fc(R)312 5151 y Fh(\010)d Ff(L)470 5165 y Fe(i)498 5151 y Fl(.)63 b(Let)38 b Ff(w)821 5165 y Fe(i)887 5151 y Fl(=)g Ff(\013)1054 5165 y Fe(i)1107 5151 y Fh(\010)25 b Ff(L)1265 5165 y Fe(i)1293 5151 y Fl(.)63 b(Next,)40 b(determine)d Ff(l)2096 5165 y Fe(w)2153 5151 y Fl(,)j(and)d(let)h Ff(\014)2592 5165 y Fe(i)2658 5151 y Fl(=)f Ff(X)2841 5166 y Fe(l)2862 5174 y Fc(w)2942 5151 y Fh(\010)25 b Ff(R)3107 5166 y Fe(l)3128 5174 y Fc(w)3207 5151 y Fh(\010)g Ff(R)3372 5165 y Fe(i)3400 5151 y Fl(,)40 b(and)d(let)1867 5416 y(17-4)p eop %%Page: 5 5 5 4 bop 150 159 a Ff(x)202 173 y Fe(i)255 159 y Fl(=)26 b Ff(\014)403 173 y Fe(i)451 159 y Fh(\010)20 b Ff(R)611 173 y Fe(i)640 159 y Fl(.)40 b(Next,)32 b(determine)d Ff(l)1404 173 y Fe(x)1448 159 y Fl(,)i(and)f(let)g Ff(\015)1859 173 y Fe(i)1913 159 y Fl(=)25 b Ff(Y)2062 174 y Fe(l)2083 182 y Fc(x)2146 159 y Fh(\010)20 b Ff(\013)2295 174 y Fe(l)2316 182 y Fc(x)2379 159 y Fh(\010)g Ff(\013)2528 173 y Fe(i)2557 159 y Fl(,)30 b(and)g(let)h Ff(y)2966 173 y Fe(i)3019 159 y Fl(=)25 b Ff(\015)3162 173 y Fe(i)3210 159 y Fh(\010)20 b Ff(\013)3359 173 y Fe(i)3387 159 y Fl(.)41 b(Finally)-8 b(,)150 271 y(determine)29 b Ff(l)601 285 y Fe(y)643 271 y Fl(,)i(and)e(let)i Ff(\016)1047 285 y Fe(i)1101 271 y Fl(=)24 b Ff(Z)1258 286 y Fe(l)1279 294 y Fc(y)1341 271 y Fh(\010)c Ff(\014)1483 286 y Fe(l)1504 294 y Fc(y)1567 271 y Fh(\010)g Ff(\014)1709 285 y Fe(i)1737 271 y Fl(.)41 b(Output)29 b Ff(\015)2173 285 y Fe(i)2221 271 y Fh(\016)21 b Ff(\016)2327 285 y Fe(i)2356 271 y Fl(.)291 384 y(No)m(w,)38 b(note)f(that)f(if)f Ff(l)1056 398 y Fe(x)1134 384 y Fl(=)f Ff(i)i Fl(then)g Ff(\015)1566 398 y Fe(i)1628 384 y Fl(=)f Ff(Y)1787 398 y Fe(i)1814 384 y Fl(,)j(and)d(if)g Ff(l)2175 398 y Fe(y)2251 384 y Fl(=)f Ff(i)i Fl(then)f Ff(\016)2675 398 y Fe(i)2738 384 y Fl(=)f Ff(Z)2905 398 y Fe(i)2934 384 y Fl(.)57 b(Th)m(us,)36 b(w)m(e)g(argue,)i(if)150 497 y(for)f(all)e Ff(m)i Fl(queries,)g Ff(l)911 511 y Fe(x)991 497 y Fl(=)f Ff(i)h Fl(and)f Ff(l)1376 511 y Fe(y)1454 497 y Fl(=)f Ff(i)i Fl(then)g(w)m(e)g(migh)m(t)f(as)h(w)m(ell)f(ha)m(v)m(e)i(nev)m (er)f(b)s(othered)f(with)g(the)150 610 y Ff(W)47 b Fl(and)35 b Ff(X)42 b Fl(parts)35 b(of)g(the)h(random)e(tap)s(e)h(and)g(just)f (ha)m(v)m(e)j(giv)m(en)e Ff(Y)55 b Fl(and)34 b Ff(Z)42 b Fl(directly)33 b(as)j(answ)m(ers;)h(in)150 723 y(other)c(w)m(ords,)h (if)e(w)m(e)h(had)f(used)h(a)g(random)f(function.)47 b(No)m(w,)35 b(w)m(e)f(can)f(pro)m(v)m(e)h(that)f(regardless)g(of)g (the)150 836 y(adv)m(ersary's)e(strategy)-8 b(,)32 b(this)d(can)i(happ) s(en)e(only)g(with)g(negligible)f(probabilit)m(y)-8 b(.)291 949 y(This)33 b(follo)m(ws)h(directly)g(from)g(the)h(pro)s(of)f(in)f (class.)54 b(T)-8 b(ranslating)34 b(notation,)i(in)e(class)g(w)m(e)i (pro)m(v)m(ed)150 1062 y(that)c(the)f(probabilit)m(y)e(that)i Ff(l)1195 1076 y Fe(w)1278 1062 y Fh(6)p Fl(=)c Ff(i)k Fl(or)g Ff(l)1577 1076 y Fe(x)1647 1062 y Fh(6)p Fl(=)c Ff(i)k Fl(for)g(an)m(y)g Ff(i)g Fl(w)m(as)h(at)g(most)f Ff(m)2772 1029 y Fg(2)2811 1062 y Ff(=)p Fl(2)2901 1029 y Fe(k)2945 1062 y Fl(.)43 b(No)m(w,)32 b(w)m(e)g(note)f(that)150 1175 y(if)d Ff(l)259 1189 y Fe(w)341 1175 y Fh(6)p Fl(=)c Ff(i)29 b Fl(and)f Ff(l)698 1189 y Fe(x)768 1175 y Fh(6)p Fl(=)d Ff(i)j Fl(then)h(the)g(probabilit)m(y)d(that)j Ff(l)1969 1189 y Fe(y)2036 1175 y Fh(6)p Fl(=)c Ff(i)k Fl(is)e(at)j(most)e(\()p Ff(i)18 b Fh(\000)e Fl(1\))p Ff(=)p Fl(2)2952 1142 y Fe(k)2997 1175 y Fl(,)29 b(so)g(the)f (probabilit)m(y)150 1288 y(that)39 b Ff(l)382 1302 y Fe(x)465 1288 y Fh(6)p Fl(=)f Ff(i)h Fl(or)g Ff(l)791 1302 y Fe(y)871 1288 y Fh(6)p Fl(=)g Ff(i)f Fl(for)h(an)m(y)f Ff(i)h Fl(is)f(at)h(most)g Ff(m)1977 1255 y Fg(3)2016 1288 y Ff(=)p Fl(2)2106 1255 y Fe(k)2150 1288 y Fl(.)65 b(This)36 b(is)i(negligible,)g(so)h Ff(LR)26 b Fh(\000)g Fl(4)p Ff(R)q Fl(\()p Ff(B)3592 1255 y Fd(0)3615 1288 y Fl(\))39 b(is)150 1401 y(indistinguishable)25 b(from)30 b(a)h(random)e(function.)291 1513 y(No)m(w,)43 b(with)38 b(one)i(quic)m(k)f(remark)g(w)m(e)h(can)f(note)i(that)f Ff(B)2306 1480 y Fd(0)2368 1513 y Fl(and)f Ff(B)k Fl(are)d(equiv)-5 b(alen)m(t,)42 b(b)s(ecause)d(for)150 1626 y(an)m(y)k(set)g(of)g (queries,)i(for)d(eac)m(h)i(random)e(tap)s(e)h Ff(B)1956 1593 y Fd(0)2021 1626 y Fl(can)g(ha)m(v)m(e)h(there)f(is)f(a)h(random)f (tap)s(e)h Ff(B)k Fl(could)150 1739 y(ha)m(v)m(e)35 b(that)f(w)m(ould)f (pro)s(duce)f(the)i(exact)h(same)f(output)f(as)h Ff(B)2305 1706 y Fd(0)2328 1739 y Fl(,)h(and)e(vice)g(v)m(ersa.)51 b(Th)m(us,)34 b Ff(LR)23 b Fh(\000)f Fl(4)p Ff(R)35 b Fl(is)150 1852 y(indistinguishable)25 b(from)30 b(a)h(random)e (function.)291 2065 y Fm(Step)i(3:)38 b Fl(No)m(w,)29 b(w)m(e)e(m)m(ust)g(construct)g(an)f Ff(LR)14 b Fh(\000)f Fl(4)p Ff(R)2090 2032 y Fd(\000)p Fg(1)2211 2065 y Fl(oracle)27 b(that)g(will)d(w)m(ork)j(with)f(the)h Ff(LR)14 b Fh(\000)f Fl(4)p Ff(R)150 2178 y Fl(oracle)35 b(and)f(whic)m(h)f(will)f(b)s(e)i (indistinguishable)29 b(from)34 b(a)h(random)f(function)f(to)j(an)m(y)e (adv)m(ersary)h(with)150 2291 y(an)30 b(oracle)h(for)f Ff(LR)21 b Fh(\000)f Fl(4)p Ff(R)31 b Fj(that)j(never)f(makes)g(r)-5 b(ep)g(etitive)33 b(queries.)291 2403 y Fl(The)23 b(w)m(a)m(y)h(to)h (construct)e Ff(LR)7 b Fh(\000)f Fl(4)p Ff(R)25 b Fl(and)e Ff(LR)7 b Fh(\000)f Fl(4)p Ff(R)1999 2370 y Fd(\000)p Fg(1)2116 2403 y Fl(sim)m(ultaneously)22 b(is)g(simple:)35 b(w)m(e)24 b(just)f(consider)150 2516 y(ourselv)m(es)f(to)h(ha)m(v)m(e) h(four)d(random)h(tap)s(es)g(whic)m(h)f(sp)s(ecify)g(the)h(outputs)g (of)h Ff(F)2744 2530 y Fg(1)2783 2516 y Fl(,)i Ff(F)2891 2530 y Fg(2)2930 2516 y Fl(,)g Ff(F)3038 2530 y Fg(3)3077 2516 y Fl(,)g(and)c Ff(F)3353 2530 y Fg(4)3393 2516 y Fl(,)j(the)f(four)150 2629 y(random)33 b(functions,)h(and)f(w)m(e)i (just)e(k)m(eep)i(trac)m(k)g(of)f(things,)g(as)g(b)s(efore.)52 b(This)32 b(giv)m(es)i(us)f(construction)150 2742 y Ff(C)38 b Fl(for)31 b Ff(LR)21 b Fh(\000)g Fl(4)p Ff(R)32 b Fl(and)f Ff(LR)21 b Fh(\000)g Fl(4)p Ff(R)1320 2709 y Fd(\000)p Fg(1)1446 2742 y Fl(w)m(orking)30 b(together,)k(and)c(this)h(clearly)f (sim)m(ulates)h Ff(LR)21 b Fh(\000)g Fl(4)31 b(using)150 2855 y(random)f(functions.)291 2968 y(Next,)g(in)d(construction)h Ff(C)1232 2935 y Fd(0)1255 2968 y Fl(,)h(instead)f(of)h(using)e(the)h (four)g(tap)s(es)h(in)e(the)h(normal)g(w)m(a)m(y)-8 b(,)30 b Ff(LR)17 b Fh(\000)f Fl(4)p Ff(R)3655 2935 y Fd(\000)p Fg(1)150 3081 y Fl(and)24 b Ff(LR)10 b Fh(\000)f Fl(4)p Ff(R)26 b Fl(do)f(what)g(construction)f Ff(B)1614 3048 y Fd(0)1662 3081 y Fl(did:)36 b(uses)25 b(the)g(v)-5 b(alues)24 b(in)g(the)h(tap)s(e)g(to)h(sp)s(ecify)d(the)i Fj(r)-5 b(ound's)150 3194 y(output)36 b Fl(rather)f(than)g(the)g Fj(function)-9 b('s)37 b(output,)g Fl(but)e(also,)h Ff(LR)24 b Fh(\000)f Fl(4)p Ff(R)2574 3161 y Fd(\000)p Fg(1)2704 3194 y Fl(uses)34 b(the)i(tap)s(es)f(in)f(opp)s(osite)150 3307 y(order.)44 b(Once)31 b(again,)h(w)m(e)g(ha)m(v)m(e)h(the)f (argumen)m(t)g(that)g(for)f(an)m(y)h(set)g(of)g(queries,)f(there)g(is)g (a)h(one-to-one)150 3420 y(corresp)s(ondence)e(b)s(et)m(w)m(een)h (random)e(tap)s(es)i(of)f Ff(C)37 b Fl(and)30 b(of)g Ff(C)2267 3387 y Fd(0)2290 3420 y Fl(.)291 3533 y(But)42 b(for)f(the)h(next)f(step)h(of)f(the)h(argumen)m(t,)j(w)m(e)d(m)m(ust)f (b)s(e)g(more)g(careful.)74 b(Basically)-8 b(,)44 b(w)m(e)e(are)150 3645 y(\\unhapp)m(y")g(-)h(ie,)i(can't)f(simply)c(b)s(e)i(outputting)g (a)h(random)f(v)-5 b(alue)42 b(-)h(if)e Ff(l)2858 3659 y Fe(x)2948 3645 y Fh(6)p Fl(=)k Ff(i)e Fl(or)g Ff(l)3289 3659 y Fe(y)3376 3645 y Fh(6)p Fl(=)i Ff(i)e Fl(on)g(a)150 3758 y Ff(LR)18 b Fh(\000)g Fl(4)p Ff(R)30 b Fl(query)-8 b(,)30 b(OR)f(if)f Ff(l)1078 3772 y Fe(x)1147 3758 y Fh(6)p Fl(=)d Ff(i)k Fl(or)h Ff(l)1441 3772 y Fe(y)1507 3758 y Fh(6)p Fl(=)25 b Ff(i)30 b Fl(on)f(a)g Ff(LR)19 b Fh(\000)e Fl(4)p Ff(R)2216 3725 y Fd(\000)p Fg(1)2340 3758 y Fl(query)-8 b(.)40 b(What)30 b(is)e(the)i(probabilit)m(y)c(that) 150 3871 y(one)31 b(of)f(these)h(bad)f(things)f(happ)s(ens?)39 b(W)-8 b(ell,)30 b(let)g(us)g(analyze)h(the)f(v)-5 b(alues)30 b(through)g(the)g(ev)-5 b(aluation.)291 3984 y(Supp)s(ose)35 b(for)h(the)g(momen)m(t)h(the)f(query)g(is)f Ff(L)1898 3998 y Fe(i)1926 3984 y Fl(,)j Ff(R)2058 3998 y Fe(i)2086 3984 y Fl(.)59 b(No)m(w,)38 b(neither)e Ff(L)2786 3998 y Fe(i)2850 3984 y Fl(nor)g Ff(R)3087 3998 y Fe(i)3151 3984 y Fl(m)m(ust)g(b)s(e)g(a)g(new)150 4097 y(v)-5 b(alue,)34 b(but)g(they)g(cannot)g(ha)m(v)m(e)h(b)s(een)e(ask)m(ed)i(together)g(b) s(efore.)51 b(There)33 b(are)h(three)g(cases,)i(therefore:)150 4210 y Ff(L)212 4224 y Fe(i)270 4210 y Fl(is)30 b(old)f(and)h Ff(R)759 4224 y Fe(i)818 4210 y Fl(is)f(new,)h Ff(L)1183 4224 y Fe(i)1241 4210 y Fl(is)g(new)g(and)f Ff(R)1765 4224 y Fe(i)1824 4210 y Fl(is)g(old,)h(and)g Ff(L)2331 4224 y Fe(i)2389 4210 y Fl(and)g Ff(R)2635 4224 y Fe(i)2694 4210 y Fl(are)g(b)s(oth)g(old.)291 4323 y(If)23 b Ff(R)444 4337 y Fe(i)496 4323 y Fl(is)f(new,)j(then)e(the)h(v)-5 b(alue)23 b Ff(F)1423 4337 y Fg(4)1463 4323 y Fl(\()p Ff(R)1567 4337 y Fe(i)1595 4323 y Fl(\))h(will)d(b)s(e)i(a)h(new)e (random)h(v)-5 b(alue,)25 b(and)e(therefore)h Ff(L)3373 4337 y Fe(i)3407 4323 y Fh(\010)6 b Ff(F)3542 4337 y Fg(4)3582 4323 y Fl(\()p Ff(R)3686 4337 y Fe(i)3715 4323 y Fl(\))150 4436 y(will)26 b(b)s(e)h(a)i(new)f(random)g(v)-5 b(alue,)28 b(and)g(so)g(therefore)h(the)g(outputs)e(from)h(the)h (second)f(round)f(on)h(will)e(all)150 4549 y(b)s(e)32 b(new)f(random)h(v)-5 b(alues.)45 b(Th)m(us,)32 b(the)g(probabilit)m(y) e(that)j(the)f(\014nal)f(outputs)h(will)d(not)k(b)s(e)e(random)h(is)150 4662 y(2)p Ff(i=)p Fl(2)316 4629 y Fe(k)360 4662 y Fl(.)291 4775 y(If)k Ff(R)457 4789 y Fe(i)521 4775 y Fl(is)f(old,)h(then)g Ff(F)1072 4789 y Fg(4)1112 4775 y Fl(\()p Ff(R)1216 4789 y Fe(i)1244 4775 y Fl(\))h(will)c(b)s(e)i(an)h(old)f(v)-5 b(alue.)57 b(Ho)m(w)m(ev)m(er,)39 b Ff(L)2658 4789 y Fe(i)2722 4775 y Fl(could)c(not)h(ha)m(v)m(e)i(b)s(een)d(used)150 4887 y(with)29 b Ff(R)426 4901 y Fe(i)484 4887 y Fl(b)s(efore,)h(so)g Ff(L)955 4901 y Fe(i)1003 4887 y Fh(\010)20 b Ff(F)1152 4901 y Fg(4)1191 4887 y Fl(\()p Ff(R)1295 4901 y Fe(i)1324 4887 y Fl(\))30 b(is)g(a)g(new)g(v)-5 b(alue.)40 b(Th)m(us,)29 b(after)i(the)f(\014rst)f(round,)g(w)m(e)i(ma)m(y)g(ha)m(v)m(e)g(an)150 5000 y(old)i(v)-5 b(alue)33 b(on)h(the)g(left)g(and)f(a)h(new)f(v)-5 b(alue)34 b(on)f(the)h(righ)m(t.)51 b(But)34 b(then)g(at)g(the)g (second)g(round)e(w)m(e)j(will)150 5113 y(ha)m(v)m(e)f(a)e(new)g(v)-5 b(alue)32 b(on)g(the)h(left)f(\(=)c Ff(L)1509 5127 y Fe(i)1559 5113 y Fh(\010)21 b Ff(F)1709 5127 y Fg(4)1749 5113 y Fl(\()p Ff(R)1853 5127 y Fe(i)1881 5113 y Fl(\)\))33 b(and)f(a)h(new)f(random)f(v)-5 b(alue)32 b(on)g(the)h(righ)m(t.)46 b(Th)m(us)1867 5416 y(17-5)p eop %%Page: 6 6 6 5 bop 150 159 a Fl(after)26 b(the)g(third)d(round,)i(w)m(e)h(will)d (ha)m(v)m(e)j(new)f(random)g(v)-5 b(alues)25 b(on)g(b)s(oth)g(sides,)g (so)h(the)f(probabilit)m(y)e(that)150 271 y(the)31 b(\014nal)e(outputs) g(will)f(not)j(b)s(e)e(random)h(is)f(2)p Ff(i=)p Fl(2)1890 238 y Fe(k)1935 271 y Fl(.)291 384 y(No)m(w)i(the)f(same)g(argumen)m(t) g(w)m(orks)g(for)g(the)g(rev)m(erse)h(direction)d(but)h(one)h(critical) f(fact)i(here)f(is)f(the)150 497 y(the)40 b(outputs)f(are)h(random)f(v) -5 b(alues)39 b(after)h(t)m(w)m(o)h(rounds.)68 b(Therefore,)42 b(all)c(the)i(places)g(there)g(can)g(b)s(e)150 610 y(relationships)31 b(the)j(adv)m(ersary)g(can)h(force)f(from)g(the)g(other)g(direction)f (con)m(tain)h(new)f(random)h(v)-5 b(alues)150 723 y(here.)43 b(If)30 b(there)h(w)m(ere)h(only)e(three)h(rounds,)f(w)m(e)i(could)e (only)g(guaran)m(tee)j(that)e(the)g(second)h(half)e(\(going)150 836 y(forw)m(ard,)d(or)f(the)g(\014rst)g(half)f(going)h(bac)m(kw)m (ards\))h(of)f(the)g(output)g(w)m(ould)f(b)s(e)h(a)g(new)g(random)f(v) -5 b(alue)26 b(eac)m(h)150 949 y(time.)291 1062 y(Th)m(us,)31 b(the)h(probabilit)m(y)d(for)i(an)m(y)h(giv)m(en)g(query)f(that)h(the)g (output)f(is)g(not)h(a)g(pair)e(of)i(new)f(random)150 1175 y(v)-5 b(alues)28 b(is)h(at)h(most)f(2)p Ff(i=)p Fl(2)1007 1142 y Fe(k)1051 1175 y Fl(.)41 b(Adding)27 b(this)h(up,)h(w)m(e)g(get)i(that)e(the)h(probabilit)m(y)c(that)k(an)m (y)g(output)e(is)h(not)150 1288 y(random)e(is)g(at)h(most)h(\()p Ff(m)1014 1255 y Fg(2)1068 1288 y Fh(\000)15 b Ff(m)p Fl(\))p Ff(=)p Fl(2)1359 1255 y Fe(k)1403 1288 y Fl(,)28 b(whic)m(h)f(is)g(still)e(negligible.)38 b(Th)m(us,)27 b(no)h(adv)m(ersary)-8 b(,)29 b(ha)m(ving)f(access)150 1401 y(to)j Ff(LR)21 b Fh(\000)f Fl(4)p Ff(R)31 b Fl(can)g(distinguish) c(b)s(et)m(w)m(een)k Ff(LR)20 b Fh(\000)g Fl(4)p Ff(R)1985 1368 y Fd(\000)p Fg(1)2110 1401 y Fl(and)30 b(a)h(random)e(function.) 291 1613 y Fm(Step)36 b(4:)42 b Fl(No)m(w,)33 b(supp)s(ose)c(there)j (is)e(an)h(adv)m(ersary)h Ff(A)2166 1580 y Fg(?)p Fe(;)p Fg(?)2288 1613 y Fl(whic)m(h)e(nev)m(er)h(mak)m(es)h(rep)s(etitiv)m(e)f (queries)150 1726 y(can)g(distinguish)c(b)s(et)m(w)m(een)32 b Ff(LR)21 b Fh(\000)f Fl(4)p Ff(;)15 b(LR)22 b Fh(\000)e Fl(4)1747 1693 y Fd(\000)p Fg(1)1873 1726 y Fl(and)30 b(a)h(random)f(p)s(erm)m(utation)g(and)g(its)h(in)m(v)m(erse.)42 b(W)-8 b(e)150 1839 y(use)30 b(a)h(non-uniform)d(h)m(ybrid)g(argumen)m (t)j(to)g(\014nish)d(o\013)j(the)f(pro)s(of.)40 b(There)30 b(are)h(\014v)m(e)f(cases:)150 1952 y Fj(c)-5 b(ase)36 b(i:)48 b Ff(A)520 1919 y Fg(?)p Fe(;)p Fg(?)644 1952 y Fl(distinguishes)30 b(b)s(et)m(w)m(een)35 b Ff(LR)23 b Fh(\000)f Fl(4)p Ff(;)15 b(LR)24 b Fh(\000)f Fl(4)2165 1919 y Fd(\000)p Fg(1)2293 1952 y Fl(and)34 b Ff(LR)23 b Fh(\000)f Fl(4)p Ff(R)q(;)15 b(LR)24 b Fh(\000)e Fl(4)p Ff(R)3239 1919 y Fd(\000)p Fg(1)3333 1952 y Fl(.)52 b(No)m(w,)35 b(w)m(e)150 2065 y(violate)f(the)f(assumption)f(that)i(the)g(functions) e Ff(F)1905 2079 y Fg(1)1945 2065 y Fl(,)i Ff(F)2062 2079 y Fg(2)2102 2065 y Fl(,)g Ff(F)2219 2079 y Fg(3)2259 2065 y Fl(,)h(and)e Ff(F)2557 2079 y Fg(4)2630 2065 y Fl(w)m(ere)h(all)e(pseudorandom.)49 b(W)-8 b(e)150 2178 y(construct)26 b(a)h(h)m(ybrid)d(argumen)m(t)i(-)h(the)f(h)m(ybrids)d (are)k Ff(LR)12 b Fh(\000)g Fl(4,)27 b Ff(LR)12 b Fh(\000)g Fl(4)p Ff(R)2642 2192 y Fg(1)2681 2178 y Ff(;)j(:)g(:)g(:)i(;)e(LR)d Fh(\000)g Fl(4)p Ff(R)3223 2192 y Fg(4)3288 2178 y Fl(=)25 b Ff(LR)12 b Fh(\000)g Fl(4)p Ff(R)q Fl(,)150 2291 y(where)41 b Ff(LR)29 b Fh(\000)e Fl(4)p Ff(R)796 2305 y Fe(i)867 2291 y Fl(is)40 b Ff(LR)29 b Fh(\000)e Fl(4)43 b(where)e(the)h(\014rst) f Ff(i)h Fl(of)g(the)g Ff(F)2370 2305 y Fe(i)2440 2291 y Fl(are)g(random)f(functions)g(instead)g(of)150 2403 y(pseudorandom.)e(No)m(w,)30 b(since)f Ff(A)1313 2370 y Fg(?)p Fe(;)p Fg(?)1433 2403 y Fl(can)h(distinguish)25 b Ff(LR)19 b Fh(\000)f Fl(4)p Ff(R)2411 2417 y Fg(0)2477 2403 y Fl(=)24 b Ff(LR)19 b Fh(\000)f Fl(4)30 b(from)f Ff(LR)19 b Fh(\000)f Fl(4)p Ff(R)3453 2417 y Fg(4)3493 2403 y Fl(,)30 b(there)150 2516 y(m)m(ust)g(b)s(e)f(some)h Ff(i)g Fl(for)g(whic)m(h)e Ff(A)i Fl(can)g(distinguish)c Ff(LR)20 b Fh(\000)e Fl(4)p Ff(R)2262 2530 y Fe(i)2321 2516 y Fl(from)29 b Ff(LR)20 b Fh(\000)e Fl(4)p Ff(R)2889 2530 y Fe(i)p Fg(+1)3008 2516 y Fl(,)30 b(whic)m(h)f(sho)m(ws)g(that) 150 2629 y Ff(A)h Fl(can)h(b)s(e)f(used)f(to)j(break)e(the)g (pseudorandomness)f(of)h Ff(F)2195 2643 y Fe(i)p Fg(+1)2314 2629 y Ff(;)h Fl(whic)m(h)e(is)g(a)i(con)m(tradiction.)150 2742 y Fj(c)-5 b(ase)34 b(ii:)42 b Ff(A)540 2709 y Fg(?)p Fe(;)p Fg(?)661 2742 y Fl(distinguishes)28 b(b)s(et)m(w)m(een)k Ff(LR)21 b Fh(\000)f Fl(4)p Ff(R)q(;)15 b(LR)22 b Fh(\000)f Fl(4)p Ff(R)2309 2709 y Fd(\000)p Fg(1)2434 2742 y Fl(and)31 b Ff(LR)21 b Fh(\000)g Fl(4)p Ff(R)q(;)15 b(R)q Fl(.)43 b(W)-8 b(e)33 b(pro)m(v)m(ed)e(this)150 2855 y(w)m(as)g(imp)s(ossible)c (in)i(step)h(3.)41 b(Con)m(tradiction.)150 2968 y Fj(c)-5 b(ase)31 b(iii:)38 b Ff(A)561 2935 y Fg(?)p Fe(;)p Fg(?)679 2968 y Fl(distinguishes)24 b(b)s(et)m(w)m(een)k Ff(LR)16 b Fh(\000)f Fl(4)p Ff(R)q(;)g(R)28 b Fl(and)f Ff(R)2290 2935 y Fd(0)2313 2968 y Ff(;)15 b(R)q Fl(.)40 b(No)m(w,)29 b(w)m(e)f(can)g(create)h(an)e Ff(A)3469 2935 y Fd(0)3520 2968 y Fl(whic)m(h)150 3081 y(distinguishes)j(b)s(et)m(w)m(een)k Ff(LR)23 b Fh(\000)f Fl(4)p Ff(R)35 b Fl(and)e Ff(R)h Fl(b)m(y)g(simply)d(creating)j(its)f(o)m(wn)g(random)g(function)f (oracle)150 3194 y(and)d(sim)m(ulating)f(the)i(execution)g(of)f Ff(A)1500 3161 y Fg(?)p Fe(;)p Fg(?)1591 3194 y Fl(.)40 b(Since)29 b(in)g(step)g(2)h(w)m(e)h(pro)m(v)m(ed)f(suc)m(h)f(an)h Ff(A)3092 3161 y Fd(0)3145 3194 y Fl(cannot)g(exist,)g Ff(A)150 3307 y Fl(cannot)h(exist)f(either.)40 b(Con)m(tradiction.)150 3420 y Fj(c)-5 b(ase)39 b(iv:)52 b Ff(A)569 3387 y Fg(?)p Fe(;)p Fg(?)696 3420 y Fl(can)37 b(distinguish)c(b)s(et)m(w)m(een)k Ff(R)1762 3434 y Fg(1)1801 3420 y Ff(;)15 b(R)1910 3434 y Fg(2)1987 3420 y Fl(and)36 b(\005)2238 3434 y Fg(1)2277 3420 y Ff(;)15 b Fl(\005)2385 3434 y Fg(2)2462 3420 y Fl(\(or)37 b(equiv)-5 b(alen)m(tly)35 b(\005)3194 3434 y Fg(1)3233 3420 y Ff(;)15 b Fl(\005)3341 3381 y Fd(\000)p Fg(1)3341 3446 y(2)3436 3420 y Fl(\).)60 b(By)37 b(a)150 3533 y(simple)29 b(h)m(ybrid)g(argumen)m(t,)j Ff(A)f Fl(can)g(either)g(distinguish)c(b)s(et)m(w)m(een)k Ff(R)2562 3547 y Fg(1)2633 3533 y Fl(and)f(\005)2878 3547 y Fg(1)2949 3533 y Fl(or)h(it)f(can)i(distinguish)150 3645 y Ff(R)219 3659 y Fg(2)289 3645 y Fl(from)e(\005)572 3659 y Fg(2)611 3645 y Fl(,)h(whic)m(h)e(violates)h(our)g(result)f(from)h(problem)f(1.) 41 b(Con)m(tradiction.)150 3758 y Fj(c)-5 b(ase)31 b(v:)39 b Ff(A)520 3725 y Fg(?)p Fe(;)p Fg(?)639 3758 y Fl(can)29 b(distinguish)c(b)s(et)m(w)m(een)k(\005)1680 3772 y Fg(1)1719 3758 y Ff(;)15 b Fl(\005)1827 3772 y Fg(2)1896 3758 y Fl(and)27 b(\005)2138 3772 y Fg(1)2178 3758 y Ff(;)15 b Fl(\005)2286 3720 y Fd(\000)p Fg(1)2286 3784 y(1)2381 3758 y Fl(.)40 b(In)27 b(order)h(to)h(do)g(this,)f Ff(A)g Fj(must)h Fl(mak)m(e)150 3871 y(rep)s(etitiv)m(e)h(queries,)f(but)h(w)m (e)h(assumed)f(it)f(did)g(not.)41 b(Therefore,)30 b(con)m(tradiction.) 291 3984 y(Finally)-8 b(,)29 b(w)m(e)i(conclude)f(that)h(no)f(suc)m(h)g Ff(A)h Fl(ma)m(y)f(exist.)41 b(This)29 b(completes)h(the)h(pro)s(of.) 150 4271 y Fi(Problem)45 b(4:)61 b(Encrypting)45 b(the)g(Secret)g(Key) 150 4474 y Fl(Let)39 b(\()p Ff(G;)15 b(E)5 b(;)15 b(D)s Fl(\))41 b(b)s(e)d(a)i(\(one-pass\))f(GM-secure)h(cryptosystem.)67 b(This)37 b(should)g(b)s(e)h(though)m(t)h(of)g(as)h(a)150 4586 y(\\secure)32 b(en)m(v)m(elop)s(e")h({)f(the)g(adv)m(ersary)h (cannot)f(see)h(its)e(con)m(ten)m(ts,)j(no)e(matter)h(what)f(the)g (message)h(is.)150 4699 y(This)38 b(problem)g(sho)m(ws)h(that)h(one)g (has)g(to)g(b)s(e)f(careful)g(with)f(this)h(in)m(tuition.)66 b(Namely)-8 b(,)43 b(w)m(e)d(ask)g(the)150 4812 y(question,)28 b(\\Can)g(I)g(trust)f(m)m(y)h(encryption)f(to)i(the)f(exten)m(t)h(that) g(I)e(can)i(publish)24 b(the)k(encryption)f(of)h(m)m(y)150 4925 y(secret)j(k)m(ey?")1867 5416 y(17-6)p eop %%Page: 7 7 7 6 bop 291 159 a Fm(\(a\))26 b Fl(\(7)h(p)s(oin)m(ts\))f(Construct)g (a)h(cryptosystem)g(\()p Ff(G)2014 126 y Fd(0)2038 159 y Ff(;)15 b(E)2150 126 y Fd(0)2174 159 y Ff(;)g(D)2292 126 y Fd(0)2316 159 y Fl(\))26 b(based)h(on)f(\()p Ff(G;)15 b(E)5 b(;)15 b(D)s Fl(\))29 b(suc)m(h)d(that)h(giv)m(en)150 271 y Ff(t)e Fl(=)g Ff(E)376 238 y Fd(0)400 271 y Fl(\()p Ff(P)13 b(K)590 238 y Fd(0)613 271 y Ff(;)i(S)5 b(K)798 238 y Fd(0)821 271 y Fl(\),)27 b(the)f(adv)m(ersary)f(can)g(easily)g (distinguish)c(the)k(encryptions)f(of)i Fj(any)f Fl(t)m(w)m(o)i (messages)150 384 y Ff(m)230 398 y Fg(0)302 384 y Fl(and)32 b Ff(m)561 398 y Fg(1)601 384 y Fl(.)48 b(\(Hin)m(t:)e Ff(t)32 b Fl(should)f(someho)m(w)j(con)m(tain)f Ff(S)5 b(K)2171 351 y Fd(0)2194 384 y Fl(.)48 b(T)-8 b(ry)32 b(to)i(imagine)e(ho)m(w)h(this)f(can)h(b)s(e)f(done)150 497 y(without)d(making)h(\()p Ff(G)915 464 y Fd(0)939 497 y Ff(;)15 b(E)1051 464 y Fd(0)1075 497 y Ff(;)g(D)1193 464 y Fd(0)1217 497 y Fl(\))30 b(insecure.\))291 710 y Fm(Solution:)40 b Fl(The)28 b(tric)m(k)g(here)g(is)f(that)i(when)e(w) m(e)i(are)g(giv)m(en)f(a)g(message)i Ff(m)e Fl(it)f(is)h(non-trivial)e (to)j(tell)150 823 y(if)c Ff(m)g Fl(is)g(the)h(secret)h(k)m(ey)f(asso)s (ciated)g(with)f(PK.)g(The)h(basic)f(idea)g(is)g(to)h(let)g Ff(E)5 b Fl(\()p Ff(P)13 b(K)q(;)i(m)p Fl(\))27 b(=)e Ff(m)h Fl(if)e Ff(m)i Fl(is)f(the)150 936 y(secret)33 b(k)m(ey)-8 b(.)47 b(One)32 b(simple)e(solution)g(is)i(to)g(try)g(a)h (bunc)m(h)e(of)h(messages)h(and)f(see)g(if)f Ff(m)h Fl(decrypts)g (them.)150 1049 y(So,)f(in)e(this)g(spirit,)f(let)150 1261 y Ff(G)221 1228 y Fd(0)245 1261 y Fl(\(1)325 1228 y Fe(k)368 1261 y Fl(\))e(=)f Ff(G)p Fl(\(1)676 1228 y Fe(k)720 1261 y Fl(\))150 1374 y Ff(E)222 1341 y Fd(0)246 1374 y Fl(\()p Ff(P)13 b(K)q(;)i(m)p Fl(\))26 b(:)150 1487 y(1.)41 b(Let)31 b Ff(m)504 1454 y Fd(0)552 1487 y Fh( )26 b Ff(M)40 b Fl(b)s(e)30 b(c)m(hosen)h(at)g(random.)150 1600 y(2.)41 b(If)30 b Ff(m)432 1567 y Fd(0)480 1600 y Fl(=)25 b Ff(D)s Fl(\()p Ff(m;)15 b(E)5 b Fl(\()p Ff(P)13 b(K)q(;)i(m)1185 1567 y Fd(0)1211 1600 y Fl(\)\),)31 b(output)f(0)20 b Fh(\016)h Ff(m)p Fl(,)31 b(else)f(output)g(1)20 b Fh(\016)h Ff(E)5 b Fl(\()p Ff(P)13 b(K)q(;)i(m)p Fl(\).)150 1713 y Ff(D)228 1680 y Fd(0)251 1713 y Fl(\()p Ff(S)5 b(K)q(;)15 b(b)22 b Fh(\016)e Ff(c)p Fl(\))26 b(:)150 1826 y(1.)41 b(If)30 b Ff(b)25 b Fl(=)g(0)31 b(output)f Ff(m)p Fl(.)150 1939 y(2.)41 b(If)30 b Ff(b)25 b Fl(=)g(1)31 b(output)f Ff(D)s Fl(\()p Ff(S)5 b(K)q(;)15 b(c)p Fl(\).)291 2264 y(No)m(w)37 b(clearly)-8 b(,)38 b(if)d(the)h(adv)m(ersary)h(is)e (giv)m(en)h Ff(E)5 b Fl(\()p Ff(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))38 b(then)e(the)g(adv)m(ersary)h(gets)g Ff(S)5 b(K)43 b Fl(in)35 b(the)150 2377 y(clear)24 b(and)g(th)m(us)g(can)h (decrypt)f(an)m(y)h(message.)40 b(Ho)m(w)m(ev)m(er,)28 b Ff(G)2219 2344 y Fd(0)2242 2377 y Ff(;)15 b(E)2354 2344 y Fd(0)2378 2377 y Ff(;)g(D)2496 2344 y Fd(0)2544 2377 y Fl(is)24 b(still)e(secure.)39 b(This)22 b(is)i(not)h(quite)150 2490 y(trivial)33 b(but)h(it)h(is)f(close.)55 b(Supp)s(ose)33 b(an)i(adv)m(ersary)h Ff(A)2033 2457 y Fd(0)2091 2490 y Fl(can)g(break)e(this)g(cryptosystem.)56 b(Then)34 b(there)150 2603 y(are)d(t)m(w)m(o)h(p)s(ossibilities.)38 b(Either)29 b(the)i(adv)m(ersary)g(can)g(also)g(break)g(\()p Ff(G;)15 b Fl(1)22 b Fh(\016)f Ff(E)5 b(;)15 b(D)s Fl(\))31 b(directly)f(\(whic)m(h)g(is)g(a)150 2716 y(con)m(tradiction\),)g(or)f (the)h(adv)m(ersary)f(manages)h(to)g(c)m(ho)s(ose)g(an)f Ff(m)g Fl(\(either)g Ff(m)2811 2730 y Fg(0)2880 2716 y Fl(or)g Ff(m)3070 2730 y Fg(1)3109 2716 y Fl(\))h(suc)m(h)e(that)i (with)150 2828 y(nonnegligible)24 b(probabilit)m(y)-8 b(,)26 b Ff(m)g Fl(can)h(decrypt)g(a)g(randomly)f(c)m(hosen)h(message)h Ff(m)2963 2796 y Fd(0)2986 2828 y Fl(.)39 b(No)m(w)28 b(w)m(e)f(could)f(use)150 2941 y Ff(A)218 2908 y Fd(0)273 2941 y Fl(to)32 b(break)f(RM)h(securit)m(y)f(of)g(\()p Ff(G;)15 b(E)5 b(;)15 b(D)s Fl(\))34 b(b)m(y)d(c)m(ho)s(osing)g(a)h (random)e(message)j(for)e(our)g Ff(m)p Fl(,)g(and)g(when)150 3054 y(giv)m(en)c(the)g(c)m(hallenge)g Ff(c)p Fl(,)h(w)m(e)g(run)d Ff(A)1379 3021 y Fd(0)1429 3054 y Fl(on)i(input)e Ff(P)13 b(K)34 b Fl(to)27 b(get)h Ff(m)2308 3068 y Fg(0)2348 3054 y Ff(;)15 b(m)2468 3068 y Fg(1)2507 3054 y Fl(.)40 b(W)-8 b(e)28 b(try)f(to)g(decrypt)g Ff(c)g Fl(with)f(eac)m(h)150 3167 y(of)31 b Ff(m)334 3181 y Fg(0)404 3167 y Fl(and)g Ff(m)662 3181 y Fg(1)732 3167 y Fl(and)f(if)g(either)h(giv)m(es)g(us)g Ff(m)f Fl(or)h Ff(r)s Fl(,)g(w)m(e)h(mak)m(e)g(up)e(our)g(mind)f(that)j (w)m(a)m(y)g(and)f(otherwise)150 3280 y(\015ip)e(a)h(coin.)41 b(I)30 b(omit)g(the)g(analysis)f(of)i(this)e(reduction,)h(but)f(it)h (should)f(b)s(e)g(fairly)g(clear.)291 3493 y Fm(\(b\))41 b Fl(\(7)i(p)s(oin)m(ts\))e(Construct)h(a)g(cryptosystem)g(\()p Ff(G)2113 3460 y Fd(00)2156 3493 y Ff(;)15 b(E)2268 3460 y Fd(00)2312 3493 y Ff(;)g(D)2430 3460 y Fd(00)2472 3493 y Fl(\))42 b(based)g(on)g(\()p Ff(G;)15 b(E)5 b(;)15 b(D)s Fl(\))44 b(suc)m(h)d(that)150 3606 y(ev)m(en)27 b(giving)e Ff(t)g Fl(=)g Ff(E)845 3573 y Fd(00)888 3606 y Fl(\()p Ff(P)13 b(K)1078 3573 y Fd(00)1121 3606 y Ff(;)i(S)5 b(K)1306 3573 y Fd(00)1348 3606 y Fl(\))27 b(to)g(the)f(adv)m(ersary)-8 b(,)28 b(the)f(sc)m(heme)g(is)e(still)g(GM-secure.)40 b(Y)-8 b(ou)27 b(should)150 3718 y(imagine)35 b(that)i(the)g(adv)m (ersary)f(learns)f Ff(t)h Fl(at)h(the)g(same)g(time)f(as)g Ff(P)13 b(K)2615 3686 y Fd(00)2657 3718 y Fl(.)59 b(\(Hin)m(t:)52 b Ff(t)36 b Fl(should)e(someho)m(w)150 3831 y(con)m(tain)d(random)e (garbage.)42 b(T)-8 b(ry)30 b(to)h(imagine)f(ho)m(w)g(w)m(e)h(can)g(do) f(this)f(but)h Fj(stil)5 b(l)33 b(b)-5 b(e)32 b(able)h(to)g(de)-5 b(crypt)34 b Ff(t)p Fl(\))291 4044 y Fm(Solution:)41 b Fl(Our)29 b(construction)h(this)f(time)i(is)e(as)i(follo)m(ws:)150 4256 y Ff(G)221 4223 y Fd(00)264 4256 y Fl(\(1)344 4223 y Fe(k)387 4256 y Fl(\))26 b(=)f Ff(G)p Fl(\(1)695 4223 y Fe(k)739 4256 y Fl(\))150 4369 y Ff(E)222 4336 y Fd(00)265 4369 y Fl(\()p Ff(P)13 b(K)q(;)i(m)p Fl(\))27 b(:)150 4482 y(1.)41 b(Let)31 b Ff(m)504 4449 y Fd(0)552 4482 y Fh( )26 b Ff(M)40 b Fl(b)s(e)30 b(c)m(hosen)h(at)g(random.)150 4595 y(2.)41 b(If)30 b Ff(m)432 4562 y Fd(0)480 4595 y Fl(=)25 b Ff(D)s Fl(\()p Ff(m;)15 b(E)5 b Fl(\()p Ff(P)13 b(K)q(;)i(m)1185 4562 y Fd(0)1211 4595 y Fl(\)\),)31 b(output)f(0,)h(else)f(output)g(1)21 b Fh(\016)f Ff(E)5 b Fl(\()p Ff(P)13 b(K)q(;)i(m)p Fl(\).)150 4708 y Ff(D)228 4675 y Fd(00)270 4708 y Fl(\()p Ff(S)5 b(K)q(;)15 b(c)p Fl(\))28 b(:)150 4821 y(1.)41 b(If)30 b Ff(c)c Fl(=)f(0)30 b(output)g Ff(S)5 b(K)i Fl(.)150 4934 y(2.)41 b(If)30 b Ff(c)c Fl(=)f(1)20 b Fh(\016)h Ff(c)683 4901 y Fd(0)737 4934 y Fl(output)30 b Ff(D)s Fl(\()p Ff(S)5 b(K)q(;)15 b(c)1366 4901 y Fd(0)1391 4934 y Fl(\).)1867 5416 y(17-7)p eop %%Page: 8 8 8 7 bop 291 159 a Fl(By)33 b(the)f(same)h(argumen)m(t)f(as)h(ab)s(o)m (v)m(e,)h(this)d(cryptosystem)i(is)e Ff(GM)10 b Fl(-secure.)47 b(On)32 b(the)g(other)h(hand,)150 271 y(this)e(time)h(the)g(adv)m (ersary)h(gets)g(a)f(0)h(when)e(giv)m(en)h(the)g(encryption)f(of)i(the) f(secret)h(k)m(ey)-8 b(,)34 b(whic)m(h)d(w)m(ould)150 384 y(clearly)h(b)s(e)f(repro)s(ducible)e(without)j(the)g(secret)h(k)m (ey)-8 b(.)48 b(Th)m(us,)32 b(if)f(the)i(adv)m(ersary)f(can)h(break)f (the)g(cryp-)150 497 y(tosystem)42 b(with)e(this,)j(it)e(can)g(break)g (it)g(without)f(it,)k(so)d(the)h(sc)m(heme)f(m)m(ust)g(still)f(b)s(e)g (secure)h(if)f(w)m(e)150 610 y(pro)m(vide)29 b(the)i(encryption)e(of)i (the)f(secret)i(k)m(ey)-8 b(.)291 723 y(Note)35 b(that)f(this)e (cryptosystem)i(is)e(faithful)g(for)h(the)g(message)i Ff(m)30 b Fl(=)g Ff(S)5 b(K)i Fl(.)50 b(Ho)m(w)m(ev)m(er,)36 b(it)d(will)e(not)150 836 y(b)s(e)f(faithful)e(for)i(an)m(y)h (\\alternate")h(secret)f(k)m(ey)-8 b(,)32 b(but)e(this)f(is)g(not)i (suc)m(h)f(a)h(big)e(deal.)291 1049 y Fm(\(c\))34 b Fl(\(6)i(p)s(oin)m (ts\))d(Supp)s(ose)g(Alice)h(and)g(Bob)h(eac)m(h)g(ha)m(v)m(e)h(k)m (eys,)g(\()p Ff(P)13 b(K)2687 1063 y Fe(A)2745 1049 y Ff(;)i(S)5 b(K)2923 1063 y Fe(A)2981 1049 y Fl(\))34 b(and)g(\()p Ff(P)13 b(K)3414 1063 y Fe(B)3475 1049 y Ff(;)i(S)5 b(K)3653 1063 y Fe(B)3715 1049 y Fl(\))150 1161 y(under)19 b(\()p Ff(G;)c(E)5 b(;)15 b(D)s Fl(\).)39 b(Pro)m(v)m(e)22 b(that)f(Alice)f(can)g(securely)g(encrypt)g Fj(her)h Fl(secret)g(k)m(ey)g(under)e Fj(Bob's)i Fl(public)c(k)m(ey)-8 b(.)150 1274 y(That)24 b(is,)g(pro)m(v)m(e)h(that)f(\()p Ff(G;)15 b(E)5 b(;)15 b(D)s Fl(\))26 b(is)d(still)f(GM-secure)j(ev)m (en)g(when)d(the)i(adv)m(ersary)h(kno)m(ws)e Ff(P)13 b(K)3418 1288 y Fe(A)3475 1274 y Ff(;)i(P)e(K)3663 1288 y Fe(B)3725 1274 y Fl(,)150 1387 y(and)30 b Ff(E)5 b Fl(\()p Ff(P)13 b(K)582 1401 y Fe(B)643 1387 y Ff(;)i(S)5 b(K)821 1401 y Fe(A)879 1387 y Fl(\).)41 b(F)-8 b(ormally)g(,)30 b(pro)m(v)m(e)i(that)150 1600 y Fh(8f)p Ff(A)314 1615 y Fe(k)357 1600 y Fh(g)26 b(2)e Ff(P)13 b(P)g(T)43 b Fh(9)p Ff(\027)31 b Fh(2)25 b Ff(neg)s(l)32 b Fh(8)p Ff(k)150 1713 y Fl(Pr)o([\()p Ff(P)13 b(K)455 1727 y Fe(A)513 1713 y Ff(;)i(S)5 b(K)691 1727 y Fe(A)749 1713 y Fl(\))26 b Fh( )f Ff(G)p Fl(\(1)1077 1680 y Fe(k)1121 1713 y Fl(\);)15 b(\()p Ff(P)e(K)1379 1727 y Fe(B)1441 1713 y Ff(;)i(S)5 b(K)1619 1727 y Fe(B)1680 1713 y Fl(\))25 b Fh( )h Ff(G)p Fl(\(1)2008 1680 y Fe(k)2052 1713 y Fl(\);)15 b Ff(t)25 b Fh( )h Ff(E)5 b Fl(\()p Ff(P)13 b(K)2557 1727 y Fe(B)2618 1713 y Ff(;)i(S)5 b(K)2796 1727 y Fe(A)2854 1713 y Fl(\);)273 1826 y(\()p Ff(\013;)15 b(m)486 1840 y Fg(0)526 1826 y Ff(;)g(m)646 1840 y Fg(1)686 1826 y Fl(\))25 b Fh( )h Ff(A)p Fl(\()p Ff(P)13 b(K)1114 1840 y Fe(A)1171 1826 y Ff(;)i(P)e(K)1359 1840 y Fe(B)1421 1826 y Ff(;)i(t)p Fl(\);)g Ff(b)26 b Fh( )f(f)p Fl(0)p Ff(;)15 b Fl(1)p Fh(g)p Fl(;)g Ff(b)2049 1793 y Fd(0)2100 1826 y Fh( )25 b Ff(A)p Fl(\()p Ff(\013;)15 b(E)5 b Fl(\()p Ff(P)13 b(K)2672 1840 y Fe(A)2732 1826 y Ff(;)i(m)2852 1841 y Fe(b)2886 1826 y Fl(\)\))26 b(:)g Ff(b)3072 1793 y Fd(0)3120 1826 y Fl(=)f Ff(b)p Fl(])h Ff(<)f Fl(1)p Ff(=)p Fl(2)c(+)f Ff(\027)6 b Fl(\()p Ff(k)s Fl(\))291 2138 y Fm(Solution:)41 b Fl(Supp)s(ose)28 b(there)j(is)e(an)h(adv)m (ersary)g Ff(A)h Fl(that)f(can)h(defeat)g Ff(GM)10 b Fl(-securit)m(y)31 b(of)f Ff(P)13 b(K)3478 2152 y Fe(A)3565 2138 y Fl(from)150 2251 y(\()p Ff(G;)i(E)5 b(;)15 b(D)s Fl(\))42 b(giv)m(en)e Ff(E)5 b Fl(\()p Ff(P)13 b(K)1065 2265 y Fe(B)1126 2251 y Ff(;)i(S)5 b(K)1304 2265 y Fe(A)1362 2251 y Fl(\).)69 b(There)40 b(are)g(t)m(w)m(o)h(cases.)70 b(Either)39 b(the)h(adv)m(ersary)g(w)m(orks)g(when)150 2364 y(giv)m(en)26 b Ff(t)f Fl(=)g Ff(E)5 b Fl(\()p Ff(P)13 b(K)792 2378 y Fe(B)854 2364 y Ff(;)i(S)5 b(K)1032 2378 y Fe(A)1089 2364 y Fl(\))27 b(but)e(do)s(esn't)h(w)m(ork)g(giv)m(en)h (the)f(encryption)f(of)h(a)h(random)e(message)i(under)150 2477 y Ff(P)13 b(K)298 2491 y Fe(B)386 2477 y Fl(\(in)26 b(whic)m(h)g(case)i(the)f(adv)m(ersary)g(breaks)g(encryptions)f(under)f Ff(P)13 b(K)2702 2491 y Fe(B)2763 2477 y Fl(\),)28 b(or)f(the)h(adv)m (ersary)f(w)m(orks)150 2589 y(with)j(the)i(encryption)e(of)i(a)g (random)e(message)j(under)d Ff(P)13 b(K)2243 2603 y Fe(B)2304 2589 y Fl(,)32 b(in)e(whic)m(h)g(case)j(the)e(adv)m(ersary)h(breaks)150 2702 y(encryptions)d(under)g Ff(P)13 b(K)1044 2716 y Fe(A)1101 2702 y Fl(.)41 b(More)31 b(formally)-8 b(,)291 2815 y(Let)35 b Ff(p)504 2829 y Fg(0)577 2815 y Fl(b)s(e)e(the)i (probabilit)m(y)c(that)k Ff(A)f Fl(w)m(orks)g(under)f(its)g(exp)s (ected)i(circumstances.)51 b Ff(p)3347 2829 y Fg(0)3418 2815 y Fh(\025)31 b Fl(1)p Ff(=)p Fl(2)24 b(+)150 2928 y(1)p Ff(=p)p Fl(\()p Ff(k)s Fl(\))29 b(for)f(some)g Ff(p)p Fl(.)40 b(Let)28 b Ff(p)1114 2942 y Fg(1)1181 2928 y Fl(b)s(e)f(the)h(probabilit)m(y)d(that)k Ff(A)f Fl(w)m(orks)f(giv)m(en)h(an)g(encryption)f(of)h(a)g(random)150 3041 y(message)j(rather)g(than)f(of)g Ff(S)5 b(K)1226 3055 y Fe(A)1284 3041 y Fl(.)291 3154 y(If)30 b Ff(p)428 3168 y Fg(1)493 3154 y Fh(\025)25 b Fl(1)p Ff(=)p Fl(2)d(+)e(1)p Ff(=q)s Fl(\()p Ff(k)s Fl(\))32 b(for)e(some)h(p)s(olynomial)d Ff(q)s Fl(,)i(then)g(w)m(e)h(can)g(use)f(the)h(follo)m(wing)e (reduction)g(to)150 3267 y(break)h Ff(GM)41 b Fl(securit)m(y:)150 3479 y Ff(R)q Fl(\()p Ff(P)13 b(K)q(;)i Fl(1)489 3446 y Fe(k)533 3479 y Fl(\))26 b(:)150 3592 y(1.)41 b(Let)31 b(\()p Ff(P)13 b(K)607 3606 y Fe(B)668 3592 y Ff(;)i(S)5 b(K)846 3606 y Fe(B)907 3592 y Fl(\))26 b Fh( )f Ff(G)p Fl(\(1)1235 3559 y Fe(k)1279 3592 y Fl(\).)150 3705 y(2.)41 b(Pic)m(k)30 b Ff(m)25 b Fh( )h Ff(M)774 3720 y Fe(k)847 3705 y Fl(and)k(get)h(\()p Ff(m)1290 3719 y Fg(0)1330 3705 y Ff(;)15 b(m)1450 3719 y Fg(1)1489 3705 y Ff(;)g(\013)p Fl(\))27 b Fh( )e Ff(A)p Fl(\()p Ff(P)13 b(K)q(;)i(P)e(K)2205 3719 y Fe(B)2267 3705 y Ff(;)i(E)5 b Fl(\()p Ff(P)13 b(K)2562 3719 y Fe(B)2624 3705 y Ff(;)i(m)p Fl(\)\).)150 3818 y(3.)41 b(Return)30 b(\()p Ff(m)686 3832 y Fg(0)725 3818 y Ff(;)15 b(m)845 3832 y Fg(1)885 3818 y Ff(;)g(\013)p Fl(\).)42 b Ff(R)q Fl(\()p Ff(c;)15 b(\013)p Fl(\))27 b(:)150 3931 y(1.)41 b(Return)30 b Ff(A)p Fl(\()p Ff(c;)15 b(\013)p Fl(\).)291 4144 y(The)28 b(analysis)e(in)h(this)g(case)i(is)e (trivial.)38 b(If)28 b(this)f(is)g(not)h(the)g(case)h(then)f Ff(p)2785 4158 y Fg(0)2840 4144 y Fh(\000)15 b Ff(p)2972 4158 y Fg(1)3036 4144 y Fh(\025)25 b Fl(1)p Ff(=q)s Fl(\()p Ff(k)s Fl(\))30 b(for)e(some)150 4257 y(p)s(olynomial)g Ff(q)s Fl(.)40 b(Therefore,)31 b(this)e(reduction)g(will)f(w)m(ork:)150 4469 y Ff(R)q Fl(\()p Ff(P)13 b(K)q(;)i Fl(1)489 4436 y Fe(k)533 4469 y Fl(\))26 b(:)150 4582 y(1.)41 b(Let)31 b(\()p Ff(P)13 b(K)607 4596 y Fe(A)665 4582 y Ff(;)i(S)5 b(K)843 4596 y Fe(A)900 4582 y Fl(\))26 b Fh( )f Ff(G)p Fl(\(1)1228 4549 y Fe(k)1272 4582 y Fl(\).)150 4695 y(2.)41 b(Let)31 b Ff(m)25 b Fh( )g Ff(M)733 4710 y Fe(k)806 4695 y Fl(and)30 b(return)f(\()p Ff(m;)15 b(S)5 b(K)1554 4709 y Fe(A)1612 4695 y Ff(;)15 b Fl(0\).)150 4808 y Ff(R)q Fl(\()p Ff(c;)g(\013)p Fl(\))27 b(:)150 4921 y(1.)41 b(Get)32 b(\()p Ff(m)554 4935 y Fg(0)593 4921 y Ff(;)15 b(m)713 4935 y Fg(1)753 4921 y Ff(;)g(\013)p Fl(\))26 b Fh( )f Ff(A)p Fl(\()p Ff(P)13 b(K)1279 4935 y Fe(A)1337 4921 y Ff(;)i(P)e(K)q(;)i(c)p Fl(\).)150 5034 y(2.)41 b(Pic)m(k)30 b Ff(b)c Fh( )f(f)p Fl(0)p Ff(;)15 b Fl(1)p Fh(g)p Fl(.)150 5147 y(3.)41 b(If)30 b Ff(b)25 b Fl(=)g Ff(A)p Fl(\()p Ff(E)5 b Fl(\()p Ff(P)13 b(K)870 5161 y Fe(A)929 5147 y Ff(;)i(m)1049 5162 y Fe(b)1083 5147 y Fl(\))p Ff(;)g(\013)p Fl(\))27 b(=)e Ff(b)30 b Fl(then)g(return)g(1,) h(else)f(return)f(0.)1867 5416 y(17-8)p eop %%Page: 9 9 9 8 bop 291 159 a Fl(Here,)29 b(the)f(analysis)f(is)g(a)h(little)f (di\013eren)m(t.)39 b(If)28 b Ff(c)g Fl(is)f(the)h(encryption)f(of)h Ff(m)f Fl(w)m(e)i(will)c(b)s(e)i(correct)i(with)150 271 y(probabilit)m(y)k(1)24 b Fh(\000)g Ff(p)830 285 y Fg(1)869 271 y Fl(.)56 b(If)35 b Ff(c)h Fl(is)f(the)h(encryption)e(of)i Ff(S)5 b(K)2084 285 y Fe(A)2141 271 y Fl(,)37 b(w)m(e)f(will)d(b)s(e)i (correct)i(with)d(probabilit)m(y)g Ff(p)3686 285 y Fg(0)3725 271 y Fl(.)150 384 y(Therefore,)c(w)m(e)h(will)d(b)s(e)i(correct)h (with)e(probabilit)m(y)748 610 y(1)p Ff(=)p Fl(2\(1)22 b Fh(\000)e Ff(p)1122 624 y Fg(1)1161 610 y Fl(\))h(+)f(1)p Ff(=)p Fl(2\()p Ff(p)1524 624 y Fg(0)1565 610 y Fl(\))25 b(=)g(1)p Ff(=)p Fl(2)d(+)e(1)p Ff(=)p Fl(2\()p Ff(p)2185 624 y Fg(0)2246 610 y Fh(\000)g Ff(p)2383 624 y Fg(1)2422 610 y Fl(\))25 b Fh(\025)g Fl(1)p Ff(=)p Fl(2)d(+)e(1)p Ff(=)p Fl(2)p Ff(q)s Fl(\()p Ff(k)s Fl(\))p Ff(;)291 777 y Fl(whic)m(h)29 b(is)h(nonnegligibly)d(b)s(etter)j(than)g(1/2.)150 1063 y Fi(Problem)45 b(5:)61 b(Insecure)45 b(Signatures)150 1266 y Fl(All)35 b(of)i(the)g(signature)f(sc)m(hemes)h(presen)m(ted)g (in)e(this)h(problem)f(are)i(insecure.)58 b(A)37 b(signature)f(sc)m (heme)150 1379 y(can)29 b(b)s(e)g(brok)m(en)g(in)e(a)j(n)m(um)m(b)s(er) d(of)j(w)m(a)m(ys:)40 b(it)29 b(can)g(b)s(e)g(sho)m(wn)f(to)i(b)s(e)e (existen)m(tially)g(forgeable,)i(it)e(can)i(b)s(e)150 1492 y(target-message)h(forgeable,)e(or)f(its)g(secret)h(k)m(ey)f(can)h (b)s(e)e(completely)h(reco)m(v)m(erable.)41 b(In)28 b(order)f(to)i (break)150 1605 y(a)34 b(signature)e(sc)m(heme,)j(di\013eren)m(t)d (\015a)m(v)m(ors)i(of)f(attac)m(ks)j(can)d(b)s(e)g(launc)m(hed:)45 b(a)33 b(public-k)m(ey)f(only)g(attac)m(k,)150 1718 y(or)e(an)h(in)m (teractiv)m(e)g(attac)m(k)i(where)d(the)g(adv)m(ersary)h(asks)f(for)g (signatures)g(on)g(messages)h(of)g(his)e(c)m(hoice.)291 1831 y(In)j(this)f(problem,)h(y)m(ou)h(are)g(ask)m(ed)g(to)h(break)e (eac)m(h)i(in)d(the)i(strongest)g(sense)g(p)s(ossible,)e(using)g(the) 150 1944 y(w)m(eak)m(est)h(attac)m(k)h(y)m(ou)e(can.)291 2057 y(On)h(the)g(other)h(hand,)f(eac)m(h)h(of)g(these)g(signature)e (sc)m(hemes)i(is)f(secure)g(in)f(a)i(w)m(eak)g(sense:)45 b(namely)-8 b(,)150 2170 y(on)33 b(input)d(a)j(public)d(k)m(ey)k(and)e (a)h(message,)i(it)d(is)g(infeasible)e(to)j(compute)g(the)g(signature.) 47 b(Pro)m(v)m(e)34 b(this)150 2283 y(w)m(eak)d(securit)m(y)f(prop)s (ert)m(y)g(for)g(eac)m(h)i(signature)e(sc)m(heme.)291 2495 y Fm(\(a\))h Fl(\(8)i(p)s(oin)m(ts\))f(Supp)s(ose)e(a)j(family)d (of)j(trap)s(do)s(or)e(p)s(erm)m(utations)g Fh(f)p Ff(p)2730 2509 y Fe(P)10 b(K)2853 2495 y Fh(g)33 b Fl(o)m(v)m(er)g Fh(f)p Fl(0)p Ff(;)15 b Fl(1)p Fh(g)3347 2462 y Fe(k)3424 2495 y Fl(is)32 b(giv)m(en.)150 2608 y(The)d(message)h(space)g(is)f Fh(f)p Fl(0)p Ff(;)15 b Fl(1)p Fh(g)1237 2575 y Fe(k)1282 2608 y Fl(.)40 b(A)30 b(public)c(k)m(ey)k(is)f(the)h(public)c(k)m(ey)k Ff(P)13 b(K)36 b Fl(of)30 b(a)g(mem)m(b)s(er)f(of)g(the)h(TDP;)150 2721 y(the)37 b(secret)h(k)m(ey)g(is)e(the)h(corresp)s(o)s(ding)e Ff(S)5 b(K)i Fl(.)60 b(The)37 b(signature)f(on)h(message)h Ff(m)f Fl(is)f(the)h(v)-5 b(alue)36 b Ff(x)h Fl(suc)m(h)150 2834 y(that)31 b Ff(p)393 2848 y Fe(P)10 b(K)516 2834 y Fl(\()p Ff(x)p Fl(\))26 b(=)e Ff(m)p Fl(.)291 3046 y Fm(Solution:)41 b Fl(If)30 b(this)g(signature)f(sc)m(heme)j(w)m(as)e (not)h(minimally)c(secure,)k(then)f(giv)m(en)g(a)h(message)h Ff(m)p Fl(,)150 3159 y(and)24 b(a)h(public)d(k)m(ey)k Ff(P)13 b(K)7 b Fl(,)25 b(w)m(e)h(can)f(e\016cien)m(tly)f(\014nd)f(an)i Ff(x)f Fl(suc)m(h)h(that)g Ff(p)2525 3173 y Fe(P)10 b(K)2648 3159 y Fl(\()p Ff(x)p Fl(\))25 b(=)g Ff(m)p Fl(.)39 b(But)25 b(this)e(is)h(exactly)150 3272 y(what)30 b(w)m(e)h Fj(c)-5 b(an)c('t)31 b Fl(do,)f(since)g Ff(p)g Fl(is)g(a)g(TDPF.)291 3385 y(On)40 b(the)h(other)f(hand,)j(this)c(signature)h(sc)m(heme)h(is) f(existen)m(tially)f(forgeable)i(under)e(a)i(k)m(ey-only)150 3498 y(attac)m(k.)56 b(The)35 b(attac)m(k?)56 b(Pic)m(k)34 b Ff(x)h Fl(at)h(random)e(and)g(compute)h Ff(m)d Fl(=)h Ff(p)2576 3512 y Fe(P)10 b(K)2698 3498 y Fl(\()p Ff(x)p Fl(\),)37 b(then)e(return)e(\()p Ff(m;)15 b(x)p Fl(\).)55 b Ff(x)150 3611 y Fl(will)28 b(b)s(e)h(a)i(v)-5 b(alid)29 b(signature)h(on)g Ff(m)p Fl(.)40 b(Though)30 b(w)m(e)h(can't)g(con)m (trol)g(what)f Ff(m)g Fl(is,)g(this)f(is)g(still)g(a)h(forgery)-8 b(.)291 3823 y Fm(\(b\))39 b Fl(\(6)h(p)s(oin)m(ts\))f(Assume)g(that)h (RSA)f(is)g(a)h(family)e(of)h(trap)s(do)s(or)g(p)s(erm)m(utations.)67 b(A)40 b(signature)150 3936 y(sc)m(heme)34 b(is)e(as)h(follo)m(ws:)45 b(the)33 b(public)e(k)m(ey)j(is)e(an)h(RSA)f(public)f(k)m(ey)i(\()p Ff(n;)15 b(e)p Fl(\),)35 b(and)e(the)g(secret)h(k)m(ey)g(is)e(the)150 4049 y(corresp)s(onding)c(RSA)h(secret)i(k)m(ey)f Ff(d)p Fl(.)41 b(The)29 b(message)i(space)g(is)d Fa(Z)2416 4016 y Fd(\003)2416 4072 y Fe(n)2459 4049 y Fl(.)40 b(The)30 b(signature)f(on)g(message)i Ff(m)f Fl(is)150 4162 y(the)h(v)-5 b(alue)29 b Ff(m)621 4129 y Fe(d)687 4162 y Fl(mo)s(d)24 b Ff(n)p Fl(.)291 4375 y Fm(Solution:)67 b Fl(This)42 b(signature)h(sc)m(heme)h(is)e(to)j(just)e(apply)f(the)h(in)m(v)m(erse) h(of)f(the)h(RSA)f(trap)s(do)s(or)150 4488 y(p)s(erm)m(utation)25 b(to)i(get)g(the)g(signature)e(of)h(a)h(message.)40 b(In)26 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(cannot)h(succeed)f(in)e(a)i(one-pass)g(GM)g(securit)m(y)f(scenario)h (ev)m(en)g(when)150 1636 y(the)39 b(adv)m(ersary)g(has)f(access)i(to)g (a)f(decryption)e(oracle)j(under)d(the)i(relev)-5 b(an)m(t)39 b(secret)g(k)m(ey)-8 b(,)42 b(though)d(of)150 1749 y(course)31 b(querying)e(the)h(oracle)h(on)f(the)h(actual)g(c)m(hallenge)f(should)e (b)s(e)i(disallo)m(w)m(ed.)291 1957 y Fp(\(a\))f Fo(\(15)i(p)s(oin)m (ts\))e(Giv)m(e)i(a)f(formal)f(de\014nition)f(of)i(a)g(public-k)m(ey)f (cryptosystem)h(whic)m(h)f(is)g(one-pass)150 2070 y(GM)i(secure)f (against)h(adaptiv)m(e)g(c)m(hosen)g(ciphertext)f(attac)m(k,)j(as)d (describ)s(ed)f(ab)s(o)m(v)m(e.)291 2279 y Fp(Solution:)52 b Fo(W)-8 b(e)37 b(w)m(an)m(t)g(to)f(use)g(the)g(idea)f(of)h (GM-securit)m(y)h(but)e(allo)m(w)h(the)g(opp)s(onen)m(t)f(access)i(to) 150 2392 y(an)42 b(oracle)g(whic)m(h)e(will)g(decrypt.)74 b(The)42 b(one)g(detail)f(to)h(remem)m(b)s(er)f(is)g(that)h(w)m(e)h (should)c(not)j(allo)m(w)150 2505 y(the)36 b(opp)s(onen)m(t)g(to)g (succeed)h(b)m(y)e(simply)f(asking)h(for)h(the)g(decryption)f(of)h(the) g(c)m(hallenge.)57 b(Th)m(us,)37 b(the)150 2618 y(follo)m(wing)30 b(de\014nition)e(is)i(the)i(righ)m(t)e(one.)44 b(\(I)31 b(omit)g(the)g(de\014nition)e(of)i(faithfulness,)e(it)i(is)f(the)h (same)h(as)150 2731 y(b)s(efore.\))41 b(\()p Fk(G;)15 b(E)5 b(;)15 b(D)s Fo(\))32 b(is)e(a)g(GM-secure)i(cryptosystem)f (against)f(CCA)g(if:)150 2938 y Fj(8f)p Fk(A)314 2905 y Fi(?)314 2965 y Fh(k)357 2938 y Fj(g)p Fk(O)s(P)13 b(P)g(T)g(F)43 b Fj(9)p Fk(\027)31 b Fj(2)25 b Fk(neg)s(l)32 b Fj(8)p Fk(k)150 3054 y Fo(Pr)o([\()p Fk(P)13 b(K)q(;)i(S)5 b(K)i Fo(\))27 b Fj( )f Fk(G)p Fo(\(1)971 3021 y Fh(k)1015 3054 y Fo(\);)15 b(\()p Fk(m)1205 3068 y Fi(0)1245 3054 y Fk(;)g(m)1365 3068 y Fi(1)1405 3054 y Fk(;)g(\013;)g(Q)1615 3068 y Fi(0)1655 3054 y Fo(\))26 b Fj( )f Fk(A)1900 3021 y Fh(D)r Fi(\()p Fh(S)t(K)q(;)p Fg(\001)p Fi(\))2165 3054 y Fo(\()p Fk(P)13 b(K)7 b Fo(\);)273 3171 y Fk(b)25 b Fj( )g(f)p Fo(0)p Fk(;)15 b Fo(1)p Fj(g)p Fo(;)g Fk(c)28 b Fj( )e Fk(E)5 b Fo(\()p Fk(P)13 b(K)q(;)i(m)1273 3186 y Fh(b)1308 3171 y Fo(\);)g(\()p Fk(b)1457 3138 y Fg(0)1482 3171 y Fk(;)g(Q)1594 3185 y Fi(1)1634 3171 y Fo(\))25 b Fj( )g Fk(A)1878 3138 y Fh(D)r Fi(\()p Fh(S)t(K)q(;)p Fg(\001)p Fi(\))2144 3171 y Fo(\()p Fk(c;)15 b(\013)p Fo(\))27 b(:)273 3284 y Fk(b)312 3251 y Fg(0)360 3284 y Fo(=)e Fk(b)20 b Fj(^)g Fk(c)26 b Fj(62)f Fk(Q)819 3298 y Fi(1)858 3284 y Fo(])h Fk(<)e Fo(1)p Fk(=)p Fo(2)e(+)e Fk(\027)6 b Fo(\()p Fk(k)s Fo(\))291 3491 y(Note)43 b(there)f(is)f(a)h (seman)m(tic)g(problem)f(with)f(calling)g(an)i(oracle)g(TM)g(p)s (olynomial)d(time,)45 b(since)150 3604 y(the)36 b(running)d(time)j(ma)m (y)g(dep)s(end)e(on)i(the)g(oracle.)58 b(Ho)m(w)m(ev)m(er,)39 b(the)d(w)m(a)m(y)h(w)m(e)f(\014x)g(this)e(is)h(to)i(sa)m(y)f(that)150 3716 y(suc)m(h)28 b(an)g Fk(A)h Fo(only)e(violates)i(the)f(ab)s(o)m(v)m (e)i(de\014nition)c(if)h(with)g(nonzero)i(probabilit)m(y)-8 b(,)27 b Fk(A)3085 3683 y Fh(D)r Fi(\()p Fh(S)t(K)q(;)p Fg(\001)p Fi(\))3350 3716 y Fo(\()p Fk(P)13 b(K)7 b Fo(\))29 b(and)150 3829 y Fk(A)218 3796 y Fh(D)r Fi(\()p Fh(S)t(K)q(;)p Fg(\001)p Fi(\))483 3829 y Fo(\()p Fk(c;)15 b(\013)p Fo(\))32 b(runs)d(in)g(p)s(olynomial)f(time.)291 4038 y Fp(\(b\))e Fo(\(15)h(p)s(oin)m(ts\))f(Recall)g(the)g(ElGamal-A)g (cryptosystem.)40 b(Alice)26 b(selects)h(at)g(random)e(\(1\))j(a)e Fk(k)s Fo(-bit)150 4151 y(prime)k Fk(p)d Fo(=)f(2)p Fk(q)f Fo(+)20 b(1)32 b(where)f Fk(q)j Fo(is)d(also)g(prime,)g(\(2\))h(an)f (elemen)m(t)h Fk(g)j Fo(in)c Fk(Z)2624 4118 y Fg(\003)2617 4173 y Fh(p)2694 4151 y Fo(of)h(order)e Fk(q)s Fo(,)i(and)f(\(3\))i(an) e Fk(x)g Fo(in)150 4264 y Ff(Z)215 4231 y Fg(\003)215 4286 y Fh(q)250 4264 y Fo(.)40 b(Then,)28 b(she)g(sets)g(her)g(public)d (k)m(ey)k(to)g(b)s(e)e(\()p Fk(p;)15 b(g)s(;)g(h)p Fo(\),)31 b(where)d Fk(h)e Fo(=)e Fk(g)2553 4231 y Fh(x)2623 4264 y Fo(mo)s(d)h Fk(p)p Fo(,)j(and)g(her)f(secret)i(k)m(ey)g(to)150 4377 y(b)s(e)h Fk(x)p Fo(.)291 4490 y(Alice's)22 b(public)d(k)m(ey)k (restricts)f(the)g(message)h(space)g(to)g(consist)e(of)i(all)e(elemen)m (ts)h(in)f Fk(Z)3197 4457 y Fg(\003)3190 4512 y Fh(p)3258 4490 y Fo(ha)m(ving)g(order)150 4603 y Fk(q)s Fo(.)52 b(T)-8 b(o)35 b(send)e(suc)m(h)h(a)h(message)g Fk(m)f Fo(to)h(Alice,)g(one)f(pic)m(ks)g(an)g Fk(r)g Fj( )e Ff(Z)2546 4617 y Fh(q)2613 4603 y Fo(and)i(computes)g(his)f(ciphertext) 150 4716 y Fk(c)26 b Fo(=)f(\()p Fk(g)392 4683 y Fh(r)456 4716 y Fo(mo)s(d)f Fk(p;)15 b(h)793 4683 y Fh(r)832 4716 y Fk(m)25 b Fo(mo)s(d)f Fk(p)p Fo(\).)291 4828 y(T)-8 b(o)31 b(decrypt)f Fk(c)c Fo(=)f(\()p Fk(a;)15 b(b)p Fo(\),)31 b(Alice)f(computes)h Fk(b=)p Fo(\()p Fk(a)1972 4795 y Fh(x)2016 4828 y Fo(\))26 b(mo)s(d)f Fk(p)p Fo(.)291 4941 y(In)34 b(this)g(problem)g(w)m(e)h(go)h(b)s(ey)m(ond)e(pro)m(ving) h(that)g(ElGamal-A)g(is)f(not)h(secure)g(against)h(adaptiv)m(e)150 5054 y(c)m(hosen)27 b(ciphertext)g(attac)m(k.)42 b(Supp)s(ose)24 b(the)j(adv)m(ersary)g(has)g(access)h(to)f(a)g(c)m(hallenge)g (ciphertext)g Fk(c)p Fo(,)h(and)150 5167 y(ma)m(y)d(ask)h(only)e Fm(one)h Fo(query)f(of)h(the)g(decryption)f(oracle,)j(whic)m(h)c(m)m (uc)m(h)i(b)s(e)g(di\013eren)m(t)f(from)g Fk(c)p Fo(.)39 b(Sho)m(w)25 b(ho)m(w)1867 5416 y(18-1)p eop %%Page: 2 2 2 1 bop 150 159 a Fo(to)27 b(design)f(a)h Fk(c)643 126 y Fg(0)693 159 y Fo(from)f Fk(c)h Fo(suc)m(h)g(that)g(giv)m(en)g(the)g (decryption)e(of)i Fk(c)2339 126 y Fg(0)2389 159 y Fo(it)f(is)g(easy)h (to)h(reco)m(v)m(er)g(the)f(decryption)150 271 y(of)k Fk(c)p Fo(.)291 384 y(Can)38 b(y)m(ou)g(strengthen)g(this)e(result)h (so)h(that)h(if)e Fk(c)h Fo(=)f(\()p Fk(a;)15 b(b)p Fo(\))39 b(and)f Fk(c)2633 351 y Fg(0)2694 384 y Fo(=)g(\()p Fk(a)2886 351 y Fg(0)2909 384 y Fk(;)15 b(b)2988 351 y Fg(0)3012 384 y Fo(\))38 b(then)g Fk(a)g Fj(6)p Fo(=)f Fk(a)3542 351 y Fg(0)3604 384 y Fo(and)150 497 y Fk(b)25 b Fj(6)p Fo(=)g Fk(b)349 464 y Fg(0)373 497 y Fo(?)40 b(If)30 b(so,)h(sho)m(w)f(ho)m(w.)291 684 y Fp(Solution:)58 b Fo(The)39 b(\014rst)f(natural)g(idea)h(here)g(is)f(to)i(m)m(ultiply)c Fk(b)j Fo(b)m(y)g(some)g Fk(m)3002 651 y Fg(0)3026 684 y Fo(,)i(then)e(ask)g(for)g(the)150 797 y(decryption)27 b(of)h(\()p Fk(a;)15 b(bm)943 764 y Fg(0)967 797 y Fo(\),)30 b(and)d(divide)f(the)i(result)g(b)m(y)g Fk(m)2109 764 y Fg(0)2160 797 y Fo(to)h(get)g Fk(m)p Fo(.)40 b(Ho)m(w)m(ev)m(er,)31 b(if)c(w)m(e)i(wish)d(to)j(c)m(hange)150 910 y Fk(a)h Fo(also,)h(w)m(e)g(can)f(ask)h(for)f(the)h(decryption)e(of)i(\()p Fk(ag)s(;)15 b(bh)p Fo(\),)32 b(whic)m(h)d(will)f(b)s(e)i(just)g(the)g (message)i Fk(m)p Fo(.)291 1097 y Fp(\(c\))g Fo(\(10)i(p)s(oin)m(ts\))e (Supp)s(ose)f(the)i(adv)m(ersary)g(has)f(access)i(to)g(a)f(ciphertext)f Fk(c)h Fo(of)g(a)g(message)g(under)150 1209 y(the)c(\(1-bit\))g(Goldw)m (asser-Micali)e(cryptosystem,)i(and)f(ma)m(y)h(ask)g(one)g(query)e(of)i (a)g(decryption)e(oracle,)150 1322 y(other)i(than)g Fk(c)p Fo(.)41 b(Sho)m(w)29 b(ho)m(w)g(to)h(design)e(a)h Fk(c)1629 1289 y Fg(0)1682 1322 y Fo(from)g Fk(c)g Fo(suc)m(h)g(that)h(giv)m(en)f (the)g(decryption)f(of)i Fk(c)3348 1289 y Fg(0)3400 1322 y Fo(it)f(is)f(easy)150 1435 y(to)e(reco)m(v)m(er)i(the)e(decryption)e (of)i Fk(c)p Fo(.)40 b(Also,)26 b(generalize)g(y)m(our)g(attac)m(k)i (on)e(the)f(1-bit)h(sc)m(heme)g(to)h(an)f(attac)m(k)150 1548 y(on)k(the)h Fk(`)p Fo(-bit)e(sc)m(heme.)291 1735 y Fp(Solution:)61 b Fo(The)40 b(answ)m(er)h(here)f(is)g(ev)m(en)h (easier.)71 b(If)40 b(w)m(e)h(let)f Fk(r)j Fo(b)s(e)d(a)h(randomly)e(c) m(hosen)i(v)-5 b(alue,)150 1848 y(an)37 b(ask)h(for)f(the)g(decryption) f(of)i Fk(r)1373 1815 y Fi(2)1412 1848 y Fk(c)p Fo(,)h(w)m(e)f(will)d (get)j(the)g(righ)m(t)e(answ)m(er)i(directly)-8 b(.)60 b(If)37 b(the)g(message)i(is)150 1961 y(man)m(y)31 b(bits)g(long,)g(w)m (e)h(need)f(only)f(do)h(this)f(with)g Fm(one)i Fo(of)f(the)h(bits)e(of) h(the)h(ciphertext,)f(since)g(then)g(the)150 2074 y(ciphertext)e(as)g (a)g(whole)f(will)f(b)s(e)h(di\013eren)m(t.)40 b(Ho)m(w)m(ev)m(er,)31 b(w)m(e)f(could)e(also)h(m)m(ultiply)d(eac)m(h)k(encrypted)f(bit)150 2187 y(b)m(y)f(a)g(random)g(quadratic)f(residue)f(mo)s(dulo)h Fk(n)g Fo(and)h(get)h(the)f(same)h(result.)38 b(There)28 b(are)g(man)m(y)g(attac)m(ks.)150 2467 y Fl(Problem)45 b(2:)61 b(Collision-resistan)l(t)48 b(hashing)150 2670 y Fo(\(30)32 b(p)s(oin)m(ts\))d(De\014ne)i(a)f(family)f(of)i (collision-resistan)m(t)d(hash)i(functions)f Fj(H)i Fo(as)f(follo)m (ws:)261 2844 y(1.)46 b(On)30 b(input)e(1)816 2811 y Fh(k)859 2844 y Fo(,)j Fj(H)g Fo(outputs)f(the)g(description)f(of)h(a)h (k)m(ey)g Fk(P)13 b(K)7 b Fo(.)261 3018 y(2.)46 b(F)-8 b(or)31 b(an)m(y)g Fk(P)13 b(K)7 b Fo(,)30 b(it)g(is)g(e\016cien)m(t)h (to)g(compute)f Fk(H)1996 3032 y Fh(P)10 b(K)2119 3018 y Fo(.)261 3193 y(3.)46 b Fk(H)453 3207 y Fh(P)10 b(K)606 3193 y Fo(is)29 b(length)h(decreasing:)41 b Fj(8)p Fk(k)s(;)15 b Fj(8)p Fk(P)e(K)31 b Fj(2)25 b(H)q Fo(\(1)2066 3160 y Fh(k)2109 3193 y Fo(\))p Fj(8)p Fk(x)g Fj(2)g(f)p Fo(0)p Fk(;)15 b Fo(1)p Fj(g)2578 3160 y Fg(\003)2620 3193 y Fk(;)g Fj(j)p Fk(H)2761 3207 y Fh(P)10 b(K)2884 3193 y Fo(\()p Fk(x)p Fo(\))p Fj(j)26 b(\024)k Fo(min\()p Fk(k)s(;)15 b Fj(j)p Fk(x)p Fj(j)p Fo(\).)261 3368 y(4.)46 b(It)31 b(is)e(hard)h(to)h(\014nd)d(collisions:)377 3573 y Fj(8f)p Fk(A)541 3588 y Fh(k)584 3573 y Fj(g)p Fk(P)13 b(P)g(T)g(F)44 b Fj(9)p Fk(\027)30 b Fj(2)25 b Fk(neg)s(l)33 b Fj(8)p Fk(k)377 3686 y Fo(Pr[)p Fk(P)13 b(K)32 b Fj( )25 b(H)q Fo(\(1)954 3653 y Fh(k)997 3686 y Fo(\);)15 b(\()p Fk(x;)g(y)s Fo(\))27 b Fj( )e Fk(A)1493 3701 y Fh(k)1536 3686 y Fo(\()p Fk(P)13 b(K)7 b Fo(\))26 b(:)f Fk(H)1913 3700 y Fh(P)10 b(K)2036 3686 y Fo(\()p Fk(x)p Fo(\))26 b(=)f Fk(H)2356 3700 y Fh(P)10 b(K)2478 3686 y Fo(\()p Fk(y)s Fo(\))21 b Fj(^)f Fk(x)25 b Fj(6)p Fo(=)g Fk(y)s Fo(])g Fk(<)g(\027)6 b Fo(\()p Fk(k)s Fo(\))291 3964 y Fp(\(a\))28 b Fo(Pro)m(v)m(e)j(that)e(the)g(existence)h(of)f(a)h (family)d(of)i(collision-resistan)m(t)f(hash)g(functions)f(implies)g (the)150 4077 y(existence)k(of)f(a)h(one-w)m(a)m(y)h(function.)291 4264 y Fp(Solution:)40 b Fo(W)-8 b(e)29 b(will)c(pro)m(v)m(e)j(that)g Fk(H)1572 4278 y Fh(P)10 b(K)1722 4264 y Fo(is)27 b(a)h(O)m(WFF,)h(and) e(then)h(b)m(y)f(problem)f(set)j(6)f(problem)e(3a,)150 4377 y(w)m(e)31 b(kno)m(w)f(that)h(O)m(WFF)h Fj(\))e Fo(O)m(WF,)i(so)e(that)h(will)d(complete)j(the)f(pro)s(of.)291 4490 y(The)k(k)m(ey)g(lies)f(in)g(the)h(fact)h(that)g Fk(H)41 b Fo(is)33 b(length-reducing)f(once)j(the)f(messages)h(are)f (long)g(enough.)150 4603 y(Actually)-8 b(,)40 b(this)d(mak)m(es)j(us)d (de\014ne)h(the)g(O)m(WFF)i(a)e(little)f(di\013eren)m(tly;)42 b(remem)m(b)s(er)37 b(that)i(while)e Fk(H)3628 4617 y Fh(P)10 b(K)150 4716 y Fo(is)33 b(de\014ned)f(to)i(w)m(ork)g(on)f(all)f (inputs,)h(in)f(a)i(O)m(WFF,)h(w)m(e)f(only)e(exp)s(ect)i(to)h(w)m(ork) e(on)h(inputs)d(of)j(a)g(\014xed)150 4828 y(length.)42 b(T)-8 b(o)32 b(mak)m(e)g(the)f(math)h(easy)-8 b(,)32 b(let)f Fk(F)1642 4842 y Fh(P)10 b(K)1797 4828 y Fo(b)s(e)30 b(de\014ned)g(as)h Fk(H)2423 4842 y Fh(P)10 b(K)2577 4828 y Fo(where)30 b(the)i(length)e(of)i(the)f(input)150 4941 y(is)e(2)p Fk(k)s Fo(.)42 b(This)28 b(de\014nes)i(an)g(e\016cien)m (tly)g(computable)g(function)f(family)-8 b(.)291 5054 y(Supp)s(ose)41 b(that)j(this)e(is)g(not)h(a)h(O)m(WFF.)g(Then)e(there) h(exists)g(an)g(adv)m(ersary)g Fk(A)h Fo(that)f(can)g(\014nd)150 5167 y(preimages)30 b(with)f(non-negligible)f(probabilit)m(y)-8 b(.)39 b(Then,)29 b(w)m(e)i(can)g(\014nd)e(hash)g(collisions)f(as)j (follo)m(ws:)1867 5416 y(18-2)p eop %%Page: 3 3 3 2 bop 150 159 a Fk(A)218 126 y Fg(0)241 159 y Fo(\()p Fk(P)13 b(K)7 b Fo(\))26 b(:)150 271 y(1.)41 b(Pic)m(k)30 b Fk(x)c Fj( )f(f)p Fo(0)p Fk(;)15 b Fo(1)p Fj(g)878 239 y Fi(2)p Fh(k)958 271 y Fo(.)41 b(Compute)30 b Fk(a)25 b Fo(=)g Fk(F)1645 285 y Fh(P)10 b(K)1768 271 y Fo(\()p Fk(x)p Fo(\))26 b(=)f Fk(H)2088 285 y Fh(P)10 b(K)2210 271 y Fo(\()p Fk(x)p Fo(\).)150 384 y(2.)41 b(Get)32 b Fk(y)c Fj( )d Fk(A)p Fo(\()p Fk(P)13 b(K)q(;)i(a)p Fo(\).)150 497 y(3.)41 b(Return)30 b(\()p Fk(x;)15 b(y)s Fo(\).)291 699 y(The)36 b(probabilit)m(y)e(analysis)g(is)h(a)i(little)e (tric)m(ky)h(here.)58 b(W)-8 b(e'd)37 b(lik)m(e)f(to)h(b)s(e)e(able)h (to)h(sa)m(y)g(that)g(since)150 812 y(there)28 b(are)g(t)m(wice)g(as)g (man)m(y)g(inputs)d(to)k Fk(F)1557 826 y Fh(P)10 b(K)1707 812 y Fo(as)28 b(there)g(are)g(outputs)f(\(at)i(least\))2887 779 y Fi(1)2927 812 y Fo(,)f(that)h(if)d Fk(y)31 b Fo(is)26 b(indeed)h(a)150 925 y(preimage)e(of)g Fk(H)708 939 y Fh(P)10 b(K)831 925 y Fo(\()p Fk(x)p Fo(\))25 b(then)g(with)f (probabilit)m(y)e(at)k(least)g(1/2,)h Fk(y)h Fj(6)p Fo(=)d Fk(x)g Fo(and)g(so)g(w)m(e)h(pro)s(duce)e(a)h(collision.)150 1038 y(Ho)m(w)m(ev)m(er,)39 b(it)c(ma)m(y)i(w)m(ell)d(b)s(e)h(that)h (for)g(man)m(y)g Fk(x)f Fo(there)h(is)f(no)g(other)h(v)-5 b(alue)35 b(that)h(pro)s(duces)f(the)h(same)150 1150 y(output.)41 b(Let)31 b Fk(G)g Fo(b)s(e)f(the)h(set)g(of)g(all)f Fk(x)g Fo(suc)m(h)g(that)i(there)e(is)g(at)h(least)g(one)g(other)g(v)-5 b(alue)30 b Fk(x)3228 1117 y Fg(0)3282 1150 y Fo(whic)m(h)f(maps)150 1263 y(to)i(the)g(same)f(output)g(under)f Fk(F)1259 1277 y Fh(P)10 b(K)1383 1263 y Fo(,)30 b(and)g(let)p 1746 1190 72 4 v 30 w Fk(G)h Fo(b)s(e)f(the)g(rest)h(of)f(the)h(v)-5 b(alues)30 b(in)f Fj(f)p Fo(0)p Fk(;)15 b Fo(1)p Fj(g)3163 1230 y Fi(2)p Fh(k)3243 1263 y Fo(.)291 1376 y(W)-8 b(e)35 b(kno)m(w)f(that)h Fk(H)41 b Fo(is)33 b(one-to-one)j(on)p 1683 1303 V 34 w Fk(G)p Fo(.)52 b(Th)m(us,)34 b(if)f(w)m(e)i(coun)m(t)f (elemen)m(ts)h(of)f Fj(f)p Fo(0)p Fk(;)15 b Fo(1)p Fj(g)3272 1343 y Fh(k)3351 1376 y Fo(\(outputs\),)150 1489 y(there)42 b(are)g(three)g(sets)h(to)f(consider.)74 b(Some)42 b(v)-5 b(alues)41 b(\(denoted)h(b)m(y)g Fk(N)10 b Fo(\))42 b(ma)m(y)h(ha)m(v)m (e)g Fm(no)f Fo(preimages.)150 1602 y(Others)29 b(ma)m(y)i(ha)m(v)m(e)g (only)e(one)h(\(denoted)h(b)m(y)f Fk(H)7 b Fo(\()p 1842 1529 V Fk(G)p Fo(\)\),)31 b(and)f(the)g(rest)g(\(denoted)g(b)m(y)g Fk(H)7 b Fo(\()p Fk(G)p Fo(\)\))32 b(ha)m(v)m(e)f(more)150 1715 y(than)f(one.)41 b(No)m(w,)1334 2021 y(2)1379 1984 y Fh(k)1505 2021 y Fo(=)83 b Fj(j)p Fk(N)10 b Fj(j)21 b Fo(+)f Fj(j)p Fk(H)7 b Fo(\()p 2047 1948 V Fk(G)q Fo(\))p Fj(j)20 b Fo(+)g Fj(j)p Fk(H)7 b Fo(\()p Fk(G)p Fo(\))p Fj(j)1334 2159 y Fo(2)1379 2121 y Fh(k)1505 2159 y Fj(\025)83 b(j)p Fk(H)7 b Fo(\()p 1802 2086 V Fk(G)q Fo(\))p Fj(j)1505 2297 y Fo(=)83 b Fj(j)p 1684 2224 V Fk(G)q Fj(j)p Fk(;)291 2490 y Fo(so)291 2603 y Fj(j)p Fk(G)p Fj(j)26 b(\025)f Fo(2)579 2570 y Fi(2)p Fh(k)678 2603 y Fj(\000)20 b Fo(2)814 2570 y Fh(k)857 2603 y Fo(.)291 2716 y(Therefore,)37 b(the)f(probabilit)m(y)d(that)j Fk(x)e Fj(2)g Fk(G)i Fo(is)f(at)h(least)g(1)24 b Fj(\000)f Fo(2)2496 2683 y Fg(\000)p Fh(k)2595 2716 y Fo(.)56 b(Giv)m(en)35 b(this,)i(the)e (probabilit)m(y)150 2829 y(that)f Fk(y)g Fj(6)p Fo(=)c Fk(x)k Fo(is)e(at)j(least)f(1/2.)52 b(But)33 b(more)h(imp)s(ortan)m (tly)-8 b(,)34 b(since)f Fk(A)g Fo(w)m(orks)h(a)g(non-negligible)d (fraction)150 2942 y(of)36 b(the)h(time,)g(it)f(m)m(ust)g(w)m(ork)g(a)h (non-negligible)c(fraction)j(of)g(the)h(time)f(when)f Fk(x)g Fj(2)f Fk(G)p Fo(,)k(since)p 3500 2868 V 36 w Fk(G)e Fo(is)g(a)150 3054 y(negligible)26 b(fraction.)39 b(This)26 b(allo)m(ws)i(us)f(to)h(conclude)g(that)g Fk(A)2252 3021 y Fg(0)2304 3054 y Fo(will)d(\014nd)h(a)i(hash)g(collision)d(at)k (least)f(half)150 3167 y(as)j(often)f(as)h Fk(A)g Fo(succeeds)f(in)f (in)m(v)m(erting)h Fk(H)7 b Fo(.)291 3373 y Fp(\(b\))35 b Fo(Supp)s(ose)f(a)j(family)d(of)i(collision-resistan)m(t)e(hash)h (functions)g Fj(H)h Fo(is)f(giv)m(en,)i(and)f(let)g Fj(H)3517 3340 y Fg(0)3575 3373 y Fo(b)s(e)g(a)150 3485 y(family)h(of)h(hash)f (functions)g(that)i(tak)m(es)g(t)m(w)m(o)h(securit)m(y)e(parameters,)i Fk(l)g Fo(and)e Fk(n)p Fo(,)i(so)e(that)h(it)f(has)g(t)m(w)m(o)150 3598 y(k)m(eys)33 b Fk(P)13 b(K)500 3613 y Fh(l)555 3598 y Fj( )29 b(H)q Fo(\(1)833 3565 y Fh(l)859 3598 y Fo(\))k(and)f Fk(P)13 b(K)1254 3612 y Fh(n)1330 3598 y Fj( )28 b(H)q Fo(\(1)1607 3565 y Fh(n)1655 3598 y Fo(\).)47 b(De\014ne)33 b Fk(H)2129 3565 y Fg(0)2122 3625 y Fh(P)10 b(K)2237 3637 y Fe(l)2260 3625 y Fh(;P)g(K)2395 3633 y Fe(n)2440 3598 y Fo(\()p Fk(x)p Fo(\))29 b(=)g Fk(H)2767 3612 y Fh(P)10 b(K)2882 3624 y Fe(l)2909 3598 y Fo(\()p Fk(H)3020 3612 y Fh(P)g(K)3135 3620 y Fe(n)3180 3598 y Fo(\()p Fk(x)p Fo(\)\).)48 b(Pro)m(v)m(e)34 b(or)150 3711 y(dispro)m(v)m(e:)49 b(there)36 b(is)e(an)h(in\014nite)d(set)k(of)f(securit)m(y)g(parameter) h(pairs)d(for)i(whic)m(h)f Fj(H)3081 3678 y Fg(0)3139 3711 y Fo(is)g(not)h(collision-)150 3824 y(resistan)m(t.)59 b(Pro)m(v)m(e)37 b(or)g(dispro)m(v)m(e:)52 b(for)36 b(all)f(pairs)g(\() p Fk(k)s(;)15 b(k)s Fo(\),)40 b Fj(H)2214 3791 y Fg(0)2273 3824 y Fo(is)35 b(a)i(family)e(of)h(collision-resistan)m(t)f(hash)150 3937 y(functions.)291 4142 y Fp(Solution:)59 b Fo(It)39 b(is)f(certainly)g(not)h(hard)g(to)g(\014nd)f(collisions)f(for)h(the)i (family)d(de\014ned)h(b)m(y)h Fk(l)j Fo(=)d(1,)150 4255 y Fk(n)25 b Fo(=)g Fk(k)s Fo(.)40 b(Since)26 b(the)i(outputs)f(will)e (b)s(e)i(of)g(length)g(1)h(or)g(0,)h(it)e(will)d(tak)m(e)30 b(only)c(constan)m(t)j(time)e(to)i(\014nd)d(t)m(w)m(o)150 4368 y(v)-5 b(alues)30 b(that)h(pro)s(duce)e(the)h(same)h(output.)291 4481 y(On)i(the)h(other)g(hand,)g Fk(l)f Fo(=)e Fk(k)s(;)15 b(n)31 b Fo(=)g Fk(k)37 b Fo(de\014nes)32 b(a)j(secure)f(family)-8 b(.)50 b(Supp)s(ose)32 b(w)m(e)i(can)g(reliably)d(\014nd)150 4594 y(collisions)24 b(for)j(this)f(family)-8 b(.)38 b(Then,)27 b(there)g(is)f(an)g Fk(A)h Fo(that)h(giv)m(en)f Fk(P)13 b(K)33 b Fo(and)26 b Fk(P)13 b(K)2876 4561 y Fg(0)2926 4594 y Fo(\(eac)m(h)28 b(generated)g(with)150 4707 y(securit)m(y)34 b(parameter)g(k\))g(can)h(\014nd)d(an)i Fk(x)g Fo(and)f(a)h Fk(y)j Fo(suc)m(h)d(that)g Fk(H)2446 4721 y Fh(P)10 b(K)2569 4707 y Fo(\()p Fk(H)2680 4722 y Fh(P)g(K)2800 4703 y Fd(0)2825 4707 y Fo(\()p Fk(x)p Fo(\)\))32 b(=)f Fk(H)3192 4721 y Fh(P)10 b(K)3315 4707 y Fo(\()p Fk(H)3426 4722 y Fh(P)g(K)3546 4703 y Fd(0)3571 4707 y Fo(\()p Fk(y)s Fo(\)\).)150 4820 y(Basic)33 b(idea:)46 b(either)33 b Fk(x)g Fo(and)g Fk(y)i Fo(will)c(b)s(e)i(a)g(collision)e (for)i Fk(H)2183 4835 y Fh(P)10 b(K)2303 4816 y Fd(0)2361 4820 y Fo(or)33 b Fk(H)2551 4835 y Fh(P)10 b(K)2671 4816 y Fd(0)2696 4820 y Fo(\()p Fk(x)p Fo(\))33 b(and)g Fk(H)3107 4835 y Fh(P)10 b(K)3227 4816 y Fd(0)3252 4820 y Fo(\()p Fk(y)s Fo(\))34 b(will)c(b)s(e)j(a)150 4933 y(collision)28 b(for)i Fk(H)714 4947 y Fh(P)10 b(K)837 4933 y Fo(.)40 b(W)-8 b(e)32 b(use)e(a)h(randomized)e(h)m(ybrid)f(reduction:)p 150 4990 1440 4 v 254 5044 a Fc(1)288 5076 y Fb(W)-6 b(e)25 b(assume)h(here)f(the)g Fa(H)31 b Fb(returns)25 b(outputs)g(of)i(constan)n(t)e(length.)35 b(F)-6 b(ortunately)g(,)25 b(w)n(e)h(can)f(tak)n(e)h(an)n(y)f(suc)n(h)g(\\non)n(uni-)150 5167 y(form")h(hash-function)f(and)h(mak)n(e)e(a)i(uniform)f(one)h(out) g(of)g(it,)g(and)g(pro)r(ceed)g(as)g(normal.)1867 5416 y Fo(18-3)p eop %%Page: 4 4 4 3 bop 150 159 a Fk(A)218 126 y Fg(0)241 159 y Fo(\()p Fk(P)13 b(K)7 b Fo(\):)150 271 y(1.)41 b(Pic)m(k)30 b Fk(b)c Fj( )f(f)p Fo(0)p Fk(;)15 b Fo(1)p Fj(g)p Fo(.)43 b(Pic)m(k)30 b Fk(P)13 b(K)1291 239 y Fg(0)1339 271 y Fj( )25 b(H)q Fo(\(1)1613 239 y Fh(k)1657 271 y Fo(\).)150 384 y(2.)41 b(If)30 b Fk(b)25 b Fo(=)g(0)31 b(get)h(\()p Fk(x;)15 b(y)s Fo(\))26 b Fj( )f Fk(A)p Fo(\()p Fk(P)13 b(K)q(;)i(P)e(K)1539 351 y Fg(0)1563 384 y Fo(\))31 b(and)f(return)f Fk(H)2160 400 y Fh(P)10 b(K)2280 381 y Fd(0)2305 384 y Fo(\()p Fk(x)p Fo(\))p Fk(;)15 b(H)2543 400 y Fh(P)10 b(K)2663 381 y Fd(0)2689 384 y Fo(\()p Fk(y)s Fo(\))p Fk(:)150 497 y Fo(3.)41 b(If)30 b Fk(b)25 b Fo(=)g(1,)31 b(return)f Fk(A)p Fo(\()p Fk(P)13 b(K)1150 464 y Fg(0)1173 497 y Fk(;)i(P)e(K)7 b Fo(\).)291 682 y(Since)42 b(the)h(t)m(w)m(o)i (cases)e(are)h(m)m(utually)d(exclusiv)m(e,)46 b(the)d(analysis)f(is)g (easy)-8 b(.)79 b(On)42 b(an)m(y)i(particular)150 795 y(execution)32 b(where)f Fk(A)g Fo(is)g(successful,)f(either)h(w)m(e)h (get)h(a)e(collision)e(for)j Fk(P)13 b(K)38 b Fo(or)31 b(for)g Fk(P)13 b(K)3160 762 y Fg(0)3183 795 y Fo(.)44 b(Th)m(us,)31 b(one)h(of)150 908 y(the)e(settings)h(for)f Fk(b)g Fo(will)e(mak)m(e)j Fk(A)1326 875 y Fg(0)1379 908 y Fo(return)f(a)g(correct)i(answ)m(er)e(and)g(one)g(will)e(b)s(e)h (incorrect.)41 b(Th)m(us,)29 b(w)m(e)150 1021 y(get)j(exactly)f(half)e (of)h(the)h(success)g(rate)g(of)f Fk(A)p Fo(,)h(whic)m(h)e(is)h(still)e (nonnegligible.)291 1215 y Fp(\(c\))39 b Fo(Supp)s(ose)f Fk(G)i Fo(is)e(a)i(PR)m(G.)g(No)m(w)g(supp)s(ose)e Fj(H)2027 1182 y Fh(G)2125 1215 y Fo(is)g(a)i(family)e(of)h(hash)g(functions)f (de\014ned)g(b)m(y)150 1328 y Fj(H)228 1295 y Fh(G)287 1328 y Fo(\(1)367 1295 y Fh(k)410 1328 y Fo(\))26 b(=)f Fj(H)q Fo(\(1)725 1295 y Fh(k)768 1328 y Fo(\))g(and)f Fk(H)1082 1295 y Fh(G)1075 1355 y(P)10 b(K)1197 1328 y Fo(\()p Fk(x)p Fo(\))26 b(=)f Fk(G)p Fo(\()p Fk(H)1623 1342 y Fh(P)10 b(K)1747 1328 y Fo(\()p Fk(x)p Fo(\)\).)39 b(Pro)m(v)m(e)26 b(or)f(dispro)m(v)m(e:)37 b Fj(H)2792 1295 y Fh(G)2875 1328 y Fo(is)24 b(a)h(family)e(of)i(collision-)150 1441 y(resistan)m(t)34 b(hash)g(functions.)50 b(Pro)m(v)m(e)36 b(or)e(dispro)m(v)m(e:)48 b Fj(H)2030 1408 y Fh(G)2123 1441 y Fo(is)33 b(a)i(family)d(of)j(functions)d(indistinguishable)150 1554 y(from)e(random.)291 1748 y Fp(Solution:)65 b Fo(It)43 b(is)e(easy)i(to)h(dispro)m(v)m(e)d(that)j Fj(H)1968 1715 y Fh(G)2069 1748 y Fo(is)d(not)i(a)g(family)e(of)h (collision-resistan)m(t)f(hash)150 1860 y(functions.)e(This)29 b(is)g(b)s(ecause)i(it)f(is)f(no)h(longer)g(length-reducing.)39 b(Sho)m(wing)29 b(this)h(is)f(trivial.)291 1973 y(Ho)m(w)m(ev)m(er,)48 b(it)43 b(is)f(not)h(so)g(easy)g(to)h(pro)m(v)m(e)g(that)f Fj(H)2085 1940 y Fh(G)2187 1973 y Fo(is)f(not)h(necessarily)f (distinguishable)c(from)150 2086 y(random.)h(Supp)s(ose)26 b Fk(G)943 2053 y Fg(0)994 2086 y Fo(is)h(a)h(PR)m(G,)g(and)f(let)h Fk(G)p Fo(\()p Fk(b)15 b Fj(\016)g Fk(x)p Fo(\))26 b(=)f Fk(b)15 b Fj(\016)g Fk(G)2323 2053 y Fg(0)2347 2086 y Fo(\()p Fk(x)p Fo(\).)41 b(If)27 b Fk(G)h Fo(is)f(not)g(a)h(PR)m(G)h (then)e(neither)150 2199 y(is)33 b Fk(G)316 2166 y Fg(0)339 2199 y Fo(:)48 b(giv)m(en)33 b(a)h(c)m(hallenge,)h(prep)s(end)d(a)i (random)e(bit)h(and)g(determine)g(whether)g(it)g(is)g(the)h(output)f (of)150 2312 y Fk(G)e Fo(or)f(a)g(random)f(string.)40 b(No)m(w)31 b(supp)s(ose)d Fj(H)1691 2279 y Fg(0)1744 2312 y Fo(is)i(a)g(collision)e(resistan)m(t)i(hash)f(function)g(family) f(and)i(let)150 2425 y Fj(H)h Fo(b)s(e)f(de\014ned)f(as)h(follo)m(ws:) 150 2610 y Fj(H)q Fo(\(1)308 2577 y Fh(k)351 2610 y Fo(\):)150 2723 y(Return)g Fj(H)538 2690 y Fg(0)561 2723 y Fo(\(1)641 2690 y Fh(k)r Fg(\000)p Fi(1)774 2723 y Fo(\))p Fk(:)h Fo(Unde\014ned)e(for)h Fk(k)e Fo(=)d(0.)150 2836 y Fk(H)226 2850 y Fh(P)10 b(K)348 2836 y Fo(\()p Fk(x)p Fo(\))26 b(:)150 2949 y(Return)k(0)20 b Fj(\016)h Fk(H)674 2916 y Fg(0)667 2975 y Fh(P)10 b(K)789 2949 y Fo(\()p Fk(x)p Fo(\).)291 3133 y(The)24 b(use)f(of)i 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y Fh(;)15 b(r)658 126 y Fg(0)655 185 y Fe(R)713 159 y Fh(;)g(b)792 126 y Fg(0)816 159 y Fl(\))25 b Ff( )h Fh(A)p Fl(\()p Fh(t)p Fl(\).)150 271 y(4.)41 b(Chec)m(k)31 b(that)g Fh(F)13 b Fl(\()p Fh(r)881 238 y Fg(0)878 298 y Fe(C)937 271 y Fh(;)i(r)1021 238 y Fg(0)1018 298 y Fe(R)1077 271 y Fh(;)g(b)1156 238 y Fg(0)1179 271 y Fl(\))26 b(=)f Fh(t)p Fl(.)40 b(If)30 b(not,)h(or)f(if)g Fh(b)25 b Fl(=)g Fh(b)2106 238 y Fg(0)2129 271 y Fl(,)31 b(ab)s(ort.)150 384 y(5.)41 b(Let)31 b Fh(u)476 400 y Fe(b)506 381 y Fc(0)563 384 y Fl(b)s(e)f(the)g(output)g (of)h Fh(C)37 b Fl(under)28 b(\()p Fh(b)1678 351 y Fg(0)1702 384 y Fh(;)15 b(r)1786 351 y Fg(0)1783 411 y Fe(C)1843 384 y Fl(\).)150 497 y(6.)41 b(T)-8 b(o)31 b(op)s(en)f(a)g(0,)h(p)s (erform)e Fh(C)37 b Fl(with)29 b(input)g Fh(u)1739 511 y Fd(0)1778 497 y Fl(.)41 b(T)-8 b(o)31 b(op)s(en)e(a)i(1,)g(p)s (erform)e Fh(C)37 b Fl(with)29 b(input)f Fh(u)3321 511 y Fd(1)3361 497 y Fl(.)291 704 y(Whenev)m(er)36 b Fh(A)f Fl(returns)f(a)i(v)-5 b(alid)33 b(preimage)i(suc)m(h)g(that)h Fh(b)2295 671 y Fg(0)2352 704 y Ff(6)p Fl(=)d Fh(b)p Fl(,)j(w)m(e)g(can)g(op)s(en)e(the)i(commitmen)m(t)150 817 y(either)c(w)m(a)m(y)-8 b(.)50 b(W)-8 b(e)34 b(succeed)g(in)d(op)s (ening)h(b)s(ecause)h(the)g(probabilit)m(y)d(that)k(the)f(sc)m(heme)h (is)d(correct)k(is)d(1)150 930 y(-)h(ie,)f(the)h(v)-5 b(alid)31 b(receiv)m(er)h(m)m(ust)h(accept)h(for)e(b)s(oth)f Fh(u)1974 944 y Fd(0)2046 930 y Fl(and)h Fh(u)2277 944 y Fd(1)2316 930 y Fl(.)47 b(Th)m(us,)32 b(if)g Fh(A)g Fl(returns)f(suc)m(h)h(preimages)150 1043 y(with)d(nonnegligible)f (probabilit)m(y)-8 b(,)28 b Fh(C)q(;)15 b(R)32 b Fl(is)d(not)i (binding.)291 1156 y(In)38 b(the)h(other)g(case,)j Fh(A)d Fl(returns)e(a)i(preimage)g(suc)m(h)f(that)h Fh(b)2420 1123 y Fg(0)2483 1156 y Ff(6)p Fl(=)f Fh(b)h Fl(with)e(negligible)g (probabilit)m(y)-8 b(.)150 1268 y(Therefore,)32 b(if)e Fh(p)719 1282 y Fd(0)790 1268 y Fl(is)h(the)h(probabilit)m(y)d(that)j Fh(b)1744 1235 y Fg(0)1795 1268 y Fl(=)27 b Fh(b)k Fl(and)g Fh(p)2187 1282 y Fd(1)2258 1268 y Fl(is)g(the)g(probabilit)m(y)f(that)i Fh(b)3212 1235 y Fg(0)3262 1268 y Ff(6)p Fl(=)27 b Fh(b)p Fl(,)33 b(then)e Fh(p)3711 1282 y Fd(0)150 1381 y Fl(is)e (non-negligible)f(and)i Fh(p)1045 1395 y Fd(1)1114 1381 y Fl(is)g(negligible.)38 b(In)30 b(this)f(case,)j(w)m(e)f(break)f(the)h (hiding)c(prop)s(ert)m(y)j(as)h(follo)m(ws.)150 1588 y(1.)41 b(As)30 b Fh(R)465 1555 y Fg(0)488 1588 y Fl(,)h(generate)h(a)f (random)e Fh(r)1358 1602 y Fe(R)1446 1588 y Fl(and)h(execute)i(the)e (commitmen)m(t)h(proto)s(col)f(with)g Fh(C)7 b Fl(,)30 b(observing)f(the)h(transcript)g Fh(t)p Fl(.)150 1701 y(2.)41 b(Get)32 b(\()p Fh(r)518 1668 y Fg(0)515 1728 y Fe(C)574 1701 y Fh(;)15 b(r)658 1668 y Fg(0)655 1728 y Fe(R)713 1701 y Fh(;)g(b)792 1668 y Fg(0)816 1701 y Fl(\))25 b Ff( )h Fh(A)p Fl(\()p Fh(t)p Fl(\).)150 1814 y(3.)41 b(Chec)m(k)31 b(that)g Fh(F)13 b Fl(\()p Fh(r)881 1781 y Fg(0)878 1841 y Fe(C)937 1814 y Fh(;)i(r)1021 1781 y Fg(0)1018 1841 y Fe(R)1077 1814 y Fh(;)g(b)1156 1781 y Fg(0)1179 1814 y Fl(\))26 b(=)f Fh(t)p Fl(.)40 b(If)30 b(not,)h(return)e(a)i(random)f(bit.)39 b(4.)j(Return)29 b Fh(b)3036 1781 y Fg(0)3059 1814 y Fl(.)291 2021 y(The)36 b(probabilit)m(y)d(this)i(will)f(succeed)i(is)f(exactly)i(1)p Fh(=)p Fl(2\(1)26 b Ff(\000)e Fh(p)2447 2035 y Fd(0)2510 2021 y Ff(\000)g Fh(p)2651 2035 y Fd(1)2690 2021 y Fl(\))g(+)g Fh(p)2890 2035 y Fd(0)2953 2021 y Ff(\000)g Fh(p)3094 2035 y Fd(1)3133 2021 y Fl(,)38 b(since)d(when)m(v)m(er)150 2133 y Fh(b)27 b Fl(=)f Fh(b)352 2100 y Fg(0)406 2133 y Fl(w)m(e)32 b(are)f(correct)i(and)d(whenev)m(er)h Fh(b)c Ff(6)p Fl(=)f Fh(b)1777 2100 y Fg(0)1832 2133 y Fl(w)m(e)31 b(are)h(incorrect,)f(if)g(the)g(preimage)g(is)f(v)-5 b(alid.)41 b(If)31 b(it)g(is)150 2246 y(in)m(v)-5 b(alid,)28 b(w)m(e)j(guess,)g(so)f(the)h(probabilit)m(y)d(w)m(e)j(are)f (successful)f(is)h(exactly)h(1/2.)291 2359 y(This)39 b(simpli\014es)f(to)j(1)p Fh(=)p Fl(2)29 b(+)e Fh(p)1344 2373 y Fd(0)1383 2359 y Fh(=)p Fl(2)h Ff(\000)f Fl(3)p Fh(p)1690 2373 y Fd(1)1730 2359 y Fh(=)p Fl(2.)73 b(Ho)m(w)m(ev)m(er,) 46 b(since)40 b Fh(p)2602 2373 y Fd(0)2682 2359 y Fl(is)g (non-negligible)f(and)h Fh(p)3609 2373 y Fd(1)3689 2359 y Fl(is)150 2472 y(negligible,)28 b(this)i(is)f(non-negligibly)e(b)s (etter)k(than)f(1/2.)42 b(Th)m(us,)30 b(the)g(hiding)e(prop)s(ert)m(y)i (is)f(brok)m(en.)291 2681 y Fm(Common)44 b(errors:)59 b Fl(I)39 b(w)m(as)h(sho)s(c)m(k)m(ed)g(grading)e(this)h(problem)e (that)j(for)g(a)f(problem)f(w)m(orth)i(so)150 2794 y(m)m(uc)m(h)34 b(of)g(the)g(problem)f(set)h(grade,)h(NO)f(ONE)g(ga)m(v)m(e)i(the)e (formal)f(reductions.)50 b(The)34 b(reductions)f(are)150 2907 y(tric)m(ky)-8 b(,)30 b(and)e(y)m(ou)h(get)h(in)m(to)e(problems)f (if)h(y)m(ou)h(c)m(ho)s(ose)h(the)f(wrong)f(thing.)39 b(F)-8 b(or)30 b(instance,)f(some)g(p)s(eople)150 3019 y(used)g(the)h(receiv)m(er's)h(output)e Fh(v)k Fl(as)d(part)g(of)g(the) g(output)f(of)h Fh(F)13 b Fl(,)31 b(whic)m(h)d(is)h(a)h(problem)f(if)f (y)m(ou're)j(trying)150 3132 y(to)41 b(break)f(binding,)g(since)f Fh(C)47 b Fl(w)m(ould)38 b(not)j(ha)m(v)m(e)g(access)h(to)e Fh(v)s Fl(.)71 b(Similarly)-8 b(,)39 b(if)g(the)i(output)e(includes)150 3245 y(the)34 b(randomness)e(of)h(the)h(receiv)m(er,)h(there)e(is)f(a)i (problem)e(with)g(that)i(same)g(reduction)e(for)h(the)h(same)150 3358 y(reason.)150 3644 y Fi(Problem)45 b(2)150 3847 y Fl(Consider)c(the)i(follo)m(wing)f(commitmen)m(t)h(sc)m(heme.)80 b(Supp)s(ose)41 b(there)i(are)h(public)c(parameters)j Fh(p)g Fl(\(a)150 3960 y(prime\),)28 b(and)g Fh(g)k Fl(and)d Fh(h)p Fl(,)g(t)m(w)m(o)h(generators)g(mo)s(dulo)d Fh(p)p Fl(,)i(and)f(that)h(these)h(are)f(c)m(hosen)g(at)g(random)f(at)i(the) 150 4072 y(start)h(and)f(made)g(a)m(v)-5 b(ailable)30 b(to)h(b)s(oth)f(parties.)291 4185 y(T)-8 b(o)33 b(commit)g(to)h(a)f (bit)f Fh(b)p Fl(,)i(the)f(committer)g(c)m(ho)s(oses)g(a)h(random)e Fh(r)j Fl(and)d(computes)h Fh(g)3242 4152 y Fe(r)3281 4185 y Fh(h)3333 4152 y Fe(b)3368 4185 y Fl(,)g(whic)m(h)f(is)150 4298 y(sen)m(t)f(to)g(the)g(v)m(eri\014er.)39 b(T)-8 b(o)31 b(op)s(en)f(the)h(commitmen)m(t,)g(the)f(committer)h(rev)m(eals) g Fh(r)h Fl(and)e Fh(b)p Fl(.)291 4507 y Fm(\(a\))j Fl(\(11)h(p)s(oin)m (ts\))f(Pro)m(v)m(e)h(that)g(this)f(is)f(a)i(secure)f(commitmen)m(t)h (sc)m(heme.)51 b(More)34 b(precisely)-8 b(,)33 b(pro)m(v)m(e)150 4620 y(that)j(this)e(sc)m(heme)i(is)f(correct,)j(that)e(it)e(is)h (hiding)d(unconditionally)-8 b(,)34 b(and)h(that)h(it)f(is)f(binding)e (if)j(the)150 4733 y(discrete)30 b(log)h(problem)d(is)i(hard.)291 4941 y Fm(Solution:)41 b Fl(That)30 b(this)g(sc)m(heme)h(is)e(correct)j (is)d(trivial.)291 5054 y(T)-8 b(o)28 b(see)f(that)h(it)e(is)g (unconditionally)e(hiding,)h(w)m(e)j(remark)e(that)i(since)e Fh(g)31 b Fl(is)26 b(a)h(generator,)i(there)f(is)e(a)150 5167 y(v)-5 b(alue)26 b Fh(\013)h Fl(suc)m(h)g(that)g Fh(h)f Fl(=)f Fh(g)1081 5134 y Fe(\013)1157 5167 y Fl(Th)m(us,)i(giv)m (en)g(a)g(commitmen)m(t)g Fh(g)2286 5134 y Fe(r)2325 5167 y Fl(,)g(the)g(probabilit)m(y)d(that)k Fh(g)3231 5134 y Fe(r)3296 5167 y Fl(is)e(pro)s(duced)1867 5416 y(19-2)p eop %%Page: 3 3 3 2 bop 150 159 a Fl(from)24 b(a)h(commitmen)m(t)h(to)f(a)g(0)h(is)d (exactly)j(the)f(probabilit)m(y)d(that)k(w)m(e)f(pic)m(k)f Fh(r)s Fl(,)i(and)e(the)h(probabilit)m(y)d(that)150 271 y Fh(g)196 238 y Fe(r)269 271 y Fl(is)34 b(pro)s(duced)f(from)h(a)h (commitmen)m(t)f(to)i(a)f(0)g(is)e(exactly)i(the)g(probabilit)m(y)d (that)j Fh(r)26 b Ff(\000)d Fh(\013)35 b Fl(is)e(c)m(hosen.)150 384 y(Since)d(these)i(probabilities)c(are)j(equal,)h(w)m(e)f(kno)m(w)h (that)g(Pr)o([)p Fh(b)27 b Fl(=)g(0)p Ff(j)p Fh(g)2536 351 y Fe(r)2575 384 y Fl(])g(=)f(Pr[)p Fh(b)h Fl(=)f(1)p Ff(j)p Fh(g)3126 351 y Fe(r)3165 384 y Fl(],)32 b(so)g Fh(c)f Fl(m)m(ust)h(b)s(e)150 497 y(unconditionally)27 b(hiding.)291 610 y(Supp)s(ose)g(a)i(committer)g Fh(C)1230 577 y Fg(0)1281 610 y Fl(can)g(decommit)g(a)g(commitmen)m(t)g Fh(c)g Fl(in)e(t)m(w)m(o)j(w)m(a)m(ys)f(with)f(nonnegligible)150 723 y(probabilit)m(y)-8 b(.)63 b(Then,)40 b(w)m(e)f(can)f(use)g(this)g (adv)m(ersary)h(to)g(solv)m(e)g(the)f(discrete)g(logarithm)g(problem)f (as)150 836 y(follo)m(ws.)79 b(Supp)s(ose)41 b(w)m(e)j(are)g(giv)m(en)f (a)h(discrete)f(logarithm)f(instance)h Fh(p;)15 b(g)s(;)g(h)45 b Fl(to)f(solv)m(e.)79 b(Since)43 b(the)150 949 y(instance)33 b(is)f(generated)i(randomly)-8 b(,)33 b(with)f(non-negligible)e (probabilit)m(y)-8 b(,)32 b Fh(h)i Fl(is)e(a)h(generator.)50 b(If)33 b Fh(h)g Fl(is)g(a)150 1062 y(generator,)40 b(then)c(w)m(e)h (can)g(pass)f Fh(p;)15 b(g)s(;)g(h)38 b Fl(to)g Fh(C)1770 1029 y Fg(0)1829 1062 y Fl(and)e(ha)m(v)m(e)i Fh(C)2300 1029 y Fg(0)2359 1062 y Fl(commit)e(to)i(a)f(v)-5 b(alue)36 b Fh(c)h Fl(and)f(ha)m(v)m(e)i(t)m(w)m(o)150 1175 y(decommitmen)m(ts)31 b Fh(r)m(;)15 b Fl(0)32 b(and)e Fh(r)1181 1142 y Fg(0)1204 1175 y Fh(;)15 b Fl(1.)43 b(If)30 b(these)i(are)f(correct)h(then)e Fh(c)d Fl(=)f Fh(g)2554 1142 y Fe(r)2618 1175 y Fl(=)g Fh(g)2761 1142 y Fe(r)2795 1118 y Fc(0)2822 1175 y Fh(h)p Fl(,)31 b(so)g Fh(h)c Fl(=)e Fh(g)3263 1142 y Fe(r)r Fg(\000)p Fe(r)3386 1118 y Fc(0)3413 1175 y Fl(,)31 b(and)f(w)m(e)150 1288 y(can)f(return)f Fh(r)20 b Ff(\000)d Fh(r)785 1255 y Fg(0)837 1288 y Fl(as)29 b(the)h(answ)m(er)f(to)g(the)g(discrete)g (logarithm)f(problem.)39 b(This)27 b(pro)m(v)m(es)j(that)f Fh(C)q(;)15 b(R)31 b Fl(is)150 1401 y(binding)c(if)j(the)g(discrete)g (logarithm)g(problem)e(is)i(hard.)291 1600 y Fm(\(b\))36 b Fl(\(11)h(p)s(oin)m(ts\))e(If)h(instead)f(the)h(committer)h (generates)g Fh(p)p Fl(,)h Fh(g)s Fl(,)g(and)e Fh(h)p Fl(,)i(is)d(the)h(sc)m(heme)h(secure?)150 1713 y(Pro)m(v)m(e)32 b(y)m(our)e(answ)m(er.)41 b(If)29 b(no,)i(is)e(there)i(a)g(w)m(a)m(y)g (to)g(\014x)f(it)g(easily?)291 1912 y Fm(Solution:)56 b Fl(No.)63 b(If)37 b(the)h(committer)g(kno)m(ws)g Fh(\013)g Fl(suc)m(h)f(that)i Fh(g)2503 1879 y Fe(\013)2590 1912 y Fl(=)e Fh(h)h Fl(then)g(the)g(committer)g(can)150 2025 y(easily)30 b(break)i(binding.)40 b(The)31 b(committer)g(simply)e(alw)m (a)m(ys)j(commits)f(to)h(a)g(1)f(and)g(remem)m(b)s(ers)g Fh(r)i Fl(suc)m(h)150 2137 y(that)h Fh(c)c Fl(=)g Fh(g)566 2104 y Fe(r)604 2137 y Fh(h)p Fl(.)49 b(Then,)34 b(to)f(decommit)g(to)h (a)g(1)f(is)g(easy)-8 b(,)35 b(and)d(to)i(decommit)f(to)h(a)f(0,)i(w)m (e)f(rev)m(eal)f Fh(r)25 b Fl(+)d Fh(\013)150 2250 y Fl(instead)30 b(of)i Fh(r)s Fl(,)f(and)f(this)h(is)f(correct.)44 b(There)31 b(is)f(no)h(easy)h(w)m(a)m(y)g(to)g(\014x)f(this,)f(as)h (the)h(committer)f(cannot)150 2363 y(pro)m(v)m(e)39 b(that)g(he)f(do)s (esn't)g(kno)m(w)h(the)f(discrete)g(logarithm)g(of)g Fh(h)h Fl(to)g(the)f(base)h Fh(g)s Fl(,)i(whic)m(h)c(is)g(what)h(w)m(e) 150 2476 y(require.)291 2675 y Fm(\(c\))26 b Fl(\(11)i(p)s(oin)m(ts\))d (If)h(instead)g(the)g(receiv)m(er)h(generates)h Fh(p)p Fl(,)f Fh(g)s Fl(,)h(and)e Fh(h)p Fl(,)i(is)d(the)h(sc)m(heme)i (secure?)39 b(Pro)m(v)m(e)150 2788 y(y)m(our)30 b(answ)m(er.)41 b(If)30 b(no,)g(is)g(there)g(a)h(w)m(a)m(y)g(to)h(\014x)d(it)h(easily?) 291 2987 y Fm(Solution:)50 b Fl(No.)56 b(The)34 b(sc)m(heme)i(is)e (supp)s(osed)f(to)j(w)m(ork)f(with)e Fh(g)39 b Fl(and)34 b Fh(h)i Fl(b)s(oth)e(b)s(eing)g(generators,)150 3100 y(but)29 b Fh(C)36 b Fl(nev)m(er)30 b(c)m(hec)m(ks)h(this)d(so)i(there) g(is)e(an)i(attac)m(k)h(that)g(w)m(e)f(can)f(launc)m(h.)40 b(Supp)s(ose)28 b(that)i Fh(g)j Fl(is)28 b(c)m(hosen)150 3213 y(to)42 b(ha)m(v)m(e)g(order)f Fh(w)46 b(<)d(p)27 b Ff(\000)g Fl(1,)45 b(but)40 b Fh(h)i Fl(is)e(c)m(hosen)i(to)g(ha)m(v) m(e)g(order)f Fh(p)27 b Ff(\000)g Fl(1.)74 b(No)m(w,)45 b(if)40 b(the)i(committer)150 3326 y(publishes)26 b Fh(g)590 3293 y Fe(r)628 3326 y Fl(,)k(this)d(will)g(ha)m(v)m(e)j(order)e Fh(w)k Fl(\(or)d(less\).)40 b(Ho)m(w)m(ev)m(er,)31 b(if)d(the)h (committer)h(publishes)25 b Fh(g)3458 3293 y Fe(r)3497 3326 y Fh(h)p Fl(,)30 b(and)150 3439 y(w)m(e)k(raise)e(this)g(to)i(the) g(p)s(o)m(w)m(er)f Fh(w)r Fl(,)h(w)m(e)g(get)g Fh(h)1693 3406 y Fe(w)1750 3439 y Fl(,)g(whic)m(h)e(is)g(not)i(1)f(since)g(the)g (order)g(of)g Fh(h)h Fl(is)e Fh(p)22 b Ff(\000)f Fl(1)31 b Fh(>)e(w)r Fl(.)150 3552 y(Therefore,)g(w)m(e)g(can)g(easily)f(test)h (to)g(see)g(whic)m(h)f(v)-5 b(alue)28 b(the)h(committer)f(committed)h (to)g(b)m(y)g(raising)e(the)150 3665 y(commitmen)m(t)k(to)g(the)f(p)s (o)m(w)m(er)h Fh(w)r Fl(.)41 b(If)30 b(the)h(result)e(is)g(1,)i(return) f(0.)41 b(If)30 b(not,)g(return)g(1.)291 3778 y(This)36 b(can)h(b)s(e)g(\014xed,)i(ho)m(w)m(ev)m(er.)63 b(What)38 b(w)m(e)f(require)g(is)f(simply)f(that)j(there)f(b)s(e)g(a)h(v)-5 b(alue)36 b Fh(\013)i Fl(suc)m(h)150 3891 y(that)e Fh(g)398 3858 y Fe(\013)481 3891 y Fl(=)d Fh(h)p Fl(,)k(whic)m(h)d(is)g(not)h (true)g(in)f(the)h(ab)s(o)m(v)m(e)i(scenario.)55 b(Th)m(us,)35 b(w)m(e)h(simply)d(require)g(some)j(w)m(a)m(y)150 4004 y(to)f(v)m(erify)e(that)i(suc)m(h)f(an)f Fh(\013)i Fl(exists.)51 b(This)33 b(can)h(b)s(e)f(ac)m(hiev)m(ed)i(as)f(follo)m(ws:)47 b(let)34 b Fh(M)45 b Fl(b)s(e)33 b(a)h(probabilistic)150 4116 y(TM)28 b(whic)m(h)f(pro)s(duces)g(a)i Fh(p;)15 b(g)s(;)g(h)30 b Fl(where)e Fh(p)g Fl(is)f(prime)g(and)h Fh(g)k Fl(and)c Fh(h)g Fl(are)h(generators,)h(and)e(ha)m(v)m(e)i Fh(R)f Fl(send)150 4229 y(not)j(only)f Fh(p)p Fl(,)g Fh(g)s Fl(,)i(and)e Fh(h)p Fl(,)i(but)e(also)g(the)h(random)f(coins)g (used)g(with)f Fh(M)42 b Fl(that)32 b(pro)s(duce)e(them.)45 b(No)m(w,)33 b Fh(C)150 4342 y Fl(c)m(hec)m(ks)c(that)e Fh(M)38 b Fl(used)26 b(with)g(those)i(random)e(coins)h(pro)s(duces)f (that)i Fh(p)p Fl(,)f Fh(g)s Fl(,)i(and)e Fh(h)p Fl(,)h(and)f(if)f(so,) i(pro)s(ceeds.)150 4455 y(\(W)-8 b(e)31 b(assume)d Fh(M)39 b Fl(w)m(ould)28 b(nev)m(er)h(pro)s(duce)f Fh(g)33 b Fl(that)c(is)f(not)h(a)h(generator;)h(therefore,)f(there)f(exists)f(an) h Fh(\013)p Fl(,)150 4568 y(as)i(required.\))150 4851 y Fi(Problem)45 b(3)150 5054 y Fl(Consider)36 b(the)i(follo)m(wing)e (zero-kno)m(wledge)j(proto)s(col)e(for)h(pro)m(ving)f(that)h(a)g(v)-5 b(alue)37 b Fh(a)h Fl(is)e(a)i(quadratic)150 5167 y(residue)29 b(mo)s(dulo)g Fh(n)p Fl(,)h(in)f(whic)m(h)1867 5416 y(19-3)p eop %%Page: 4 4 4 3 bop 291 159 a Fl(Input:)39 b Fh(n)30 b Fl(and)g Fh(a)p Fl(.)41 b(Let)31 b Fh(\013)1173 126 y Fd(2)1238 159 y Ff(\021)25 b Fh(a)g Fl(mo)s(d)g Fh(n)30 b Fl(b)s(e)f(the)i(square)f(ro) s(ot)h(of)f Fh(a)p Fl(.)261 349 y(1.)46 b Fh(P)40 b Fl(c)m(ho)s(oses)28 b(a)g(random)e(prime)g Fh(p)f Fl(=)g(2)p Fh(q)16 b Fl(+)e(1)27 b(where)g Fh(q)j Fl(is)c(a)h(random)g Fh(k)s Fl(-bit)f(prime,)h(and)f (a)i(random)377 462 y Fh(g)34 b Fl(or)d(order)e Fh(q)s Fl(,)i(and)f(randomly)e(c)m(ho)s(oses)k Fh(\014)j Fl(and)30 b(sets)h Fh(h)25 b Fl(=)g Fh(g)2471 429 y Fe(\014)2544 462 y Fl(mo)s(d)g Fh(p)p Fl(.)40 b Fh(P)k Fl(sends)29 b Fh(p;)15 b(q)s(;)g(g)s(;)g(h)32 b Fl(to)f Fh(V)20 b Fl(.)261 642 y(2.)46 b Fh(V)j Fl(computes)30 b(a)f(random)f(string)g Fh(c)i Fl(of)f(length)f Fh(k)k Fl(and)d(a)g(random)g Fh(r)s Fl(,)g(and)f(sends)g Fh(x)d Fl(=)g Fh(g)3389 609 y Fe(r)3428 642 y Fh(h)3480 609 y Fe(c)3544 642 y Fl(to)30 b Fh(P)13 b Fl(.)261 822 y(3.)46 b Fh(P)40 b Fl(c)m(ho)s(oses)27 b Fh(k)j Fl(random)25 b(v)-5 b(alues)26 b Fh(r)1509 836 y Fd(1)1549 822 y Fh(;)15 b(:)g(:)g(:)h(;)f(r)1791 837 y Fe(k)1861 822 y Fl(and)26 b(computes)h Fh(s)2478 836 y Fe(i)2531 822 y Fl(=)e Fh(r)2671 789 y Fd(2)2668 848 y Fe(i)2736 822 y Fl(and)h(sends)f Fh(s)3191 836 y Fd(1)3231 822 y Fh(;)15 b(:)g(:)g(:)h(;)f(s)3475 837 y Fe(k)3544 822 y Fl(to)28 b Fh(V)20 b Fl(.)261 1003 y(4.)46 b Fh(V)51 b Fl(sends)29 b Fh(r)k Fl(and)d Fh(c)g Fl(to)i Fh(P)13 b Fl(.)261 1296 y(5.)46 b Fh(P)h Fl(c)m(hec)m(ks)35 b(that)g Fh(x)c Fl(=)g Fh(g)1198 1263 y Fe(r)1236 1296 y Fh(h)1288 1263 y Fe(c)1323 1296 y Fl(.)51 b(If)34 b(so,)h Fh(P)47 b Fl(sends)33 b(to)h Fh(V)21 b Fl(,)35 b(for)e(eac)m(h)i Fh(i)p Fl(,)g(the)g(v)-5 b(alue)33 b Fh(t)3108 1310 y Fe(i)3167 1296 y Fl(=)e Fh(r)3310 1310 y Fe(i)3338 1296 y Fh(\013)3396 1263 y Fe(c)3427 1273 y Fb(i)3458 1296 y Fl(,)j(where)377 1409 y Fh(c)26 b Fl(=)f Fh(c)577 1423 y Fd(1)637 1409 y Ff(\016)20 b Fh(:)15 b(:)g(:)22 b Ff(\016)e Fh(c)933 1424 y Fe(k)976 1409 y Fl(.)41 b Fh(P)i Fl(also)31 b(sends)e Fh(\014)36 b Fl(to)31 b Fh(V)261 1589 y Fl(6.)46 b Fh(V)55 b Fl(c)m(hec)m(ks)36 b(that)f Fh(t)1004 1556 y Fd(2)1004 1614 y Fe(i)1075 1589 y Ff(\021)c Fh(s)1220 1603 y Fe(i)1248 1589 y Fh(a)1296 1556 y Fe(c)1327 1566 y Fb(i)1391 1589 y Fl(for)k(eac)m(h)g Fh(i)p Fl(,)h(and)e(that)h Fh(h)d Fl(=)f Fh(g)2449 1556 y Fe(\014)2497 1589 y Fl(.)53 b(If)34 b(so,)i Fh(V)54 b Fl(accepts,)37 b(otherwise)d Fh(V)377 1702 y Fl(rejects.)291 1892 y(This)27 b(de\014nes)g(a)i Fh(P)s(;)15 b(V)48 b Fl(whic)m(h)27 b(y)m(ou)i(will)d(pro)m(v)m(e)j(is) e(a)i(zero-kno)m(wledge)g(pro)s(of)f(system.)40 b(It)28 b(should)f(b)s(e)150 2005 y(clear)f(that)g Fh(P)s(;)15 b(V)46 b Fl(has)25 b(completeness)h(1.)39 b(The)25 b(idea)h(of)f(this)g (construction)g(is)g(to)h(simply)d(parallelize)h(the)150 2118 y(ZKP)31 b(w)m(e)h(presen)m(ted)g(in)e(class)i(for)f(quadratic)h (residuosit)m(y)-8 b(.)44 b(In)31 b(order)g(to)h(pro)m(v)m(e)h (zero-kno)m(wledgeness)150 2231 y(w)m(e)e(need)f(to)h(add)f(the)g (commitmen)m(t)h Fh(x)f Fl(in)m(to)h(the)f(proto)s(col,)h(as)g(y)m(ou)f (will)e(see)j(when)e(pro)m(ving)h(this.)291 2429 y Fm(\(a\))j Fl(\(17)h(p)s(oin)m(ts\))f(Pro)m(v)m(e)h(that)g Fh(P)s(;)15 b(V)54 b Fl(has)33 b(negligible)e(soundness)h(\(ie,)i Fh(V)53 b Fl(can)34 b(only)e(b)s(e)h(con)m(vinced)150 2542 y(with)c(negligible)f(probabilit)m(y)g(if)h Fh(a)i Fl(is)e(a)i(quadratic)f(non-residue)e(mo)s(dulo)h Fh(n)p Fl(\).)291 2739 y Fm(Solution:)41 b Fl(Supp)s(ose)28 b Fh(a)e Ff(62)f Fh(QR)q Fl(\()p Fh(n)p Fl(\).)40 b(Then,)30 b(for)g(an)m(y)g(set)h(of)f(v)-5 b(alues)30 b Fh(s)2718 2753 y Fd(1)2757 2739 y Fh(;)15 b(:)g(:)g(:)i(;)e(s)3002 2754 y Fe(k)3075 2739 y Fl(the)30 b(pro)m(v)m(er)h(sends,)150 2852 y(there)g(is)f(exactly)i(one)f Fh(c)g Fl(for)g(whic)m(h)f(the)h (pro)m(v)m(er)g(can)g(pass)g(the)g(test:)43 b(the)31 b Fh(c)g Fl(for)g(whic)m(h)f(the)h Fh(i)p Fl(th)g(bit)f(of)150 2965 y Fh(c)h Fl(is)e(equal)h(to)h(the)g(opp)s(osite)e(of)i(the)f (quadratic)g(c)m(haracter)i(of)f Fh(s)2391 2979 y Fe(i)2418 2965 y Fl(.)291 3078 y(If)39 b(w)m(e)g(can)h(argue)f(that)h Fh(s)f Fl(is)f(c)m(hosen)h(indep)s(enden)m(tly)e(of)i Fh(c)p Fl(,)j(w)m(e)d(will)e(b)s(e)h(done,)j(since)e(then)g(the)150 3191 y(probabilit)m(y)19 b(that)j(this)e(particular)g Fh(c)i Fl(is)e(c)m(hosen)i(is)e(1)p Fh(=)p Fl(2)2015 3158 y Fe(k)2059 3191 y Fl(,)k(so)d Fh(V)42 b Fl(can)22 b(only)e(b)s(e)h(con)m(vinced)g(with)f(negligible)150 3304 y(probabilit)m(y)-8 b(.)37 b(This)24 b(is)g(fairly)g(easy)i(to)g (do,)h(ho)m(w)m(ev)m(er,)h(b)m(y)d(the)h(same)g(logic)f(as)h(in)e(the)h (previous)f(problem.)150 3417 y(F)-8 b(or)34 b(an)m(y)f Fh(x)p Fl(,)h(for)e(eac)m(h)j Fh(c)e Fl(there)g(is)f(exactly)i(one)f(v) -5 b(alue)32 b Fh(r)k Fl(suc)m(h)c(that)i Fh(g)2603 3384 y Fe(r)2641 3417 y Fh(h)2693 3384 y Fe(c)2758 3417 y Fl(=)29 b Fh(x)p Fl(,)34 b(namely)e(the)h(discrete)150 3530 y(logarithm)39 b(of)i Fh(x=h)837 3497 y Fe(c)912 3530 y Fl(to)g(the)g(base)f Fh(g)s Fl(.)71 b(Th)m(us,)42 b(the)e(probabilit)m(y)e(that)j Fh(c)g Fl(is)e(c)m(hosen)i(is)e(equal)h (to)h(the)150 3643 y(probabilit)m(y)26 b(that)i Fh(c)h Fl(is)e(c)m(hosen)i(giv)m(en)f Fh(x)p Fl(,)h(so)f Fh(x)g Fl(is)f(indep)s(enden)m(t)f(of)i Fh(c)p Fl(.)41 b(Since)27 b(no)h(other)g(v)-5 b(alues)28 b(seen)g(up)150 3755 y(to)j(that)g(p)s (oin)m(t)e(in)g(the)i(proto)s(col)f(dep)s(end)f(on)h Fh(c)p Fl(,)h(the)f(v)-5 b(alues)30 b Fh(s)2321 3769 y Fd(1)2360 3755 y Fh(;)15 b(:)g(:)g(:)i(;)e(s)2605 3770 y Fe(k)2677 3755 y Fl(that)31 b(the)g(pro)m(v)m(er)g(sends)e(m)m(ust) 150 3868 y(b)s(e)h(indep)s(enden)m(t)e(of)i Fh(c)p Fl(.)291 3981 y(Ho)m(w)m(ev)m(er,)42 b(one)37 b(crucial)f(di\013erence)h(b)s(et) m(w)m(een)h(this)e(problem)g(and)h(the)g(previous)f(one)i(is)e(that)i (in)150 4094 y(this)26 b(problem,)g Fh(P)771 4061 y Fg(0)794 4094 y Fl(,)i(who)f(ma)m(y)g(b)s(e)g(dishonest,)f(c)m(ho)s(oses)i Fh(p;)15 b(q)s(;)g(g)s Fl(,)29 b(and)d Fh(h)p Fl(,)i(and)f(so)g (therefore)g(could)f(c)m(heat)150 4207 y(if)j Fh(g)34 b Fl(is)c(not)g(a)h(generator.)42 b(Ho)m(w)m(ev)m(er,)33 b(as)d(w)m(e)h(noted)g(ab)s(o)m(v)m(e,)g(the)g(requiremen)m(t)e(w)m(e)i (need)f(is)g(that)h(there)150 4320 y(b)s(e)k(an)h Fh(\013)g Fl(suc)m(h)g(that)g Fh(g)964 4287 y Fe(\013)1049 4320 y Fl(=)e Fh(h)p Fl(,)k(whic)m(h)c(is)h(guaran)m(teed)i(to)f(b)s(e)g (the)g(case)h(if)d Fh(P)49 b Fl(con)m(vinces)36 b Fh(V)20 b Fl(,)38 b(since)d Fh(P)150 4433 y Fl(m)m(ust)30 b(rev)m(eal)h Fh(\014)36 b Fl(correctly)30 b(at)i(step)e(5.)291 4546 y(This)f(concludes)g(the)i(pro)s(of.)291 4744 y Fm(\(b\))f Fl(\(17)i(p)s(oin)m(ts\))e(Pro)m(v)m(e)i(that)f(if)e(the)i(discrete)f (logarithm)g(problem)f(is)h(hard,)g(then)g Fh(P)s(;)15 b(V)51 b Fl(is)30 b(zero-)150 4857 y(kno)m(wledge.)291 5054 y Fm(Solution:)43 b Fl(T)-8 b(o)31 b(pro)m(v)m(e)h(that)g Fh(P)s(;)15 b(V)51 b Fl(is)30 b(zero-kno)m(wledge,)j(w)m(e)f(need)f(to) g(giv)m(e)h(a)f(sim)m(ulator)f Fh(M)10 b Fl(.)43 b(Here)150 5167 y(is)29 b(the)i(co)s(de)g(for)f Fh(M)10 b Fl(,)30 b(giv)m(en)h(a)g(p)s(ossibly)c(c)m(heating)k(v)m(eri\014er)f Fh(V)2301 5134 y Fg(0)2324 5167 y Fl(:)1867 5416 y(19-4)p eop %%Page: 5 5 5 4 bop 261 159 a Fl(1.)46 b(Cho)s(ose)30 b Fh(p;)15 b(q)s(;)g(g)s(;)32 b Fl(and)e Fh(h)g Fl(as)h Fh(P)43 b Fl(w)m(ould)30 b(and)f(generates)j(a)f(random)e(tap)s(e)i Fh(R)q Fl(.)261 346 y(2.)46 b(Run)30 b Fh(V)649 313 y Fg(0)702 346 y Fl(with)f(random)h(tap)s(e)h Fh(R)g Fl(and)e(\014rst)h (inputs)e Fh(p;)15 b(q)s(;)g(g)s(;)g(h)p Fl(,)33 b(un)m(til)28 b Fh(V)2843 313 y Fg(0)2897 346 y Fl(giv)m(es)j Fh(x)p Fl(.)261 534 y(3.)46 b(Cho)s(ose)30 b Fh(k)k Fl(random)c(v)-5 b(alues)29 b Fh(r)1422 501 y Fg(0)1419 558 y Fd(1)1459 534 y Fh(;)15 b(:)g(:)g(:)h(;)f(r)1704 501 y Fg(0)1701 561 y Fe(k)1775 534 y Fl(and)29 b(send)h Ff(f)p Fh(r)2248 501 y Fg(0)2245 559 y Fe(i)2274 490 y Fd(2)2313 534 y Ff(g)h Fl(to)g Fh(V)2573 501 y Fg(0)2596 534 y Fl(.)261 721 y(4.)46 b(Receiv)m(e)34 b Fh(r)h Fl(and)d Fh(c)p Fl(.)47 b(If)32 b Fh(g)1214 688 y Fe(r)1253 721 y Fh(h)1305 688 y Fe(c)1369 721 y Ff(6)p Fl(=)c Fh(x)k Fl(then)h(send)e(an)i(\\ab)s (ort")g(message)h(to)f Fh(V)2976 688 y Fg(0)3032 721 y Fl(and)f(run)f(it)h(un)m(til)e(it)377 834 y(\014nishes,)f(at)i(whic)m (h)e(p)s(oin)m(t)g(w)m(e)i(return)e(its)h(output.)261 1022 y(5.)46 b(Rewind)26 b(to)i(the)g(b)s(eginning.)37 b(Run)26 b Fh(V)1685 989 y Fg(0)1736 1022 y Fl(with)g(random)g(tap)s(e) i Fh(R)g Fl(and)f(\014rst)f(inputs)g Fh(p;)15 b(q)s(;)g(g)s(;)g(h)p Fl(,)30 b(un)m(til)377 1135 y Fh(V)451 1102 y Fg(0)504 1135 y Fl(giv)m(es)h Fh(x)p Fl(.)261 1322 y(6.)46 b(Generate,)32 b(for)e(eac)m(h)i Fh(i)p Fl(,)f(a)g(random)e Fh(t)1665 1336 y Fe(i)1724 1322 y Fl(and)g(let)i Fh(s)2075 1336 y Fe(i)2128 1322 y Fl(=)25 b Fh(t)2257 1289 y Fd(2)2257 1348 y Fe(i)2296 1322 y Fh(=a)2389 1289 y Fe(c)2420 1299 y Fb(i)2450 1322 y Fl(.)41 b(Send)29 b Ff(f)p Fh(s)2826 1336 y Fe(i)2855 1322 y Ff(g)h Fl(to)h Fh(V)3115 1289 y Fg(0)3138 1322 y Fl(.)261 1510 y(7.)46 b(Receiv)m(e)34 b Fh(r)753 1477 y Fg(0)808 1510 y Fl(and)e Fh(c)1026 1477 y Fg(0)1050 1510 y Fl(.)47 b(If)32 b Fh(g)1261 1477 y Fe(r)1295 1454 y Fc(0)1322 1510 y Fh(h)1374 1477 y Fe(c)1405 1454 y Fc(0)1461 1510 y Ff(6)p Fl(=)c Fh(x)33 b Fl(then)f(send)g(an)g(\\ab)s(ort")h(message)h(to)f Fh(V)3069 1477 y Fg(0)3124 1510 y Fl(and)f(run)g(it)g(un)m(til)377 1623 y(it)e(\014nishes,)f(at)i(whic)m(h)e(p)s(oin)m(t)g(w)m(e)i(return) e(its)h(output.)261 1811 y(8.)46 b(If)30 b Fh(c)507 1778 y Fg(0)556 1811 y Ff(6)p Fl(=)25 b Fh(c)p Fl(,)31 b(fail.)261 1998 y(9.)46 b(Giv)m(e)27 b Ff(f)p Fh(t)664 2012 y Fe(i)692 1998 y Ff(g)g Fl(and)e Fh(\014)32 b Fl(to)27 b Fh(V)1199 1965 y Fg(0)1222 1998 y Fl(,)g(and)f(run)f(it)g(un)m(til)g(it)h (\014nishes,)f(at)i(whic)m(h)e(p)s(oin)m(t)g(w)m(e)h(return)f(its)h (output.)291 2211 y(Here,)36 b(w)m(e)f(assume)g(that)g(the)g(\\ab)s (ort")g(message)g(is)f(exp)s(ected)h(if)e Fh(V)55 b Fl(returns)33 b(an)i Fh(r)i Fl(and)d(a)h Fh(c)f Fl(that)150 2324 y(don't)d(c)m(hec)m (k)h(out.)41 b(Note)32 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6.875/18.425: Cryptography and Cryptanalysis

Fall 2001 homepage Here you can find all manner of information relating to 6.875/18.425.

Contact Information

Instructor:   Anna Lysyanskaya,  anna@theory.lcs.mit.edu
Office:       NE43 - 372

TA:           Moses Liskov,  mliskov@theory.lcs.mit.edu
Office:       NE43 - 334
Anna's office hours are by appointment.
Moses' regular office hours are 3PM-5PM on Mondays, in room 34-302 (and also by appointment)

Handouts

All handouts are in postscript format.

Problem Sets and Solutions

Scribe Notes

More detailed list

To sign up to scribe a lecture, email Anna at anna@theory.lcs.mit.edu. Available lectures:

Announcements

Students looking for collaborators, check out this page. Anyone wanting to be added to or removed from the list, email Moses at mliskov@theory.lcs.mit.edu index.html~0100664003326500237220000001272207403474337011741 0ustar annata68756.875: Cryptography and Cryptanalysis, Fall 2001

6.875/18.425: Cryptography and Cryptanalysis

Fall 2001 homepage Here you can find all manner of information relating to 6.875/18.425.

Contact Information

Instructor:   Anna Lysyanskaya,  anna@theory.lcs.mit.edu
Office:       NE43 - 372

TA:           Moses Liskov,  mliskov@theory.lcs.mit.edu
Office:       NE43 - 334
Anna's office hours are by appointment.
Moses' regular office hours are 3PM-5PM on Mondays, in room 34-302 (and also by appointment)

Handouts

All handouts are in postscript format.

Problem Sets and Solutions

Scribe Notes

More detailed list

To sign up to scribe a lecture, email Anna at anna@theory.lcs.mit.edu. Available lectures:

Announcements

Students looking for collaborators, check out this page. Anyone wanting to be added to or removed from the list, email Moses at mliskov@theory.lcs.mit.edu lectures.html0100664003325600237220000000557107374546216013040 0ustar mliskovta68756.875: Cryptography and Cryptanalysis, Fall 2001

6.875/18.425: Cryptography and Cryptanalysis Lecture notes

To sign up to scribe a lecture, email Anna at anna@theory.lcs.mit.edu.

Lecture
Date
Topics
Scribe
1 9/5/2001 Introduction, one-time pad, public-key model, RSA, Diffie-Hellman, computational assumptions model Moses Liskov
2 9/10/2001 Defining security of public-key cryptosystems Susan Hohenberger
3 9/12/2001 GM security, Legendre symbols and Jacobi symbols, quadratic residuosity assumption Jennifer Mulligan
4 9/19/2001 Goldwasser-Micali cryptosystem for 1-bit messages, proof of security Emil Sit
5 9/24/2001 GM cryptosystem for l-bit mesaages, proof of security Stephen Weis
6 9/26/2001 GM-security of bit-by-bit encryption (hybrid argument) Emina Torlak
7 10/1/2001 One-pass security definitions, Semantic security Nick Feamster
8 10/3/2001 Equivalence of GM-security and Semantic security Eamon Walsh
9 10/10/2001 Cryptographic Primitives, Hardcore Bits Denis Cebikins
10 10/15/2001 OWFs, OWPs, Hardcore Bits, and Goldreich-Levin Paisa Seeluangsawat
11 10/17/2001 OWP -> PRG, Candidate OWPs and hardcore bits Alice Chan
12 10/22/2001 Factoring based OWPF and OWF candidates, PRFs Jelena Spasojevic
13 10/24/2001 More on PRFs, PRG -> PRF (GGM construction) Marshall Tappen
14 10/29/2001 PRG -> PRF, Naor-Reingold PRG, Closeness of distributions Abhinav Kumar
15 10/31/2001 Notions of Indistinguishibility, Naor-Reingold continued. Joshua Marron
16 11/5/2001 DDH self-reducibility, PRPs. Luby-Rackoff construction (PRF -> PRP). Kolya Zeldovich
17 11/7/2001 PRF -> PRP -- Proof of Luby-Rackoff construction. Anna Lysyanskaya

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y(=)83 b Fn(P)13 b(r)s Fr([)p Fn(m)2027 512 y Fl(P)2085 498 y Fr(])p Fn(P)g(r)s Fr([)p Fn(s)25 b Fr(=)g Fn(c)20 b Fm(\010)g Fn(m)p Fm(j)p Fn(m)2749 512 y Fl(P)2808 498 y Fr(])1653 636 y(=)83 b Fn(P)13 b(r)s Fr([)p Fn(m)2027 650 y Fl(P)2085 636 y Fr(]2)2155 599 y Fk(\000)p Fl(k)291 838 y Fr(since)30 b Fn(s)g Fr(is)f(c)m(hosen)i (indep)s(enden)m(tly)c(of)k Fn(m)p Fr(.)41 b(Ho)m(w)m(ev)m(er,)32 b(w)m(e)f(also)f(ha)m(v)m(e)i(that)1162 1153 y Fn(P)13 b(r)s Fr([)p Fn(c)1341 1167 y Fl(C)1400 1153 y Fr(])83 b(=)1761 1067 y Fj(X)1662 1272 y Fl(m)1724 1253 y Fi(0)1747 1272 y Fk(2f)p Fh(0)p Fl(;)p Fh(1)p Fk(g)1954 1253 y Fg(k)2008 1153 y Fn(P)13 b(r)s Fr([)p Fn(m)2228 1116 y Fk(0)2228 1176 y Fl(P)2286 1153 y Fr(])p Fn(P)g(r)s Fr([)p Fn(c)2490 1167 y Fl(C)2549 1153 y Fm(j)p Fn(m)2654 1116 y Fk(0)2654 1176 y Fl(P)2713 1153 y Fr(])1508 1426 y(=)1761 1340 y Fj(X)1662 1545 y Fl(m)1724 1526 y Fi(0)1747 1545 y Fk(2f)p Fh(0)p Fl(;)p Fh(1)p Fk(g)1954 1526 y Fg(k)2008 1426 y Fn(P)g(r)s Fr([)p Fn(m)2228 1389 y Fk(0)2228 1449 y Fl(P)2286 1426 y Fr(]2)2356 1389 y Fk(\000)p Fl(k)1508 1699 y Fr(=)83 b(2)1707 1662 y Fk(\000)p Fl(k)1920 1613 y Fj(X)1820 1818 y Fl(m)1882 1799 y Fi(0)1905 1818 y Fk(2f)p Fh(0)p Fl(;)p Fh(1)p Fk(g)2112 1799 y Fg(k)2166 1699 y Fn(P)13 b(r)s Fr([)p Fn(m)2386 1662 y Fk(0)2386 1722 y Fl(P)2444 1699 y Fr(])1508 1960 y(=)83 b(2)1707 1923 y Fk(\000)p Fl(k)291 2162 y Fr(Th)m(us,)30 b(w)m(e)h(get)1357 2502 y Fn(P)13 b(r)s Fr([)p Fn(m)1577 2516 y Fl(P)1635 2502 y Fm(j)p Fn(c)1699 2516 y Fl(C)1759 2502 y Fr(])83 b(=)2030 2441 y Fn(P)13 b(r)s Fr([)p Fn(m)2250 2455 y Fl(P)2329 2441 y Fm(^)20 b Fn(c)2449 2455 y Fl(C)2508 2441 y Fr(])p 2030 2481 504 4 v 2150 2565 a Fn(P)13 b(r)s Fr([)p Fn(c)2329 2579 y Fl(C)2389 2565 y Fr(])1867 2761 y(=)2030 2700 y(2)2075 2667 y Fk(\000)p Fl(k)2174 2700 y Fn(P)g(r)s Fr([)p Fn(m)2394 2714 y Fl(P)2452 2700 y Fr(])p 2030 2740 447 4 v 2182 2824 a(2)2227 2797 y Fk(\000)p Fl(k)1867 2925 y Fr(=)83 b Fn(P)13 b(r)s Fr([)p Fn(m)2241 2939 y Fl(P)2299 2925 y Fr(])p Fn(;)291 3127 y Fr(whic)m(h)29 b(completes)i(the)g(pro)s(of.)291 3338 y(This)26 b(is)g(w)m(onderful!) 38 b(Since)27 b(the)g(ciphertext)g(is)g(indep)s(enden)m(t)e(of)j(the)f (message,)j(Ev)m(e)e(cannot)g(learn)150 3451 y(the)f(message)h(from)e (observing)g(the)g(ciphertext.)40 b(But)27 b(there's)g(a)g(problem:)37 b Fn(s)29 b Fp(c)-5 b(an)30 b(only)g(b)-5 b(e)29 b(use)-5 b(d)30 b(onc)-5 b(e!)150 3564 y Fr(In)27 b(our)h(scenario,)h(only)e (one)h(message)i(is)d(b)s(eing)f(sen)m(t)j(so)f(this)f(is)h(not)g(a)h (securit)m(y)e(problem,)h(but)f(rather)150 3677 y(an)f(e\016ciency)g (problem.)37 b(If)26 b(Alice)f(and)h(Bob)g(m)m(ust)g(get)h(together)g (once)g(for)f(eac)m(h)h(message)g(they)f(are)g(to)150 3790 y(send,)33 b(wh)m(y)g(b)s(other)g(ha)m(ving)f(cryptograph)m(y?)50 b(No)m(w,)35 b(Alice)d(and)h(Bob)g(could)g(get)h(together)h(once)f(and) 150 3903 y(establish)26 b Fp(many)i Fr(secret)h(k)m(eys,)f(but)f(this)f (is)h(certainly)f(a)i(problem)e(with)g(this)h(system.)39 b(But)28 b(ma)m(yb)s(e)f(w)m(e)150 4015 y(can)k(do)f(b)s(etter)g(than)h (this?)150 4302 y Fo(3)135 b(Shannon's)44 b(Theorem)150 4504 y Fr(As)26 b(it)f(turns)f(out,)k(w)m(e)e(can't.)40 b(Shannon's)24 b(Theorem)i(is)e(that)j(if)d(Alice)h(and)h(Bob)g(share)f (only)g(a)h(secret)h(of)150 4617 y(length)f Fn(k)s Fr(,)i(there)g(is)e Fp(no)h Fr(metho)s(d)g(whic)m(h)e(can)j(securely)e(comm)m(unicate)i (messages)f(of)h(length)e Fn(k)17 b Fr(+)c(1.)40 b(W)-8 b(e)150 4730 y(should)35 b(b)s(e)i(careful)f(ab)s(out)h(sa)m(ying)g (\\securely)g(comm)m(unicate")h(without)f(kno)m(wing)f(what)h(it)g (means.)150 4843 y(F)-8 b(or)30 b(this)e(theorem,)i(a)f(system)g(is)f (\\secure")i(if)e(it)h(meets)g(the)h(ab)s(o)m(v)m(e)g(de\014nition,)d (namely)h(that)i(for)f(an)m(y)150 4956 y(message)i Fn(m)g Fr(and)e(ciphertext)h Fn(c)p Fr(,)h Fn(P)13 b(r)s Fr([)p Fn(m)1533 4970 y Fl(P)1592 4956 y Fr(])25 b(=)g Fn(P)13 b(r)s Fr([)p Fn(m)1958 4970 y Fl(P)2016 4956 y Fm(j)p Fn(c)2080 4970 y Fl(C)2140 4956 y Fr(].)150 5167 y Fs(Theorem)34 b(2)h(\(Shannon\):)40 b Fr(If)30 b Fm(jMj)c Fn(>)f Fm(jS)7 b(j)31 b Fr(then)f Fm(9)p Fn(m;)15 b(c;)41 b(P)13 b(r)s Fr([)p Fn(m)2443 5181 y Fl(P)2501 5167 y Fr(])25 b Fm(6)p Fr(=)g Fn(P)13 b(r)s Fr([)p Fn(m)2867 5181 y Fl(P)2926 5167 y Fm(j)p Fn(c)2990 5181 y Fl(C)3049 5167 y Fr(].)1889 5416 y(1-3)p eop %%Page: 4 4 4 3 bop 150 159 a Fp(Pr)-5 b(o)g(of.)54 b Fr(Supp)s(ose)33 b(not.)53 b(That)34 b(is,)h(supp)s(ose)e(that)i Fm(8)p Fn(m;)15 b(c;)47 b(P)13 b(r)s Fr([)p Fn(m)2414 173 y Fl(P)2472 159 y Fr(])33 b(=)e Fn(P)13 b(r)s Fr([)p Fn(m)2852 173 y Fl(P)2911 159 y Fm(j)p Fn(c)2975 173 y Fl(C)3034 159 y Fr(].)53 b(Then,)35 b(w)m(e)g(kno)m(w)150 271 y(that)878 497 y Fm(8)p Fn(m;)15 b(m)1129 460 y Fk(0)1177 497 y Fm(2)25 b(M)p Fn(;)15 b(s)26 b Fm(2)e(S)7 b Fn(;)41 b Fm(9)p Fn(s)1788 460 y Fk(0)1836 497 y Fr(:)25 b Fn(E)5 b Fr(\()p Fn(m;)15 b(s)p Fr(\))26 b(=)f Fn(E)5 b Fr(\()p Fn(m)2500 460 y Fk(0)2524 497 y Fn(;)15 b(s)2607 460 y Fk(0)2630 497 y Fr(\))21 b Fm(^)f Fn(s)2810 460 y Fk(0)2858 497 y Fm(6)p Fr(=)25 b Fn(s:)291 655 y Fr(If)42 b(this)f(w)m(as)i(not)g (true,)i(then)e(there)f(w)m(ould)f(b)s(e)h(a)h(con)m(tradiction,)j (since)41 b Fn(P)13 b(r)s Fr([)p Fn(m)3189 622 y Fk(0)3189 682 y Fl(P)3248 655 y Fr(])45 b Fn(>)g Fr(0)e(\(since)150 768 y Fn(m)230 735 y Fk(0)278 768 y Fm(2)25 b(M)p Fr(,)31 b(it)f(is)f(a)i(p)s(ossible)d(message\),)k(but)e Fn(P)13 b(r)s Fr([)p Fn(m)1926 735 y Fk(0)1926 795 y Fl(P)1984 768 y Fm(j)p Fr(\()p Fn(E)5 b Fr(\()p Fn(m;)15 b(s)p Fr(\)\))2384 782 y Fl(C)2445 768 y Fr(])26 b(=)e(0)i Fm(6)p Fr(=)f Fn(P)13 b(r)s Fr([)p Fn(m)2978 735 y Fk(0)2978 795 y Fl(P)3036 768 y Fr(].)291 881 y(Clearly)-8 b(,)41 b Fn(s)679 848 y Fk(0)743 881 y Fm(6)p Fr(=)f Fn(s)f Fr(if)g(they)g(exist.)68 b(Otherwise,)41 b Fn(E)5 b Fr(\()p Fn(m;)15 b(s)p Fr(\))42 b(=)e Fn(E)5 b Fr(\()p Fn(m)2632 848 y Fk(0)2656 881 y Fn(;)15 b(s)p Fr(\),)42 b(and)d(then)g(Alice)g (cannot)150 994 y(compute)25 b Fn(D)s Fr(\()p Fn(E)5 b Fr(\()p Fn(m;)15 b(s)p Fr(\))p Fn(;)g(s)p Fr(\).)41 b(No)m(w)25 b(w)m(e)h(can)f(see)g(there)h(m)m(ust)e(b)s(e)h(as)g(man)m (y)g(k)m(eys)h(as)f(there)g(are)g(plain)m(texts,)150 1107 y(th)m(us)30 b(con)m(tradicting)g(that)h Fm(jMj)26 b Fn(>)f Fm(jS)7 b(j)p Fr(.)41 b(This)29 b(completes)h(the)h(pro)s(of.) 150 1391 y Fo(4)135 b(The)44 b(Computationally)j(Bounded)d(Adv)l (ersary)h(Mo)t(del)150 1594 y Fr(Shannon's)31 b(theorem)j(pro)m(v)m(es) g(that)f(w)m(e)h(cannot)f(do)g(b)s(etter)g(than)g(the)g(one-time)h(pad) e(if)g(w)m(e)i(w)m(an)m(t)f(se-)150 1707 y(curit)m(y)-8 b(.)63 b(Ho)m(w)m(ev)m(er,)41 b(the)d(securit)m(y)g(the)g(one-time)g (pad)f(pro)m(vides)f(allo)m(ws)h(the)h(adv)m(ersary)g(to)h(b)s(e)e(v)m (ery)150 1820 y(p)s(o)m(w)m(erful)23 b(indeed.)38 b(So)24 b(instead)g(of)h(simply)d(going)j(home,)h(w)m(e)g(ough)m(t)f(to)h(see)f (if)f(w)m(e)h(can)g(do)f(b)s(etter)h(in)f(the)150 1932 y(mo)s(del)30 b(where)g(w)m(e)i(only)e(allo)m(w)g(Ev)m(e)i(to)f(run)f (for)h(p)s(olynomial)d(time.)42 b(This)29 b(is)h(a)h(more)g(realistic)f (mo)s(del)150 2045 y(an)m(yw)m(a)m(y)k(-)f(after)g(all,)g(the)f(NSA)h (ma)m(y)g(b)s(e)f(v)m(ery)h(ric)m(h)f(and)g(p)s(o)m(w)m(erful,)g(but)g (they)h(are)g(computationally)150 2158 y(b)s(ounded!)38 b(Ho)m(w)m(ev)m(er,)32 b(to)d(b)s(e)g(fair,)f(w)m(e)i(should)d(also)i (assume)g(that)g(Alice)g(and)f(Bob)h(are)h(computation-)150 2271 y(ally)c(b)s(ounded.)38 b(It)27 b(w)m(ouldn't)f(b)s(e)h(a)g(v)m (ery)h(useful)e(mo)s(del)g(if)g(w)m(e)i(assumed)e(that)i(the)f(adv)m (ersary)h(w)m(as)g Fp(less)150 2384 y Fr(p)s(o)m(w)m(erful)h(than)h (the)h(pla)m(y)m(ers.)291 2585 y(Note)44 b(that)e(observ)-5 b(ation)42 b(1)h(still)d(holds)h(when)g(the)h(adv)m(ersary)h(is)e (computationally)g(b)s(ounded.)150 2698 y(If)c(Ev)m(e)h(kno)m(ws)f(ev)m (erything)g(that)g(Alice)g(kno)m(ws,)i(Ev)m(e)f(can)f(simply)e(run)h (the)h(decryption)f(algorithm)150 2811 y(\(whic)m(h)30 b(is)f(e\016cien)m(t,)i(since)f(Alice)g(is)g(also)g(b)s(ounded\),)f (and)h(reco)m(v)m(er)i(the)f(message.)42 b(But,)31 b(what)f(ab)s(out) 150 2924 y(observ)-5 b(ation)30 b(2?)291 3125 y(Observ)-5 b(ation)40 b(2)h(do)s(es)g(not)g(hold,)h(since)e(Ev)m(e)i(can)f(no)g (longer)f(try)h(ev)m(ery)g(message.)74 b(It)41 b(is)f(v)m(ery)150 3238 y(reasonable)h(to)h(assume)f(that)h(the)f(message)i(space)e(is)g (exp)s(onen)m(tially)e(large)i(\(for)h(example,)i(if)c(the)150 3351 y(p)s(ossible)31 b(messages)k(are)f(all)f(in)f Fm(f)p Fr(0)p Fn(;)15 b Fr(1)p Fm(g)1498 3318 y Fl(k)1543 3351 y Fr(\),)35 b(so)f(Ev)m(e)h(cannot)f(simply)d(run)i(this)f(attac)m(k.) 54 b(Ho)m(w)m(ev)m(er,)37 b(just)150 3463 y(b)s(ecause)e(observ)-5 b(ation)34 b(2)h(do)s(es)f(not)h(hold)e(do)s(es)h Fp(not)h Fr(mean)g(there)g(are)g(de\014nitely)d(secure)j(encryption)150 3576 y(sc)m(hemes!)51 b(Unfortunately)-8 b(,)35 b(there)f(is)f(a)h(lot) g(w)m(e)g(don't)g(kno)m(w)f(in)g(complexit)m(y)g(theory)-8 b(,)36 b(so)e(w)m(e)g(ha)m(v)m(e)h(to)150 3689 y(mak)m(e)d(some)f (assumptions.)40 b(One)30 b(assumption)f(w)m(e)i(will)d(probably)h(ha)m (v)m(e)j(to)f(mak)m(e)h(is)d(that)j Fn(P)38 b Fm(6)p Fr(=)26 b Fn(N)10 b(P)j Fr(.)150 3802 y(\(F)-8 b(or)32 b(example,)e(if)f(Ev)m(e)i(kno)m(ws)g(ev)m(erything)f(Bob)h(kno)m(ws)f (except)i(for)e Fn(m)p Fr(,)g(it)g(w)m(ould)f(b)s(e)h(easy)h(to)h(v)m (erify)150 3915 y(that)j(a)g(string)f Fn(w)j Fr(con)m(tained)e(all)f (of)h(Alice's)f(priv)-5 b(ate)34 b(information,)g(since)g(decryption)g (w)m(ould)g(w)m(ork.)150 4028 y(Th)m(us,)44 b(if)d(Ev)m(e)i(can)g(run)d (an)j(NP)f(algorithm,)i(she)e(can)g(break)h(the)f(cryptosystem\).)77 b(Ho)m(w)m(ev)m(er,)47 b(in)150 4141 y(practice)31 b(w)m(e)g(m)m(ust)f (mak)m(e)h(ev)m(en)g(more)g(assumptions.)291 4342 y(So,)g(can)f(w)m(e)h (construct)g(a)g(cryptosystem)f(where)g(Alice)g(and)g(Bob)h(share)f(no) g(secret?)150 4626 y Fo(5)135 b(Di\016e-Hellman)47 b(Key)e(Exc)l(hange) 150 4828 y Fr(So)26 b(ho)m(w)g(can)h(w)m(e)g(construct)f(a)h (cryptosystem)f(where)g(Alice)g(and)f(Bob)i(don't)f(share)g(an)m(y)h (secrets?)39 b(The)150 4941 y(\014rst)f(result)f(to)i(answ)m(er)g(this) e(question)h(w)m(as)g(\\New)h(Directions)f(in)f(Cryptograph)m(y)-8 b(,")41 b(b)m(y)e(Whit\014eld)150 5054 y(Di\016e)d(and)f(Martin)g (Hellman,)h(published)c(in)j(1976.)59 b(They)35 b(do)h(not)g(precisely) e(solv)m(e)j(this)d(problem,)150 5167 y(but)c(solv)m(e)g(a)h(similar)d (one)j(with)e(t)m(w)m(o)i(di\013erences:)1889 5416 y(1-4)p eop %%Page: 5 5 5 4 bop 261 159 a Fr(1.)46 b(Alice)37 b(and)g(Bob)g(ma)m(y)h(exc)m (hange)h(m)m(ultiple)c(messages)k(\(Ev)m(e)f(gets)g(to)h(see)f(all)e (the)h(messages\),)377 271 y(and)261 457 y(2.)46 b(A)m(t)32 b(the)f(end)f(of)h(the)g(proto)s(col,)g(Alice)f(and)g(Bob)h(ha)m(v)m(e) h(a)f(shared)f(secret.)43 b(There)30 b(is)g(no)h(message)377 570 y(comm)m(unicated.)291 778 y(Ho)m(w)m(ev)m(er,)44 b(part)39 b(of)g(the)g(idea)g(is)f(that)i(once)g(Alice)e(and)h(Bob)g (share)g(a)g(secret,)k(the)c(problem)f(of)150 891 y(encrypting)27 b(a)h(message)h(b)s(ecomes)f(m)m(uc)m(h)g(easier.)40 b(This)26 b(is)g(called)h(a)i Fp(key)h(exchange)g(pr)-5 b(oto)g(c)g(ol)32 b Fr(since)27 b(the)150 1004 y(end)i(result)g(is)g (that)h(Alice)f(and)g(Bob)h(ha)m(v)m(e)h(a)f(shared)f(secret)i(k)m(ey) -8 b(.)42 b(So,)30 b(ho)m(w)g(do)s(es)f(the)h(Di\016e-Hellman)150 1117 y(proto)s(col)g(\(hereafter)h(referred)f(to)h(as)g(\\DH"\))h(w)m (ork?)41 b(Let)30 b Fn(k)k Fr(b)s(e)c(a)g(common)h(\\securit)m(y)g (parameter.")261 1325 y(1.)46 b(Alice)30 b(pic)m(ks)g(a)h(random)e Fn(k)s Fr(-bit)h(prime)f Fn(p)p Fr(.)261 1511 y(2.)46 b(Alice)27 b(pic)m(ks)h(a)g(random)f(generator)i Fn(g)f Fm(2)d Fn(Z)1862 1478 y Fk(\003)1855 1533 y Fl(p)1901 1511 y Fr(,)k(that)f(is,)g(a)g(n)m(um)m(b)s(er)e Fn(g)j Fm(2)c(f)p Fr(1)p Fn(;)15 b(:)g(:)g(:)j(;)d(p)g Fm(\000)g Fr(1)p Fm(g)28 b Fr(suc)m(h)f(that)377 1624 y(the)k(t)m(w)m(o)h(sets)e Fm(f)p Fr(1)p Fn(;)15 b(:)g(:)g(:)j(;)d(p)20 b Fm(\000)g Fr(1)p Fm(g)31 b Fr(and)f Fm(f)p Fn(g)s(;)15 b(g)1808 1591 y Fh(2)1850 1624 y Fn(;)g(:)g(:)g(:)h(;)f(g)2097 1591 y Fl(p)p Fk(\000)p Fh(1)2228 1624 y Fm(g)31 b Fr(are)g(equal.)261 1810 y(3.)46 b(Alice)30 b(pic)m(ks)g(a)h(random)e(n)m(um)m(b)s(er)g Fn(x)c Fm(2)g Fn(Z)1807 1777 y Fk(\003)1800 1832 y Fl(p)1846 1810 y Fr(.)261 1996 y(4.)46 b(Alice)30 b(sends)g Fn(p;)15 b(g)s(;)31 b Fr(and)f Fn(a)25 b Fr(=)g Fn(g)1431 1963 y Fl(x)1501 1996 y Fr(mo)s(d)f Fn(p)30 b Fr(to)h(Bob.)261 2182 y(5.)46 b(Bob)31 b(pic)m(ks)f(a)g(random)g(n)m(um)m(b)s(er)f Fn(y)f Fm(2)d Fn(Z)1764 2149 y Fk(\003)1757 2204 y Fl(p)1803 2182 y Fr(.)261 2368 y(6.)46 b(Bob)31 b(sends)e Fn(b)d Fr(=)f Fn(g)1018 2335 y Fl(y)1085 2368 y Fr(mo)s(d)f Fn(p)30 b Fr(to)i(Alice.)261 2554 y(7.)46 b(Alice)30 b(computes)h Fn(s)24 b Fr(=)h Fn(b)1214 2521 y Fl(x)1284 2554 y Fr(mo)s(d)f Fn(p)p Fr(,)30 b(and)g(Bob)h(computes)f Fn(s)25 b Fr(=)g Fn(a)2568 2521 y Fl(y)2635 2554 y Fr(mo)s(d)f Fn(p)p Fr(.)291 2762 y(Since)30 b Fn(a)d Fr(=)g Fn(g)748 2729 y Fl(x)823 2762 y Fr(and)k Fn(b)c Fr(=)f Fn(g)1210 2729 y Fl(y)1283 2762 y Fr(w)m(e)32 b(see)g(that)g(\()p Fn(g)1845 2729 y Fl(x)1889 2762 y Fr(\))1924 2729 y Fl(y)1993 2762 y Fr(=)26 b Fn(g)2136 2729 y Fl(xy)2245 2762 y Fr(=)g(\()p Fn(g)2423 2729 y Fl(y)2466 2762 y Fr(\))2501 2729 y Fl(x)2576 2762 y Fr(so)32 b(Alice)e(and)h(Bob)g(do)h(agree)g(on)150 2875 y(their)d(v)-5 b(alues.)150 3117 y Ff(5.1)112 b(E\016ciency)37 b(of)h(DH)150 3289 y Fr(Since)33 b(w)m(e)i(are)g(in)d(the)j (computationally)e(b)s(ounded)f(mo)s(del,)i(w)m(e)h(not)f(only)g(need)g (to)h(mak)m(e)g(sure)e(that)150 3402 y(the)f(proto)s(col)g(is)f (correct,)j(but)e(that)g(it)g(is)f(e\016cien)m(t.)46 b(There)32 b(are)g(basically)f(three)h(things)f(w)m(e)h(need)g(to)150 3515 y(v)m(erify)e(can)g(b)s(e)g(done)g(e\016cien)m(tly)-8 b(.)150 3724 y Fs(1.)44 b(Generate)27 b(a)g(random)g Fn(k)s Fs(-bit)g(prime.)38 b Fr(Alice)23 b(do)s(es)h(this)f(in)f(step)i (1.)39 b(W)-8 b(e)25 b(can)f(certainly)f(generate)150 3837 y Fn(k)s Fr(-bit)i(n)m(um)m(b)s(ers)g(e\016cien)m(tly)-8 b(,)27 b(and)e(the)i(prime)d(n)m(um)m(b)s(er)g(theorem)j(tells)e(us)g (that)h(the)h(probabilit)m(y)c(that)k(a)150 3950 y(random)h Fn(k)s Fr(-bit)g(n)m(um)m(b)s(er)f(is)g(prime)g(is)h(ab)s(out)g(1)p Fn(=k)s Fr(.)41 b(Ho)m(w)m(ev)m(er,)31 b(the)e(problem)e(of)h (determining)f(whether)150 4063 y(a)j(giv)m(en)f(n)m(um)m(b)s(er)g(is)f (prime)g(is)h(a)h(di\016cult)d(one.)41 b(As)29 b(it)g(turns)g(out,)h (there)f(are)h(deterministic)e(metho)s(ds)150 4176 y(whic)m(h)41 b(solv)m(e)i(this)e(problem,)j(but)e(probabilistic)d(metho)s(ds)i(are)i (far)f(simpler.)75 b(It)42 b(su\016ces)g(for)g(our)150 4289 y(purp)s(oses)33 b(to)j(state)g(that)f(there)g(are)h(algorithms)d (whic)m(h)h(are)h Fn(R)q(P)13 b Fr(,)36 b(for)f(generating)g(a)g (random)f Fn(k)s Fr(-bit)150 4402 y(prime)e(n)m(um)m(b)s(er,)h(that)h (is,)f(their)f(running)f(time)i(is)f(exp)s(ected)i(to)g(b)s(e)e(p)s (olynomial)f(\(in)h(1)3241 4369 y Fl(k)7 b Fh(1)3324 4402 y Fr(,)34 b(and)f(their)150 4514 y(output)d(is)f(alw)m(a)m(ys)i(a) g Fn(k)s Fr(-bit)f(prime)f(n)m(um)m(b)s(er.)150 4724 y Fs(2.)83 b(Generate)47 b(a)g(random)f(generator)i(for)f Fn(Z)2001 4691 y Fk(\003)1994 4746 y Fl(p)2040 4724 y Fs(.)72 b Fr(This)39 b(is)h(also)h(solv)-5 b(able)40 b(e\016cien)m(tly)-8 b(,)44 b(but)d(the)150 4837 y(metho)s(ds)30 b(are)g(complicated)g(and)g(w)m(e)h(need)f(not)h(get)g(in)m(to)g(them)f (no)m(w.)p 150 4897 1440 4 v 254 4953 a Fe(1)288 4985 y Fd(W)-6 b(e)27 b(write)h(1)658 4953 y Fg(k)723 4985 y Fd(instead)g(of)g Fc(k)h Fd(b)r(ecause)f(the)e(p)r(olynomial)i(in)f (p)r(olynomial)g(time)f(is)i(considered)g(on)f(the)f Fb(length)i Fd(of)g(the)150 5076 y(inputs.)47 b(The)30 b(length)g(of)h(1)959 5044 y Fg(k)1028 5076 y Fd(is)f Fc(k)r Fd(,)i(so)f(this)f(giv)n(es)g(us)g(what)h(w)n(e)f(w)n(an)n(t;)j (on)d(the)f(other)h(hand,)h(the)f(length)g(of)h Fc(k)h Fd(is)f(log)14 b Fc(k)r Fd(,)150 5167 y(whic)n(h)26 b(is)g Fb(not)g Fd(what)g(w)n(e)g(w)n(an)n(t.)1889 5416 y Fr(1-5)p eop %%Page: 6 6 6 5 bop 150 159 a Fs(3.)44 b(Giv)m(en)28 b Fn(x;)15 b(y)s(;)g(n)p Fs(,)29 b(compute)e Fn(x)1335 126 y Fl(y)1401 159 y Fr(mo)s(d)e Fn(n)p Fs(.)38 b Fr(This)22 b(is)g(a)i(bit)f(tric)m(ky)-8 b(.)39 b(First)23 b(of)h(all,)g(it)g(is)e(not)i(p)s(olynomial-)150 271 y(time)d(to)h(compute)g Fn(x)862 238 y Fl(y)925 271 y Fr(and)e(then)h(reduce)h(this)e(mo)s(dulo)g Fn(n)p Fr(,)j(since)d Fn(x)2423 238 y Fl(y)2486 271 y Fr(is)h(to)s(o)h(long)f (for)g(a)h(p)s(olynomial-time)150 384 y(algorithm)29 b(to)j(compute.)1039 351 y Fh(2)1119 384 y Fr(So,)e(w)m(e)h(migh)m(t)f (try)h(this)e(simple)f(algorithm:)150 567 y(Mo)s(dExp\()p Fn(x;)15 b(y)s(;)g(n)p Fr(\):)150 680 y(1.)41 b(Set)31 b Fn(a)25 b Fr(:=)g(1.)150 793 y(2.)41 b(F)-8 b(or)31 b Fn(b)26 b Fr(:=)f(1)30 b(to)i Fn(y)h Fr(do)355 906 y(2a.)41 b Fn(a)25 b Fr(:=)h Fn(a)20 b Fm(\003)h Fn(x)355 1019 y Fr(2b.)40 b Fn(a)25 b Fr(:=)h Fn(a)f Fr(mo)s(d)f Fn(n)150 1132 y Fr(3.)41 b(Output)29 b Fn(a)291 1315 y Fr(Ho)m(w)m(ev)m(er,)j(this)c(is)g(not)h(e\016cien)m(t)h(either.)40 b(Step)29 b(2)g(will)e(run)g Fn(y)32 b Fr(times,)d(whic)m(h)f(is)g(not) i(p)s(olynomial)c(in)150 1428 y(the)31 b(length)e(of)i Fn(y)s Fr(.)40 b(So,)31 b(w)m(e)g(ha)m(v)m(e)g(to)g(b)s(e)f(a)h(little) e(more)i(clev)m(er.)41 b(Let)31 b Fn(y)2553 1442 y Fl(i)2611 1428 y Fr(b)s(e)e(the)i Fn(i)p Fr(th)g(bit)e(of)h Fn(y)s Fr(.)150 1610 y(Mo)s(dExp\()p Fn(x;)15 b(y)s(;)g(n)p Fr(\):)150 1723 y(1.)41 b(Set)31 b Fn(a)25 b Fr(:=)g(1.)150 1836 y(2.)41 b(Set)31 b Fn(s)461 1850 y Fh(1)525 1836 y Fr(:=)25 b Fn(x)p Fr(.)150 1949 y(3.)41 b(F)-8 b(or)31 b Fn(b)26 b Fr(:=)f(1)30 b(to)i Fn(k)23 b Fm(\000)d Fr(1)31 b(do)355 2062 y(3a.)41 b Fn(s)554 2077 y Fl(b)p Fh(+1)703 2062 y Fr(:=)26 b Fn(s)868 2077 y Fl(b)922 2062 y Fm(\003)21 b Fn(s)1031 2077 y Fl(b)355 2175 y Fr(3b.)40 b Fn(s)559 2190 y Fl(b)p Fh(+1)709 2175 y Fr(:=)25 b Fn(s)873 2190 y Fl(b)p Fh(+1)1022 2175 y Fr(mo)s(d)g Fn(n)150 2288 y Fr(4.)41 b(F)-8 b(or)31 b Fn(b)26 b Fr(:=)f(1)30 b(to)i Fn(k)h Fr(do)355 2401 y(4a.)41 b(If)30 b Fn(y)647 2416 y Fl(b)706 2401 y Fr(==)25 b(1)31 b(then)559 2514 y Fn(a)25 b Fr(:=)h Fn(a)20 b Fm(\003)h Fn(s)931 2529 y Fl(b)990 2514 y Fr(mo)s(d)j Fn(n)150 2627 y Fr(5.)41 b(Output)29 b Fn(a)p Fr(.)291 2810 y(Y)-8 b(ou)26 b(can)f(v)m(erify)g(that)h(this)e (is)g(correct)i(if)f Fn(y)j Fr(is)c(no)h(more)g(than)g Fn(k)k Fr(bits)24 b(long,)i(and)e(that)i(it)f(is)f(e\016cien)m(t.)150 3048 y Ff(5.2)112 b(Securit)m(y)37 b(of)g(DH)150 3220 y Fr(The)32 b(next)i(question)e(is,)h(ho)m(w)g(secret)h(is)e Fn(g)1624 3187 y Fl(xy)1705 3220 y Fr(,)i(the)f(secret)h(that)g(Alice)e (and)h(Bob)g(share)g(at)g(the)g(end)g(of)150 3333 y(the)j(proto)s(col?) 58 b(Di\016e)37 b(and)e(Hellman)g(w)m(eren't)i(exactly)g(sure,)g(but)f (they)g(couldn't)g(\014nd)f(an)h(e\016cen)m(t)150 3446 y(w)m(a)m(y)k(for)e(Ev)m(e)h(to)g(break)g(the)g(sc)m(heme.)66 b(So,)40 b(the)f(form)m(ulated)f(the)h(answ)m(er)f(to)h(the)g(question) f(of)h(the)150 3559 y(securit)m(y)30 b(of)h(their)e(sc)m(heme)i(lik)m (e)f(this:)291 3751 y(\\If)38 b(Ev)m(e)h(can)f(e\016cien)m(tly)g(reco)m (v)m(er)h(the)g(secret,)i(she)c(can)i(solv)m(e)f(the)g(follo)m(wing)f (problem:)54 b(Giv)m(en)150 3864 y Fn(p;)15 b(g)s(;)g(a;)29 b Fr(and)e Fn(b)p Fr(,)h(compute)g Fn(c)g Fr(suc)m(h)f(that)h(there)g (is)e(an)i Fn(x)f Fr(and)g(a)h Fn(y)i Fr(where)d Fn(a)f Fr(=)f Fn(g)2838 3831 y Fl(x)2907 3864 y Fr(mo)s(d)g Fn(p)p Fr(,)j Fn(b)d Fr(=)g Fn(g)3412 3831 y Fl(y)3480 3864 y Fr(mo)s(d)f Fn(p)p Fr(,)150 3977 y(and)30 b Fn(c)25 b Fr(=)g Fn(g)533 3944 y Fl(xy)640 3977 y Fr(mo)s(d)g Fn(p)p Fr(.")291 4090 y(No)m(w,)31 b(if)e(w)m(e)i(assume)e(that)i(this) e(problem)f(is)h(hard)g(to)i(solv)m(e,)g(there)f(can)g(b)s(e)g(no)g (successful)e(adv)m(er-)150 4203 y(sary)-8 b(.)40 b(This)26 b(amoun)m(ts)h(to)h(Di\016e)g(and)e(Hellman)h(sa)m(ying)g (\\Di\016e-Hellman)g(is)f(secure)i(if)e(Di\016e-Hellman)150 4316 y(is)31 b(secure")i(but)f(they)g(can)h(get)g(a)m(w)m(a)m(y)h(with) d(that)i(since)e(they)i(w)m(ere)g(the)f(\014rst)f(ones)i(to)g(come)g (up)e(with)150 4429 y(this)e(kind)g(of)h(sc)m(heme.)940 4396 y Fh(3)291 4622 y Fr(Ho)m(w)m(ev)m(er,)39 b(there)c(are)h(t)m(w)m 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1326 y(2.)46 b(Alice)30 b(computes)h Fn(')p Fr(\()p Fn(n)p Fr(\))26 b(=)f(\()p Fn(p)20 b Fm(\000)g Fr(1\)\()p Fn(q)k Fm(\000)c Fr(1\),)31 b(the)g(Euler)e(Phi)g(function.) 2838 1293 y Fh(5)261 1514 y Fr(3.)46 b(Alice)32 b(pic)m(ks)f(a)i (random)e Fn(e)i Fr(suc)m(h)e(that)i Fn(g)s(cd)p Fr(\()p Fn(e;)15 b(')p Fr(\()p Fn(n)p Fr(\)\))32 b(=)c(1)k(\(that)h(is,)f Fn(e)h Fr(and)e Fn(')p Fr(\()p Fn(n)p Fr(\))i(are)g(relativ)m(ely)377 1627 y(prime.)261 1814 y(4.)46 b(Alice)30 b(computes)h Fn(d)f Fr(suc)m(h)g(that)h Fn(ed)26 b Fr(=)f(1)h(mo)s(d)e Fn(')p Fr(\()p Fn(n)p Fr(\).)261 2002 y(5.)46 b(Alice)30 b(publishes)d Fn(n;)15 b(e)31 b Fr(as)f(her)g(public)e(k)m(ey)-8 b(,)32 b(and)d(k)m(eeps)i Fn(d)g Fr(as)g(her)e(secret)j(k)m(ey)-8 b(.)291 2214 y(No)m(w,)35 b(for)e(Bob)h(to)g(encrypt)f(a)g(secret)h Fn(s)p Fr(,)g(he)f(simply)e(computes)i Fn(c)e Fr(=)f Fn(s)2791 2182 y Fl(e)2852 2214 y Fr(mo)s(d)25 b Fn(n)p Fr(.)49 b(Alice)32 b(decrypts)150 2327 y(b)m(y)c(computing)g Fn(s)d Fr(=)g Fn(c)925 2294 y Fl(d)991 2327 y Fr(mo)s(d)f Fn(n)p Fr(.)40 b(T)-8 b(o)29 b(see)g(that)g(the)g Fn(s)f Fr(Alice)g(gets)h(is)f(the)g(same)h(as)g(the)f Fn(s)g Fr(Bob)h(used,)f(w)m(e)150 2440 y(call)i(on)g(a)h(fact)g(from)f(n)m(um) m(b)s(er)f(theory:)150 2653 y Fp(F)-7 b(act)28 b(1.)39 b Fr(F)-8 b(or)25 b(an)m(y)g Fn(s)f Fr(suc)m(h)g(that)h Fn(g)s(cd)p Fr(\()p Fn(s;)15 b(n)p Fr(\))26 b(=)f(1)1748 2620 y Fh(6)1788 2653 y Fr(,)h Fn(s)1882 2620 y Fl(')p Fh(\()p Fl(n)p Fh(\))2055 2653 y Fr(=)f(1)g(mo)s(d)g Fn(n)p Fr(.)38 b(This)23 b(is)g(true)h(for)h(an)m(y)f Fn(n)p Fr(,)i(actually)-8 b(,)150 2766 y(not)31 b(just)e(n)m(um)m(b)s (ers)g(whic)m(h)g(are)i(the)g(pro)s(ducts)e(of)h(t)m(w)m(o)i(primes.) 291 2879 y(No)m(w,)f(since)f Fn(ed)c Fr(=)f(1)h(mo)s(d)e Fn(')p Fr(\()p Fn(n)p Fr(\),)32 b(there)e(is)g(an)g(in)m(teger)h Fn(l)h Fr(suc)m(h)e(that)h Fn(ed)26 b Fr(=)f Fn(l)r(')p Fr(\()p Fn(n)p Fr(\))c(+)e(1,)31 b(so)1539 3196 y(\()p Fn(s)1617 3158 y Fl(e)1653 3196 y Fr(\))1688 3158 y Fl(d)1812 3196 y Fr(=)83 b 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b(this)g(is)g(v)n(ery)f(unlik)n(ely)-6 b(,)25 b(so)h(w)n(e)g(w)n(on't)g (w)n(orry)g(ab)r(out)g(it.)1889 5416 y Fr(1-7)p eop %%Page: 8 8 8 7 bop 150 159 a Ff(6.1)112 b(Securit)m(y)37 b(of)g(RSA)150 330 y Fr(Ho)m(w)j(hard)e(is)g(it)h(for)f(Ev)m(e)i(to)g(learn)e Fn(s)h Fr(when)f(she)h(sees)g(only)f Fn(c;)15 b(n;)40 b Fr(and)f Fn(e)p Fr(?)66 b(W)-8 b(ell,)42 b(since)c(RSA)h(w)m(as)150 443 y(one)32 b(of)g(the)g(v)m(ery)g(early)f(pap)s(ers,)g(they)h(to)s(o) g(mak)m(e)h(an)f(analysis)e(of)i(their)f(securit)m(y)g(similar)e(to)j (that)h(of)150 556 y(Di\016e-Hellman.)40 b(Namely)-8 b(,)31 b(they)f(sa)m(y:)150 768 y(\\If)h(Ev)m(e)g(can)h(reco)m(v)m(er)g Fn(s)f Fr(then)f(there)i(is)e(an)g(e\016cien)m(t)i(solution)d(to)j (this)e(problem:)40 b(Giv)m(en)31 b Fn(n;)15 b(e)p Fr(,)31 b(and)g Fn(u)p Fr(,)150 881 y(compute)g Fn(v)i Fr(suc)m(h)d(that)h Fn(v)1045 848 y Fl(e)1108 881 y Fr(=)25 b Fn(u)g Fr(mo)s(d)g Fn(n)p Fr(.")291 1094 y(This)e(problem)f(is)i(called)f(the)i(RSA)f 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b(to)h(factoring,)g(y)m(ou)f(w)m(ould)g(ha)m(v)m(e)h(an)f (instan)m(t)h(PhD!\))1889 5416 y(1-8)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF scribe/scribe_template.tex0100664003325600234200000005057607347175751015310 0ustar mliskovtoc\iffalse **************************** INSTRUCTIONS ***************************** 1. Fill in the parts that say ********* INSERT ____ HERE ***** below. You will have to come up with a good lecture title. 2. Remove everything after that line down to the last line, which says \end{document} Make sure you leave the \end{document} in place. 3. Some stylistic points: Use ``\section{SECTION NAME}'' to mark each seperate section of the lecture. You can use \subsection and \subsubsection to further break down the topics. 4. To insert a footnote, use ``\footnote{FOOTNOTE TEXT}'' but make sure you do NOT leave a space between the word or punctuation you wish the footnote mark to follow and the backslash. If you do it will look funny. 5. Use ``\bigskip \noindent'' to start any new theorem, proof, remark, fact, et cetera. This puts extra space in to make things easier on the eye. 6. Use boldface ({\bf TEXT}) for announcing theorems, lemmas, or correlaries, and italics ({\it TEXT}) for proofs, remarks, facts, et cetera. 7. If you want to have a sequence of equations where the ='s are aligned, do the following: \begin{eqnarray*} x &=& y\\ y &=& z\\ z &=& w \end{eqnarray*} For this command, you do NOT have to put things in math mode. 8. Try to keep your notation consistent with notation used in the previous scribe notes. 9. Most LaTeX things you'd want to do I used at some point in this file, which is the scribe notes for lecture 1. If you see something in there you want to duplicate you should be able to find it in this file. If you want to do something else, consult a LaTeX reference.. it's the best way to learn! But if you must, email the TA for help. ********************************* END OF INSTRUCTIONS ******************** \fi \documentclass[11pt]{article} \usepackage{amsfonts} \usepackage{latexsym} \setlength{\oddsidemargin}{.25in} \setlength{\evensidemargin}{.25in} \setlength{\textwidth}{6in} \setlength{\topmargin}{-0.4in} \setlength{\textheight}{8.5in} \newcommand{\handout}[5]{ \renewcommand{\thepage}{#1-\arabic{page}} \noindent \begin{center} \framebox{ \vbox{ \hbox to 5.78in { {\bf 6.875J/18.425J Cryptography and Cryptanalysis} \hfill #2 } \vspace{4mm} \hbox to 5.78in { {\Large \hfill #5 \hfill} } \vspace{2mm} \hbox to 5.78in { {\it #3 \hfill #4} } } } \end{center} \vspace*{4mm} } \newcommand{\ho}[5]{\handout{#1}{#2}{Instructor: #3}{Teaching Assistant: #4}{Handout #1: #5}} \newcommand{\scribe}[6]{\handout{#1}{#2}{Instructor: #3} {Teaching Assistant: #4}{Lecture #1: #5} Scribed by: #6} \begin{document} \scribe{********* INSERT LECTURE NUMBER HERE *******} {******* INSERT LECTURE DATE HERE******} {Anna Lysyanskaya}{Moses Liskov} {******* INSERT LECTURE TITLE HERE*****} {************ INSERT YOUR NAME HERE ******************} \section {The Problem} Suppose there are two people, Alice and Bob, and Bob wants to send a love-letter to Alice. Unfortunately, there is an evesdropper, Eve, who wants to read the love-letter. We want to find some way for Bob to send the letter to Alice without Eve being able to understand it. To formally describe the process, \begin{enumerate} \item Bob chooses a message $m$. Since we want Eve to be unsure of what message is being sent, the message must be chosen from a probability distribution. We use the symbol ${\cal M}$ to represent the {\em message space}, and write $$m \leftarrow {\cal M}$$ \item Bob applies the encryption algorithm $E$ to $m$, the {\em plaintext}, to get $c$, the {\em ciphertext}. What inputs does $E$ take? Well, {\it i.} Certainly $m$ will be an input. {\it ii.} Perhaps a shared secret key $s$ will be an input. {\it iii.} Perhaps a piece of public info $PK$ will be an input. {\it iv.} Perhaps $E$ will be randomized, so a random string $r$ may be another input. So Bob computes $c \leftarrow E(m, s, PK, r)$ and sends $c$ to Alice. \item Alice has a decryption algorithm $D$. What inputs does $D$ take? {\it i.} Certainly, $c$ will be an input. {\it ii.} Perhaps a shared secret key $s$ will be an input. {\it iii.} Perhaps a piece of public info $PK$ will be an input. {\it iv.} Perhaps Alice has a secret of her own, $SK$. Alice finally computes $m \leftarrow D(c, s, PK, SK)$. \end{enumerate} Since Eve gets to observe the communication between Alice and Bob, she clearly gets to know $c$. However, we are sure she doesn't get to know $m$, because otherwise the whole exercise would be pointless. But what about the other information? \bigskip\noindent {\it Observation 1:} Alice must know {\em something} Eve doesn't know. If Eve knows everything Alice knows, Eve can just compute $m \leftarrow D(c, s, PK, SK)$. Thus, Eve either learns $m$ or Alice can't decrypt $c$. \bigskip What about the rest of the information? Is it okay for Eve to know everything {\em Bob} knows? First of all, we have to define what Eve is. At the moment, let's assume nothing about Eve except that she is an algorithm. Specifically, we are not assuming any bounds on her computational power - she can take arbitrary amounts of time or use arbitrary amounts of space. \bigskip\noindent {\it Observation 2:} Bob must know {\em something} Eve doesn't know if encryption is to be achieved when Eve has no computational bound on her power. If Eve knows everything Bob knows, she can just try every possible message $m \in {\cal M}$, calculating $E(m, s, PK, r)$ every time until she gets $c$. Whatever $m$ produced $c$ would be the correct message (there cannot be any ambivalence if Alice can decrypt). \bigskip Note that for this method of attack to work, Eve really has to be a powerful adversary. Such a thing would not be possible if Eve were computationally bounded since there are probably exponentially many possible messages to try. Observation 1 and observation 2 lead us to conclude: \bigskip\noindent {\it Observation 3:} If Eve unbounded, Alice and Bob must both have secrets. \bigskip So the question is now, if Alice and Bob share a secret, can they achieve privacy? As we will see, the answer is a qualified ``yes.'' \section {The One-Time Pad} Here is the one-time pad cryptosystem. We assume that a parameter $k$ is known to everyone beforehand, and that ${\cal M}$ draws messages from the space $\{0,1\}^k$. Let ${\cal S}$ be the uniform distribution on $\{0,1\}^k$. Now, if we let $s \leftarrow {\cal S}$ be the shared secret key, we can define encryption and decryption as follows: $$E(m, s) = m \oplus s,$$ and $$D(c, s) = c \oplus s,$$ where $x \oplus y$ refers to the bitwise XOR operation. The correctness of the decryption should be obvious. We will now prove that this system is secure. Let $m_P$ be the event that message $m$ is the one chosen from ${\cal M}$ to send (P stands for ``plaintext''). Let $c_C$ be the event that ciphertext $c$ is the one sent from Bob to Alice. \bigskip\noindent {\bf Theorem 1 (Vernon):} $\forall m \; \forall c, Pr[m_P] = Pr[m_P|c_C]$. That is, the events $m_P$ and $c_C$ are independent events. \bigskip\noindent {\it Proof.} First, we note that $Pr[m_P|c_C] = Pr[m_P \wedge c_C] / Pr[c_C].$ Now, we can derive \begin{eqnarray*} Pr[m_P \wedge c_C] & = & Pr[m_P] Pr[c_C | m_P] \\ & = & Pr[m_P] Pr[s = c \oplus m | m_P] \\ & = & Pr[m_P] 2^{-k} \end{eqnarray*} since $s$ is chosen independently of $m$. However, we also have that \begin{eqnarray*} Pr[c_C] & = & \sum_{m' \in \{0,1\}^k} Pr[m'_P] Pr[c_C | m'_P]\\ & = & \sum_{m' \in \{0,1\}^k} Pr[m'_P] 2^{-k}\\ & = & 2^{-k} \sum_{m' \in \{0,1\}^k} Pr[m'_P]\\ & = & 2^{-k} \end{eqnarray*} Thus, we get \begin{eqnarray*} Pr[m_P | c_C] & = & \frac{Pr[m_P \wedge c_C]}{Pr[c_C]} \\ & = & \frac{2^{-k} Pr[m_P]}{2^{-k}} \\ & = & Pr[m_P], \end{eqnarray*} which completes the proof. \bigskip This is wonderful! Since the ciphertext is independent of the message, Eve cannot learn the message from observing the ciphertext. But there's a problem: {\em $s$ can only be used once!} In our scenario, only one message is being sent so this is not a security problem, but rather an efficiency problem. If Alice and Bob must get together once for each message they are to send, why bother having cryptography? Now, Alice and Bob could get together once and establish {\em many} secret keys, but this is certainly a problem with this system. But maybe we can do better than this? \section {Shannon's Theorem} As it turns out, we can't. Shannon's Theorem is that if Alice and Bob share only a secret of length $k$, there is {\em no} method which can securely communicate messages of length $k+1$. We should be careful about saying ``securely communicate'' without knowing what it means. For this theorem, a system is ``secure'' if it meets the above definition, namely that for any message $m$ and ciphertext $c$, $Pr[m_P] = Pr[m_P | c_C]$. \bigskip\noindent {\bf Theorem 2 (Shannon):} If $|{\cal M}| > |{\cal S}|$ then $\exists m, c, \; Pr[m_P] \neq Pr[m_P | c_C]$. \bigskip\noindent {\it Proof.} Suppose not. That is, suppose that $\forall m, c, \; Pr[m_P] = Pr[m_P | c_C]$. Then, we know that $$\forall m, m' \in {\cal M}, s \in {\cal S}, \; \exists s': E(m,s) = E(m', s') \wedge s' \neq s.$$ If this was not true, then there would be a contradiction, since $Pr[m'_P] > 0$ (since $m' \in {\cal M}$, it is a possible message), but $Pr[m'_P | (E(m,s))_C] = 0 \neq Pr[m'_P]$. Clearly, $s' \neq s$ if they exist. Otherwise, $E(m,s) = E(m',s)$, and then Alice cannot compute $D(E(m,s), s)$. Now we can see there must be as many keys as there are plaintexts, thus contradicting that $|{\cal M}| > |{\cal S}|$. This completes the proof. \section {The Computationally Bounded Adversary Model} Shannon's theorem proves that we cannot do better than the one-time pad if we want security. However, the security the one-time pad provides allows the adversary to be very powerful indeed. So instead of simply going home, we ought to see if we can do better in the model where we only allow Eve to run for polynomial time. This is a more realistic model anyway - after all, the NSA may be very rich and powerful, but they are computationally bounded! However, to be fair, we should also assume that Alice and Bob are computationally bounded. It wouldn't be a very useful model if we assumed that the adversary was {\em less} powerful than the players. \bigskip Note that observation 1 still holds when the adversary is computationally bounded. If Eve knows everything that Alice knows, Eve can simply run the decryption algorithm (which is efficient, since Alice is also bounded), and recover the message. But, what about observation 2? \bigskip Observation 2 does not hold, since Eve can no longer try every message. It is very reasonable to assume that the message space is exponentially large (for example, if the possible messages are all in $\{0,1\}^k$), so Eve cannot simply run this attack. However, just because observation 2 does not hold does {\em not} mean there are definitely secure encryption schemes! Unfortunately, there is a lot we don't know in complexity theory, so we have to make some assumptions. One assumption we will probably have to make is that $P \neq NP$. (For example, if Eve knows everything Bob knows except for $m$, it would be easy to verify that a string $w$ contained all of Alice's private information, since decryption would work. Thus, if Eve can run an NP algorithm, she can break the cryptosystem). However, in practice we must make even more assumptions. \bigskip So, can we construct a cryptosystem where Alice and Bob share no secret? \section {Diffie-Hellman Key Exchange} So how can we construct a cryptosystem where Alice and Bob don't share any secrets? The first result to answer this question was ``New Directions in Cryptography,'' by Whitfield Diffie and Martin Hellman, published in 1976. They do not precisely solve this problem, but solve a similar one with two differences: \begin{enumerate} \item Alice and Bob may exchange multiple messages (Eve gets to see all the messages), and \item At the end of the protocol, Alice and Bob have a shared secret. There is no message communicated. \end{enumerate} However, part of the idea is that once Alice and Bob share a secret, the problem of encrypting a message becomes much easier. This is called a {\em key exchange protocol} since the end result is that Alice and Bob have a shared secret key. So, how does the Diffie-Hellman protocol (hereafter referred to as ``DH'') work? Let $k$ be a common ``security parameter.'' \begin{enumerate} \item Alice picks a random $k$-bit prime $p$. \item Alice picks a random generator $g \in Z_p^*$, that is, a number $g \in \{1, \ldots, p-1\}$ such that the two sets $\{1, \ldots, p-1\}$ and $\{g, g^2, \ldots, g^{p-1}\}$ are equal. \item Alice picks a random number $x \in Z_p^*$. \item Alice sends $p, g,$ and $a = g^x \bmod p$ to Bob. \item Bob picks a random number $y \in Z_p^*$. \item Bob sends $b = g^y \bmod p$ to Alice. \item Alice computes $s = b^x \bmod p$, and Bob computes $s = a^y \bmod p$. \end{enumerate} Since $a = g^x$ and $b = g^y$ we see that $(g^x)^y = g^{xy} = (g^y)^x$ so Alice and Bob do agree on their values. \subsection {Efficiency of DH} Since we are in the computationally bounded model, we not only need to make sure that the protocol is correct, but that it is efficient. There are basically three things we need to verify can be done efficiently. \bigskip \noindent {\bf 1. Generate a random $k$-bit prime.} Alice does this in step 1. We can certainly generate $k$-bit numbers efficiently, and the prime number theorem tells us that the probability that a random $k$-bit number is prime is about $1/k$. However, the problem of determining whether a given number is prime is a difficult one. As it turns out, there are deterministic methods which solve this problem, but probabilistic methods are far simpler. It suffices for our purposes to state that there are algorithms which are $RP$, for generating a random $k$-bit prime number, that is, their running time is expected to be polynomial (in $1^k$\footnote{We write $1^k$ instead of $k$ because the polynomial in polynomial time is considered on the {\em length} of the inputs. The length of $1^k$ is $k$, so this gives us what we want; on the other hand, the length of $k$ is $\log k$, which is {\em not} what we want.}, and their output is always a $k$-bit prime number. \bigskip \noindent {\bf 2. Generate a random generator for $Z_p^*$.} This is also solvable efficiently, but the methods are complicated and we need not get into them now. \bigskip \noindent {\bf 3. Given $x, y, n$, compute $x^y \bmod n$.} This is a bit tricky. First of all, it is not polynomial-time to compute $x^y$ and then reduce this modulo $n$, since $x^y$ is too long for a polynomial-time algorithm to compute.\footnote{The length of $x^y$ is $y$ times the length of $x$, which is not polynomial in the lengths of $x$ and $y$.} So, we might try this simple algorithm: \begin{tabbing} XXX\=\kill ModExp($x, y, n$):\\ 1. Set $a := 1$.\\ 2. For $b := 1$ to $y$ do\\ \>2a. $a := a * x$\\ \>2b. $a := a \bmod n$\\ 3. Output $a$ \end{tabbing} However, this is not efficient either. Step 2 will run $y$ times, which is not polynomial in the length of $y$. So, we have to be a little more clever. Let $y_i$ be the $i$th bit of $y$. \begin{tabbing} XXX\=XXX\=\kill ModExp($x, y, n$):\\ 1. Set $a := 1$.\\ 2. Set $s_1 := x$.\\ 3. For $b := 1$ to $k-1$ do\\ \>3a. $s_{b+1} := s_b * s_b$\\ \>3b. $s_{b+1} := s_{b+1} \bmod n$\\ 4. For $b := 1$ to $k$ do\\ \>4a. If $y_b == 1$ then\\ \>\>$a := a * s_b \bmod n$\\ 5. Output $a$. \end{tabbing} You can verify that this is correct if $y$ is no more than $k$ bits long, and that it is efficient. \subsection {Security of DH} The next question is, how secret is $g^{xy}$, the secret that Alice and Bob share at the end of the protocol? Diffie and Hellman weren't exactly sure, but they couldn't find an efficent way for Eve to break the scheme. So, the formulated the answer to the question of the security of their scheme like this: \bigskip ``If Eve can efficiently recover the secret, she can solve the following problem: Given $p, g, a,$ and $b$, compute $c$ such that there is an $x$ and a $y$ where $a = g^x \bmod p$, $b = g^y \bmod p$, and $c = g^{xy} \bmod p$.'' Now, if we assume that this problem is hard to solve, there can be no successful adversary. This amounts to Diffie and Hellman saying ``Diffie-Hellman is secure if Diffie-Hellman is secure'' but they can get away with that since they were the first ones to come up with this kind of scheme.\footnote{This problem is called the Diffie-Hellman problem; the assumption that it is hard is called the Diffie-Hellman assumption, or the DHA.} \bigskip However, there are two other main shortcomings of DH. First of all, it requires many messages to be exchanged, not just the single one. Secondly, it merely provides a shared secret, but does not allow Bob to send Alice a secret message. \section {The RSA Cryptosystem} The first improvement on DH was made in 1977 by Rivest, Shamir and Adleman (``RSA'' comes from the first letters of their names). In their system, Alice and Bob need not send multiple messages, and Bob has control over their shared secret.\footnote{In a sense, Bob can communicate the message this way, but for now we assume Bob draws the ``message'' from a uniform distribution.} However, it is not quite a simple Bob-sends-Alice-one-message system. Instead, Alice comes up with a ``public key'' which Bob uses to send her message to Bob. However, it is fine if Eve knows Alice's public key. The setup ({\em key generation}) works as follows: \begin{enumerate} \item Alice picks two random $k$-bit prime numbers, $p$ and $q$. Let $n = pq$. \item Alice computes $\varphi(n) = (p-1)(q-1)$, the Euler Phi function.\footnote{$\varphi(n)$ is the number of integers in $\{1, \ldots, n-1\}$ which are relatively prime to $n$. For $n = pq$, the formula $\varphi(n) = (p-1)(q-1)$ is correct.} \item Alice picks a random $e$ such that $gcd(e, \varphi(n)) = 1$ (that is, $e$ and $\varphi(n)$ are relatively prime. \item Alice computes $d$ such that $ed = 1 \bmod \varphi(n)$. \item Alice publishes $n, e$ as her public key, and keeps $d$ as her secret key. \end{enumerate} Now, for Bob to encrypt a secret $s$, he simply computes $c = s^e \bmod n$. Alice decrypts by computing $s = c^d \bmod n$. To see that the $s$ Alice gets is the same as the $s$ Bob used, we call on a fact from number theory: \bigskip\noindent {\it Fact 1.} For any $s$ such that $gcd(s,n) = 1$\footnote{This condition does in fact give us a problem if someone tries to encrypt an $s$ which is not relatively prime to $n$. However, this is very unlikely, so we won't worry about it.}, $s^{\varphi(n)} = 1 \bmod n$. This is true for any $n$, actually, not just numbers which are the products of two primes. Now, since $ed = 1 \bmod \varphi(n)$, there is an integer $l$ such that $ed = l\varphi(n) + 1$, so \begin{eqnarray*} (s^e)^d & = & s^{ed} \\ & = & s^{l\varphi(n) + 1} \\ & = & s \cdot s^{l\varphi(n)} \\ & = & s \cdot (s^{\varphi(n)})^l \\ & = & s \cdot 1^l \\ & = & s \end{eqnarray*} Encryption, decryption, and key generation are all efficient. The only operation which we haven't examined is the problem that given $e, \varphi(n)$, we must find a $d$ such that $ed = 1 \bmod \varphi(n).$ However, this merely amounts to computing an inverse modulo $\varphi(n)$, which is an easy operation. \subsection {Security of RSA} How hard is it for Eve to learn $s$ when she sees only $c, n,$ and $e$? Well, since RSA was one of the very early papers, they too make an analysis of their security similar to that of Diffie-Hellman. Namely, they say: \bigskip \noindent ``If Eve can recover $s$ then there is an efficient solution to this problem: Given $n, e$, and $u$, compute $v$ such that $v^e = u \bmod n$.'' \bigskip This problem is called the RSA problem, and this assumption is called the RSA assumption. However, RSA did a bit more than just give this ``RSA is secure if RSA is secure'' as their security analysis. What they noted was that if Eve can factor $n$, she can recover the secret. Since factoring large numbers is a well-studied problem which has not been solved, it seems reasonable to assume that factoring is hard. However, there is room for the possibility that factoring is hard and yet the RSA problem is not. 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Fl(\))i(de\014nes)f(a)h(probabilit)m(y)c(distribution)g(if)i Fh(A)i Fl(is)e(a)i(probabilistic)c(algorithm.)40 b(More)30 b(precisely)-8 b(,)150 4173 y(let)37 b Fh(A)356 4140 y Fe(0)380 4173 y Fl(\()p Fh(x;)15 b(R)q Fl(\))38 b(b)s(e)e(the)i (deterministic)d(algorithm)h(that)i(represen)m(ts)g(the)f(b)s(eha)m (vior)g(of)g Fh(A)g Fl(on)h(input)d Fh(x)150 4286 y Fl(and)e(random)f (tap)s(e)i Fh(R)q Fl(.)49 b(The)33 b(uniform)e(distribution)e(on)k Fh(R)h Fl(induces)e(a)i(probabilit)m(y)c(distribution)g(on)150 4399 y(the)h(output)f(of)g Fh(A)776 4366 y Fe(0)800 4399 y Fl(\()p Fh(x;)15 b(R)q Fl(\).)41 b(W)-8 b(e)32 b(call)d(that)i (distribution)c Fh(A)p Fl(\()p Fh(x)p Fl(\).)291 4512 y Fh(x)j Ff( )g Fh(A)p Fl(\()p Fh(y)s Fl(\))k(denotes)g(that)g Fh(x)f Fl(is)g(a)g(sample)g(from)g(the)g(distribution)d Fh(A)p Fl(\()p Fh(x)p Fl(\);)36 b(it)d(also)g(means)g(that)h Fh(x)150 4625 y Fl(w)m(as)d(generated)g(b)m(y)f(running)e Fh(A)j Fl(on)f(input)e Fh(y)s Fl(.)291 4738 y Fh(x)d Ff( )h Fh(F)13 b Fl(,)30 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)e Fh(A)p Fl(\(3\);)15 b Fh(z)31 b Ff( )25 b Fh(B)5 b Fl(\()p Fh(y)s Fl(\)\))150 540 y(denotes)31 b(that)h(\014rst)e(w)m(e)i(ran)f(algorithm)f Fh(A)h Fl(on)g(input)e(3)i (to)h(generate)h(\()p Fh(x;)15 b(y)s Fl(\),)32 b(and)f(then)f(w)m(e)i (ran)f Fh(B)k Fl(on)150 653 y(input)28 b Fh(y)34 b Fl(to)d(generate)h Fh(z)t Fl(.)291 766 y(\()p Ff(f)p Fh(x)423 780 y Fb(i)478 766 y Ff( )25 b Fh(A)662 780 y Fb(i)690 766 y Fl(\()p Fh(y)770 780 y Fb(i)798 766 y Fl(\))p Ff(g)p Fl(1)i Ff(\024)e Fh(i)h Ff(\024)g Fh(n)55 b Fl(:)h Ff(f)p Fh(x)1488 780 y Fb(i)1517 766 y Ff(g)p Fh(;)15 b Fl(1)27 b Ff(\024)e Fh(i)h Ff(\024)f Fh(n)p Fl(\))30 b(denotes)h(the)g(distribution)c(on)j (the)h(v)-5 b(alues)30 b Ff(f)p Fh(x)3676 780 y Fb(i)3705 766 y Ff(g)150 879 y Fl(that)h(w)m(ere)g(obtained)f(b)m(y)g(running)d (algorithms)j Fh(A)1914 893 y Fg(1)1953 879 y Fh(;)15 b(:)g(:)g(:)i(;)e(A)2223 893 y Fb(n)2301 879 y Fl(on)30 b(inputs)e Fh(y)2750 893 y Fg(1)2789 879 y Fh(;)15 b(:)g(:)g(:)i(;)e(y) 3036 893 y Fb(n)3083 879 y Fl(.)291 992 y(Let)39 b Fh(p)g Fl(b)s(e)f(a)h(b)s(o)s(olean)f(function.)65 b(\()p Ff(f)p Fh(x)1659 1006 y Fb(i)1727 992 y Ff( )39 b Fh(A)1925 1006 y Fb(i)1953 992 y Fl(\()p Fh(y)2033 1006 y Fb(i)2061 992 y Fl(\))p Ff(g)p Fl(1)i Ff(\024)e Fh(i)g Ff(\024)g Fh(n)78 b Fl(:)g Fh(p)p Fl(\()p Fh(x)2886 1006 y Fb(n)2933 992 y Fl(\)\))40 b(means)e(that)i Fh(p)p Fl(\()p Fh(x)3668 1006 y Fb(n)3715 992 y Fl(\))150 1105 y(is)k(TR)m(UE)h(after)g(the)h(v) -5 b(alue)44 b Fh(x)1263 1119 y Fb(n)1355 1105 y Fl(w)m(as)h(obtained)g (b)m(y)f(running)f(algorithms)g Fh(A)2959 1119 y Fg(1)2999 1105 y Fh(;)15 b(:)g(:)g(:)i(;)e(A)3269 1119 y Fb(n)3361 1105 y Fl(on)45 b(inputs)150 1218 y Fh(y)195 1232 y Fg(1)234 1218 y Fh(;)15 b(:)g(:)g(:)i(;)e(y)481 1232 y Fb(n)528 1218 y Fl(.)291 1331 y(Finally)-8 b(,)29 b(the)i(statemen)m(t)1147 1522 y(Pr[)p Ff(f)p Fh(x)1367 1536 y Fb(i)1421 1522 y Ff( )25 b Fh(A)1605 1536 y Fb(i)1633 1522 y Fl(\()p Fh(y)1713 1536 y Fb(i)1741 1522 y Fl(\))p Ff(g)p Fl(1)i Ff(\024)e Fh(i)h Ff(\024)e Fh(n)56 b Fl(:)f Fh(p)p Fl(\()p Fh(x)2465 1536 y Fb(n)2512 1522 y Fl(\)])26 b(=)f Fh(\013)150 1713 y Fl(means)j(that)g(the)g(probabilit)m(y)d(that)j Fh(p)p Fl(\()p Fh(x)1563 1727 y Fb(n)1610 1713 y Fl(\))g(is)f(TR)m(UE)h(after) g(the)g(v)-5 b(alue)27 b Fh(x)2700 1727 y Fb(n)2775 1713 y Fl(w)m(as)h(obtained)f(b)m(y)h(running)150 1826 y(algorithms)h Fh(A)668 1840 y Fg(1)708 1826 y Fh(;)15 b(:)g(:)g(:)h(;)f(A)977 1840 y Fb(n)1055 1826 y Fl(on)31 b(inputs)d Fh(y)1505 1840 y Fg(1)1544 1826 y Fh(;)15 b(:)g(:)g(:)i(;)e(y)1791 1840 y Fb(n)1837 1826 y Fl(,)31 b(is)e Fh(\013)p Fl(.)150 2110 y Fi(3)135 b(A)44 b(Sp)t(eci\014cation)h(for)g(a)g(Cryptosystem) 150 2313 y Fl(Let)40 b(us)f(\014rst)g(giv)m(e)i(sp)s(eci\014cation)d (of)i(the)g(algorithms)f(that)h(are)g(part)g(of)g(the)g(cryptosystem.) 69 b(It)40 b(is)150 2426 y(imp)s(ortan)m(t)25 b(that)h(these)g (algorithms)f(are)h Fj(e\016cient)p Fl(.)39 b(As)26 b(is)e(usually)g (the)i(case)h(in)d(theoretical)i(computer)150 2539 y(science,)37 b(b)m(y)f(\\e\016cien)m(t,")i(w)m(e)e(mean)g(p)s(olynomial-time.)54 b(But)35 b(p)s(olynomial-time)e(in)i(what?)55 b(An)36 b(im-)150 2652 y(p)s(ortan)m(t)g(parameter)g(of)g(a)g(cryptosystem)h (is)d(its)i Fj(se)-5 b(curity)38 b(p)-5 b(ar)g(ameter)38 b Fh(k)s Fl(.)57 b(All)35 b(the)h(algorithms)e(that)150 2765 y(comprise)h(the)h(cryptosystem)g(m)m(ust)g(ha)m(v)m(e)h(run-time) d(p)s(olynomial)f(in)i Fh(k)s Fl(.)57 b(The)35 b(adv)m(ersaries)h (against)150 2877 y(whic)m(h)22 b(w)m(e)j(hop)s(e)e(to)h(ac)m(hiev)m(e) h(securit)m(y)-8 b(,)26 b(also)d(ha)m(v)m(e)i(run-time)e(p)s(olynomial) e(in)h(this)h(securit)m(y)g(parameter.)150 3077 y Fm(Key)35 b(generation)45 b Fl(The)20 b(k)m(ey)h(generation)g(algorithm,)g (denoted)g(b)m(y)f Fh(G)p Fl(,)j(is)c(a)i(probabilistic)c(p)s (olynomial-)377 3190 y(time)33 b(algorithm)e(that)i(tak)m(es)h(as)f (its)f(sole)g(input)f(the)i(securit)m(y)f(parameter)h Fh(k)s Fl(.)47 b(Ho)m(w)m(ev)m(er,)36 b(since)377 3303 y(traditionally)29 b(w)m(e)i(sa)m(y)g(that)g(an)g(algorithm)f(is)f(p)s (olynomial-time)f(if)i(its)g(running)e(time)i(is)g(p)s(oly-)377 3415 y(nomial)d(in)h(the)h(size)f(of)h(its)f(input,)f Fh(G)p Fl('s)i(input)e Fh(k)32 b Fl(will)25 b(b)s(e)j(enco)s(ded)h(in)e (unary)-8 b(,)29 b(denoted)f(1)3485 3382 y Fb(k)3528 3415 y Fl(.)40 b(The)377 3528 y(output)g(of)h Fh(G)g Fl(is)e(the)i(k)m(ey)g(pair,)h(\()p Fh(P)13 b(K)q(;)i(S)5 b(K)i Fl(\).)73 b(Using)40 b(the)g(notation)h(in)m(tro)s(duced)e(ab)s (o)m(v)m(e,)44 b(w)m(e)377 3641 y(write:)1675 3754 y(\()p Fh(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))27 b Ff( )e Fh(G)p Fl(\(1)2373 3717 y Fb(k)2417 3754 y Fl(\))377 3950 y(W)-8 b(e)29 b(sa)m(y)f(that)g(the)f(secret)i(k)m(ey)f Fh(S)5 b(K)34 b Fj(c)-5 b(orr)g(esp)g(onds)31 b Fl(to)d(the)f(public)e(k)m(ey) j Fh(P)13 b(K)34 b Fl(if)26 b(\()p Fh(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))29 b(can)f(b)s(e)377 4062 y(pro)s(duced)h(b)m(y)h(running)e Fh(G)p Fl(\(1)1390 4029 y Fb(k)1434 4062 y Fl(\).)150 4246 y Fm(Encryption)46 b Fl(The)24 b(encryption)e(algorithm)i Fh(E)29 b Fl(is)23 b(a)h(probabilistic)d(p)s(olynomial-time)g (algorithm)i(that)377 4359 y(tak)m(es)28 b(as)f(input)e(the)h(message)i Fh(m)e Fl(from)g(the)h(message)h(space)f Ff(M)p Fl(,)h(and)d(the)i (public)d(k)m(ey)j Fh(P)13 b(K)34 b Fl(and)377 4471 y(pro)s(duces)29 b(as)i(output)f(the)h(ciphertext)f Fh(c)p Fl(:)1764 4715 y Fh(c)c Ff( )f Fh(E)5 b Fl(\()p Fh(m;)15 b(P)e(K)7 b Fl(\))150 4941 y Fm(Decryption)46 b Fl(The)23 b(decryption)f(algorithm) h Fh(D)j Fl(is)c(a)i(probabilistic)19 b(p)s(olynomial-time)i(algorithm) h(that)377 5054 y(tak)m(es)39 b(as)e(input)f(the)h(ciphertext)g Fh(c)g Fl(and)g(the)g(secret)h(k)m(ey)g Fh(S)5 b(K)44 b Fl(and)37 b(pro)s(duces)f(as)h(output)g(the)377 5167 y(message)32 b Fh(m)p Fl(:)1889 5416 y(2-2)p eop %%Page: 3 3 3 2 bop 1767 363 a Fh(m)25 b Ff( )g Fh(D)s Fl(\()p Fh(c;)15 b(S)5 b(K)i Fl(\))291 604 y(There)30 b(are)h(t)m(w)m(o)h(things)d(that) i(w)m(e)g(require)e(of)h(a)h(cryptosystem:)286 792 y Ff(\017)46 b Fl(The)39 b(decryption)g(should)f(b)s(e)h(correct,)k (i.e.,)g(if)c Fh(S)5 b(K)46 b Fl(corresp)s(onds)38 b(to)j(public)c(k)m (ey)j Fh(P)13 b(K)7 b Fl(,)42 b(then)377 905 y Fh(D)s Fl(\()p Fh(E)5 b Fl(\()p Fh(b;)15 b(P)e(K)7 b Fl(\))p Fh(;)15 b(S)5 b(K)i Fl(\))45 b(=)e Fh(b)p Fl(.)72 b(\(W)-8 b(e)43 b(can)e(b)s(e)g(more)g(general)g(and)f(relax)h(this)f(condition) g(to)i(just)377 1018 y(asking)30 b(that)h(this)e(happ)s(ens)g(with)g (high)g(probabilit)m(y)-8 b(.\))286 1205 y Ff(\017)46 b Fl(The)28 b(cryptosystem)h(m)m(ust)f(b)s(e)g Fj(se)-5 b(cur)g(e)p Fl(.)41 b(The)28 b(rest)g(of)h(this)e(lecture,)i(and)f(sev) m(eral)h(up)s(coming)e(lec-)377 1318 y(tures,)g(will)c(deal)i(with)g (coming)g(up)g(with)g(an)g(appropriate)g(formalization)g(of)h(this)e (requiremen)m(t.)150 1605 y Fi(4)135 b(T)-11 b(o)l(w)l(ards)45 b(De\014ning)g(Securit)l(y)150 1808 y Fl(F)-8 b(or)38 b(to)s(da)m(y)-8 b(,)41 b(let)c(us)g(fo)s(cus)g(on)h(de\014ning)d (securit)m(y)j(for)f(the)h(one-bit)f(message)i(space:)55 b(Bob)38 b(can)g(only)150 1921 y(send)31 b(Alice)f(one)i(bit)e(of)i (information:)41 b(\\I)32 b(lo)m(v)m(e)g(y)m(ou")g(\(enco)s(ded)g(b)m (y)f(a)h(0\))g(or)f(\\I)h(hate)g(y)m(ou")g(\(enco)s(ded)150 2034 y(b)m(y)e(a)h(1\).)150 2277 y Fa(4.1)112 b(First)37 b(T)-9 b(ry)150 2449 y Fl(In)28 b(lecture)h(1,)h(w)m(e)f(explored)f (the)i(p)s(ossibilit)m(y)25 b(of)k(Ev)m(e)h(b)s(eing)d(computationally) h(un)m(b)s(ounded.)38 b(W)-8 b(e)30 b(also)150 2561 y(assumed)g(that)h (the)f(ciphertext,)h Fh(c)p Fl(,)f(do)s(es)g(not)h(help)e(in)g (determining)f(the)j(message,)h Fh(m)p Fl(,)e(i.e.,)822 2766 y Ff(8)p Fh(m;)15 b Ff(8)p Fh(c;)g Fl(Pr[)p Fh(m)30 b Fl(w)m(as)h(sen)m(t)g Ff(j)f Fh(c)h Fl(w)m(as)g(receiv)m(ed])25 b(=)g(Pr[)p Fh(m)30 b Fl(w)m(as)h(sen)m(t)q(])291 2970 y(Whic)m(h)e(in)f(our)g(case)j(with)d(just)g(one-bit)h(messages)h(c)m (hosen)g(uniformly)c(at)k(random,)f(corresp)s(onds)150 3083 y(to)1091 3287 y Ff(8)p Fh(b;)15 b Ff(8)p Fh(c;)g Fl(Pr[)p Fh(b)30 b Fl(w)m(as)h(sen)m(t)g Ff(j)f Fh(c)h Fl(w)m(as)g(receiv)m(ed])26 b(=)f(1)p Fh(=)p Fl(2)291 3491 y(As)i(w)m(e)g(sa)m(w)g(in)e(the)i(previous)e(lecture,)j(this)d (de\014nition)f(is)i(to)s(o)h(strong.)40 b(In)26 b(the)h(public-k)m(ey) e(setting)150 3604 y(it)31 b(is)g(actually)g(unattainable.)44 b(The)31 b(reason)h(for)g(this)f(is)f(that)j(it)e(implies)e(that)j(an)m (y)g(giv)m(en)g(ciphertext)150 3717 y Fh(c)37 b Fl(can)g(b)s(e)e(an)i (enco)s(ding)e(of)i(a)g(1)g(as)f(w)m(ell)g(as)h(a)f(0.)60 b(Since)35 b(Alice)h(and)g(Bob)h(ha)m(v)m(e)h(no)e(shared)g(secrets,) 150 3830 y(this)26 b(w)m(ould)g(b)s(e)g(in)g(con)m(tradiction)h(to)h (the)f(correctness)h(of)f(decryption)f(requiremen)m(t!)39 b(What)28 b(happ)s(ens)150 3943 y(is)33 b(that)h(if)e Fh(c)i Fl(could)f(b)s(e)g(obtained)f(b)m(y)i(running)d Fh(E)5 b Fl(\(0)p Fh(;)15 b(P)e(K)7 b Fl(\))35 b(as)f(w)m(ell)e(as)i(b) m(y)f(running)e Fh(E)5 b Fl(\(1)p Fh(;)15 b(P)e(K)7 b Fl(\),)36 b(then)150 4056 y(0)26 b(=)f Fh(D)s Fl(\()p Fh(E)5 b Fl(\(0)p Fh(;)15 b(P)e(K)7 b Fl(\)\))27 b(=)e Fh(c)h Fl(=)f Fh(D)s Fl(\()p Fh(E)5 b Fl(\(1)p Fh(;)15 b(P)e(K)7 b Fl(\)\))27 b(=)e(1)31 b(whic)m(h)e(is)g(con)m(tradiction.) 291 4169 y Fm(What)36 b(is)g(the)f(lesson?)45 b Fl(The)31 b(main)f(lesson)g(is)h(that)h Fh(c)f Fj(uniquely)i(determines)f Fh(b)p Fl(.)44 b(Therefore,)31 b(w)m(e)150 4282 y(cannot)h(get)h(a)f (satisfactory)h(de\014nition)c(insisting)g(that)j(the)g(probabilit)m(y) d(of)j(a)g(bit)f(ha)m(ving)g(b)s(een)g(sen)m(t)150 4395 y(remain)h(the)g(same)i(ev)m(en)f(when)f(conditioning)e(on)j(a)g(giv)m (en)f(ciphertext.)48 b(Instead,)33 b(w)m(e)g(ha)m(v)m(e)h(to)f(fo)s (cus)150 4508 y(on)d(the)h(probabilit)m(y)d(that)j(Ev)m(e)g(will)c (\014gure)j(out)h(whic)m(h)e(bit)g(w)m(as)i(sen)m(t.)150 4751 y Fa(4.2)112 b(Second)38 b(T)-9 b(ry)150 4923 y Fl(Let)31 b(us)f(try)g(another)g(de\014nition,)f(whic)m(h)g(incorp)s (orates)g(the)i(lesson)f(learned)f(from)h(our)g(\014rst)f(try:)839 5127 y(Pr[)p Fh(b)c Ff( )h(f)p Fl(0)p Fh(;)15 b Fl(1)p Ff(g)p Fl(;)g Fh(c)28 b Ff( )d Fh(E)5 b Fl(\()p Fh(b;)15 b(P)e(K)7 b Fl(\);)15 b Fh(b)2041 5089 y Fe(0)2091 5127 y Ff( )25 b Fh(E)5 b(v)s(e)p Fl(\()p Fh(c)p Fl(\))28 b(:)d Fh(b)g Fl(=)g Fh(b)2754 5089 y Fe(0)2778 5127 y Fl(])g(=)g(1)p Fh(=)p Fl(2)1889 5416 y(2-3)p eop %%Page: 4 4 4 3 bop 291 159 a Fl(The)29 b(\014rst)g(thing)f(that)j(is)d(funn)m(y)g (ab)s(out)i(this)e(de\014nition)f(is:)40 b(what)29 b(is)g Fh(P)13 b(K)7 b Fl(?)39 b(Should)28 b(this)g(hold)g(for)150 271 y(all)i Fh(P)13 b(K)7 b Fl(?)42 b(Or)31 b(for)g(some)g Fh(P)13 b(K)7 b Fl(?)43 b(The)30 b(most)i(natural)e(w)m(a)m(y)i(of)g (getting)g(the)f(public)d(k)m(ey)k(accoun)m(ted)h(for,)150 384 y(is)c(to)j(tak)m(e)f(the)g(probabilit)m(y)d(o)m(v)m(er)k(the)e(c)m (hoice)h(of)g Fh(P)13 b(K)7 b Fl(,)30 b(as)h(follo)m(ws:)431 569 y(Pr)o([\()p Fh(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))27 b Ff( )e Fh(G)p Fl(\(1)1251 532 y Fb(k)1295 569 y Fl(\);)15 b Fh(b)26 b Ff( )f(f)p Fl(0)p Fh(;)15 b Fl(1)p Ff(g)p Fl(;)g Fh(c)29 b Ff( )c Fh(E)5 b Fl(\()p Fh(b;)15 b(P)e(K)7 b Fl(\);)15 b Fh(b)2450 532 y Fe(0)2500 569 y Ff( )25 b Fh(E)5 b(v)s(e)p Fl(\()p Fh(c)p Fl(\))27 b(:)f Fh(b)f Fl(=)g Fh(b)3163 532 y Fe(0)3186 569 y Fl(])h(=)f(1)p Fh(=)p Fl(2)291 736 y(It)42 b(turns)e(out)h(that)h(this)e(de\014nition) f(of)i(securit)m(y)g(is)g(to)s(o)h(strong.)73 b(Supp)s(ose)40 b(Ev)m(e)i(is)e(using)g(the)150 849 y(follo)m(wing)29 b(algorithm:)150 1025 y Fm(Ev)m(e's)35 b(Algorithm)461 1200 y Fl(1.)46 b(Run)30 b Fh(G)g Fl(and)g(obtain)g(\()p Fh(P)13 b(K)1527 1167 y Fe(0)1550 1200 y Fh(;)i(S)5 b(K)1735 1167 y Fe(0)1759 1200 y Fl(\).)461 1334 y(2.)46 b(If)30 b Fh(P)13 b(K)32 b Fl(=)25 b Fh(P)13 b(K)1099 1301 y Fe(0)1122 1334 y Fl(,)31 b(then)f(obtain)g Fh(b)1707 1301 y Fe(0)1760 1334 y Fl(b)m(y)g(running)e Fh(D)s Fl(\()p Fh(c;)15 b(S)5 b(K)i Fl(\).)461 1468 y(3.)46 b(Otherwise,)30 b(output)g(a)g(random)g(guess)g Fh(b)2017 1435 y Fe(0)2041 1468 y Fl(.)291 1643 y(Let)25 b(us)e(estimate)i(Ev)m(e's)g(success)f (in)f(outputting)g(the)h(correct)i(bit.)37 b(Since)23 b Fh(G)i Fl(is)e(a)i(p)s(olynomial-time)150 1756 y(algorithm,)38 b(it)f(needs)g(a)h(p)s(olynomial)d(n)m(um)m(b)s(er)h(of)h(random)g (bits.)60 b(Let)38 b(us)e(denote)i(this)f(p)s(olynomial)150 1869 y(b)s(ound)23 b(b)m(y)i Fh(r)s Fl(\()p Fh(k)s Fl(\).)39 b(With)24 b(probabilit)m(y)e(2)1507 1836 y Fe(\000)p Fb(r)r Fg(\()p Fb(k)r Fg(\))1694 1869 y Fl(,)k(the)f(random)f(bits)g (Ev)m(e)i(used)e(when)f(running)g Fh(G)i Fl(coincide)150 1982 y(with)k(the)i(random)e(bits)g(used)h(b)m(y)g(Alice)g(when)g (generating)g(her)g(k)m(ey)h(pair.)291 2095 y(Supp)s(ose)f Fh(P)13 b(K)34 b Fl(=)27 b Fh(P)13 b(K)1084 2062 y Fe(0)1107 2095 y Fl(.)44 b(Then)31 b(Ev)m(e's)h Fh(b)1694 2062 y Fe(0)1749 2095 y Fl(will)d(alw)m(a)m(ys)j(corresp)s(ond)e(to)i(the)g (bit)f Fh(b)g Fl(that)h(w)m(as)g(b)s(eing)150 2208 y(sen)m(t:)41 b(otherwise,)30 b(the)h(decryption)e(w)m(ould)g(not)i(b)s(e)f (faithful!)291 2321 y(Let)41 b Fh(p)f Fl(denote)g(the)h(probabilit)m(y) c(that)k(Ev)m(e's)g Fh(P)13 b(K)2104 2288 y Fe(0)2167 2321 y Fl(is)39 b(the)h(same)h(as)f Fh(P)13 b(K)7 b Fl(.)70 b(W)-8 b(e)41 b(just)f(sa)m(w)g(that)150 2434 y Fh(p)25 b Ff(\025)g Fl(2)362 2401 y Fe(\000)p Fb(r)r Fg(\()p Fb(k)r Fg(\))549 2434 y Fl(.)40 b(No)m(w)31 b(let's)g(estimate)g(Ev)m (e's)g(success)g(probabilit)m(y)d Fh(q)s Fl(:)150 2714 y Fh(q)86 b Fl(=)d(Pr)o([)p Fh(P)13 b(K)32 b Fl(=)25 b Fh(P)13 b(K)984 2676 y Fe(0)1007 2714 y Fl(])i(Pr[Ev)m(e)31 b(is)f(correct)q Ff(j)p Fh(P)13 b(K)32 b Fl(=)25 b Fh(P)13 b(K)2169 2676 y Fe(0)2192 2714 y Fl(])21 b(+)f(Pr)o([)p Fh(P)13 b(K)32 b Ff(6)p Fl(=)25 b Fh(P)13 b(K)2882 2676 y Fe(0)2905 2714 y Fl(])i(Pr[Ev)m(e)31 b(is)f(correct)q Ff(j)p Fh(P)13 b(K)32 b Ff(6)p Fl(=)25 b Fh(P)13 b(K)4067 2676 y Fe(0)4090 2714 y Fl(])277 2852 y(=)83 b Fh(p)20 b Ff(\003)g Fl(1)h(+)f(\(1)h Ff(\000)f Fh(p)p Fl(\))g Ff(\003)h Fl(1)p Fh(=)p Fl(2)26 b(=)f(1)p Fh(=)p Fl(2)d(+)e Fh(p=)p Fl(2)26 b Ff(\025)f Fl(1)p Fh(=)p Fl(2)c(+)f(2)2133 2814 y Fe(\000)p Fg(\()p Fb(r)r Fg(\()p Fb(k)r Fg(\)+1\))291 3019 y Fm(What)30 b(is)g(the)g(lesson?)40 b Fl(The)26 b(lesson)f(is)h(that)h(w)m(e)f(cannot)h(hop)s(e)f(to)h(prev)m(en)m(t)f (the)h(adv)m(ersary)f(from)150 3132 y(ha)m(ving)36 b Fj(any)g Fl(adv)-5 b(an)m(tage)38 b(o)m(v)m(er)g(randomly)c(guessing)i (the)g(message)h(sen)m(t.)58 b(Ho)m(w)m(ev)m(er,)40 b(notice)d(that)f (in)150 3245 y(the)i(ab)s(o)m(v)m(e)h(example,)h(the)e(adv)m(ersary's)g (adv)-5 b(an)m(tage)40 b(is)d(so)h(small)e(that)j(w)m(e)f(don't)g (care:)57 b(if)36 b(w)m(e)j(only)150 3358 y(ha)m(v)m(e)32 b(p)s(olynomial)27 b(time)j(to)h(liv)m(e,)f(w)m(e)h(will)d(nev)m(er)j (ev)m(en)g(notice)f(this)g(adv)-5 b(an)m(tage.)150 3595 y Fa(4.3)112 b(Third)37 b(T)-9 b(ry)150 3767 y Fl(Let)42 b(us)f(w)m(eak)m(en)h(the)g(de\014nition)d(to)j(allo)m(w)f(the)h(adv)m (ersary)f(to)i(ha)m(v)m(e)f(an)g(adv)-5 b(an)m(tage)43 b(o)m(v)m(er)g(random)150 3880 y(guessing,)26 b(but)f(only)f(a)i(v)m (ery)g(small)e(adv)-5 b(an)m(tage.)40 b(A)26 b(small)e(adv)-5 b(an)m(tage)27 b(is)e(an)g(adv)-5 b(an)m(tage)27 b(that)f(v)-5 b(anishes)150 3993 y(faster)31 b(than)f(an)m(y)h(in)m(v)m(erse)f(p)s (olynomial)e({)i(so)h(it)f(cannot)h(ev)m(en)g(b)s(e)f(observ)m(ed)g(in) f(p)s(olynomial)f(time!)291 4105 y(A)j(function)e Fh(f)40 b Fl(is)30 b(called)g Fj(ne)-5 b(gligible)30 b Fl(if)g(for)h(all)e(in)m (tegers)i Fh(c)p Fl(,)h(there)e(exists)h(an)f(in)m(teger)i Fh(K)37 b Fl(suc)m(h)30 b(that)150 4218 y(for)g(all)f Fh(k)g Ff(\025)c Fh(K)7 b Fl(,)30 b Fh(f)10 b Fl(\()p Fh(k)s Fl(\))26 b Fh(<)e Fl(1)p Fh(=k)1162 4185 y Fb(c)1198 4218 y Fl(.)41 b(Let)31 b(negl)f(denote)h(the)f(set)h(of)g(negligible)d (functions.)291 4331 y(So)i(no)m(w)h(the)f(new)g(de\014nition)e(is:) 1821 4611 y Ff(9)p Fh(f)10 b Fl(\()p Fh(k)s Fl(\))25 b Ff(2)g Fl(negl)951 4749 y(Pr)o([\()p Fh(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))27 b Ff( )f Fh(G)p Fl(\(1)1772 4712 y Fb(k)1815 4749 y Fl(\);)15 b Fh(b)26 b Ff( )g(f)p Fl(0)p Fh(;)15 b Fl(1)p Ff(g)p Fl(;)921 4887 y Fh(c)26 b Ff( )f Fh(E)5 b Fl(\()p Fh(b;)15 b(P)e(K)7 b Fl(\);)15 b Fh(b)1557 4850 y Fe(0)1607 4887 y Ff( )26 b Fh(E)5 b(v)s(e)p Fl(\()p Fh(c)p Fl(\))27 b(:)e Fh(b)h Fl(=)f Fh(b)2271 4850 y Fe(0)2294 4887 y Fl(])83 b Ff(\024)g Fl(1)p Fh(=)p Fl(2)21 b(+)f Fh(f)10 b Fl(\()p Fh(k)s Fl(\))291 5054 y(This)27 b(de\014nition)f(is)i(almost)g(satisfactory)-8 b(,)31 b(except)e(that)g(w)m(e)g(don't)g(kno)m(w)g(Ev)m(e's)g(algorithm)e(at)j (the)150 5167 y(time)g(when)f(w)m(e)i(design)f(the)g(cryptosystem.)41 b(This)29 b(needs)h(to)h(b)s(e)f(accoun)m(ted)h(for.)1889 5416 y(2-4)p eop %%Page: 5 5 5 4 bop 150 159 a Fi(5)135 b(De\014nition)45 b(of)h(Securit)l(y)f (\(Indistinguishabilit)l(y\))150 361 y Fl(The)32 b(only)f(thing)g(that) h(remains)f(to)i(b)s(e)e(de\014ned)g(in)g(further)g(detail,)h(is)f(the) h(notion)f(of)i(the)f(algorithm)150 474 y(Ev)m(e)q(.)38 b(W)-8 b(e)23 b(w)m(an)m(t)g(it)e(to)i(capture)g(all)e(computation)h (that)g(is)f(doable)h(b)m(y)g(a)g(realistic)f(device.)38 b(F)-8 b(or)23 b(example,)150 587 y(a)35 b(probabilistic)c(p)s (olynomial)h(time)i(T)-8 b(uring)33 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Fl(family)e(of)i(T)-8 b(uring)29 b(mac)m(hines.)41 b(Let)150 1265 y Fh(P)13 b(P)g(T)g(F)43 b Fl(stand)30 b(for)g(\\probabilistic)e (p)s(olynomial)f(time)j(family")g(of)g(T)-8 b(uring)29 b(mac)m(hines.)291 1378 y(Since)24 b(w)m(e)h(don't)g(kno)m(w)f(Ev)m (e's)i(algorithm)d(ahead)i(of)g(time,)h(w)m(e)f(ha)m(v)m(e)h(to)f(quan) m(tify)f(o)m(v)m(er)i(all)d(p)s(ossible)150 1491 y(probabilitic)k(p)s (olynomial-time)h(families.)291 1603 y(The)i(\014nal)f(de\014nition)f (is)h(as)i(follo)m(ws:)1196 1921 y Ff(8)p Fh(P)13 b(P)g(T)g(F)g Ff(f)p Fh(E)5 b(v)s(e)1732 1936 y Fb(k)1775 1921 y Ff(g9)p Fh(f)10 b Fl(\()p Fh(k)s Fl(\))26 b Ff(2)f Fl(negl)951 2058 y(Pr)o([\()p Fh(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))27 b Ff( )f Fh(G)p Fl(\(1)1772 2021 y Fb(k)1815 2058 y Fl(\);)15 b Fh(b)26 b Ff( )g(f)p Fl(0)p Fh(;)15 b Fl(1)p Ff(g)p Fl(;)921 2196 y Fh(c)26 b Ff( )f Fh(E)5 b Fl(\()p Fh(b;)15 b(P)e(K)7 b Fl(\);)15 b Fh(b)1557 2159 y Fe(0)1607 2196 y Ff( )26 b Fh(E)5 b(v)s(e)p Fl(\()p Fh(c)p Fl(\))27 b(:)e Fh(b)h Fl(=)f Fh(b)2271 2159 y Fe(0)2294 2196 y Fl(])83 b Ff(\024)g Fl(1)p Fh(=)p Fl(2)21 b(+)f Fh(f)10 b Fl(\()p Fh(k)s Fl(\))291 2400 y(This)36 b(de\014nition)f(w)m(as)i (\014rst)g(giv)m(en)g(b)m(y)h(Sha\014)e(Goldw)m(asser)h(and)g(Silvio)e (Micali)h(in)g(their)h(seminal)150 2513 y(w)m(ork)c(en)m(titled)g (\\Probabilistic)e(Encryption,")i(whic)m(h)f(y)m(ou)i(will)d(receiv)m (e)j(as)g(a)g(handout)e(in)g(the)i(next)150 2626 y(lecture.)69 b(In)39 b(the)h(next)g(lecture,)j(w)m(e)d(will)d(also)j(see)h(the)f (Goldw)m(asser-Micali)f(cryptosystem)h(whic)m(h)150 2739 y(satis\014es)30 b(this)f(de\014nition.)1889 5416 y(2-5)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF scribe/lecture3.ps0100644003326500237220000102725407351704015013103 0ustar annata6875%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: lect3.dvi %%Pages: 5 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSWebPage: (www.radicaleye.com) 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b(c)m(hosen)i(using) e(algorithm)h Fk(G)p Fo(;)i(a)f(bit)e Fk(b)i Fo(is)e(c)m(hosen)i(at)g (random,)g(and)f(encrypted)g(under)e Fk(P)13 b(K)40 b Fo(using)150 2980 y(algorithm)33 b Fk(E)5 b Fo(;)37 b(the)e(resulting)d (ciphertext)i Fk(c)h Fo(is)e(giv)m(en)i(to)g(the)f(adv)m(ersary;)j(the) e(adv)m(ersary's)f(guess)h Fk(b)3727 2947 y Fi(0)150 3093 y Fo(coincides)29 b(with)g(the)i(encrypted)f(bit)f Fk(b)i Fo(with)e(probabilit)m(y)f(only)h(negligibly)f(b)s(etter)i(than) g(1)p Fk(=)p Fo(2.)150 3255 y(Or)g(more)g(precisely:)291 3468 y Fj(8)p Fk(ppt)p Fj(f)p Fk(A)580 3483 y Fh(k)622 3468 y Fj(g9)h Fo(negligible)d(function)h Fk(f)10 b Fo(\()p Fk(x)p Fo(\))30 b(suc)m(h)g(that)h Fj(8)p Fk(k)291 3631 y(P)13 b(r)s Fo([\()p Fk(P)g(K)q(;)i(S)5 b(K)i Fo(\))27 b Fj( )e Fk(G)p Fo(\(1)1129 3598 y Fh(k)1173 3631 y Fo(\);)15 b Fk(b)26 b Fj( )f Fo(0)p Fk(;)15 b Fo(1)q(;)g Fk(c)26 b Fj( )f Fk(E)5 b Fo(\()p Fk(b;)15 b(P)e(K)7 b Fo(\);)15 b Fk(b)2236 3598 y Fi(0)2287 3631 y Fj( )25 b Fk(A)2471 3646 y Fh(k)2514 3631 y Fo(\()p Fk(c;)15 b(P)e(K)7 b Fo(\))316 3743 y(:)26 b Fk(b)406 3710 y Fi(0)454 3743 y Fo(=)f Fk(b)p Fo(])h Fj(\024)746 3708 y Ff(1)p 746 3723 36 4 v 746 3775 a(2)811 3743 y Fo(+)20 b Fk(f)10 b Fo(\()p Fk(k)s Fo(\))150 4030 y Fl(3)135 b(Quadratic)45 b(Residuosit)l(y)i(Assumption)150 4233 y Fp(De\014nition)35 b(of)g(the)g(Legendre)g(Sym)m(b)s(ol:)291 4346 y Fo(Let)c Fk(p)f Fo(b)s(e)g(a)h(prime)d(n)m(um)m(b)s(er.)291 4463 y Fe(\020)355 4528 y Fh(a)p 355 4543 38 4 v 356 4595 a(p)403 4463 y Fe(\021)505 4512 y Fk(:)482 4564 y Fo(=)578 4377 y Fe(8)578 4459 y(<)578 4623 y(:)771 4451 y Fo(0:)1182 b(if)29 b Fk(GC)7 b(D)s Fo(\()p Fk(a;)15 b(p)p Fo(\))26 b Fj(6)p Fo(=)f(1)771 4564 y(1:)193 b(if)29 b Fk(GC)7 b(D)s Fo(\()p Fk(a;)15 b(p)p Fo(\))26 b(=)f(1)31 b(and)f Fk(a)g Fo(is)f(a)i(square)f(mo)s(d)g Fk(p)701 4677 y Fj(\000)p Fo(1:)g(if)g Fk(GC)7 b(D)s Fo(\()p Fk(a;)15 b(p)p Fo(\))26 b(=)f(1)30 b(and)g Fk(a)g Fo(is)g(not)g(a)h(square)f(mo) s(d)g Fk(p)150 4832 y Fp(De\014nition)35 b(of)g(Jacobi)h(Sym)m(b)s(ol)e (for)i(RSA)f(Mo)s(duli:)291 4889 y Fe(\000)345 4926 y Fh(a)p 343 4941 43 4 v 343 4994 a(n)396 4889 y Fe(\001)485 4911 y Fk(:)462 4962 y Fo(=)558 4861 y Fe(\020)623 4926 y Fh(a)p 623 4941 38 4 v 624 4994 a(p)670 4861 y Fe(\021)16 b(\020)804 4926 y Fh(a)p 804 4941 V 806 4994 a(q)851 4861 y Fe(\021)150 5093 y Fp(Quadratic)34 b(Residuosit)m(y)i (Assumption:)1889 5416 y Fo(3-2)p eop %%Page: 3 3 3 2 bop 291 159 a Fo(If)32 b Fk(n)g Fo(is)f(an)h(RSA)g(mo)s(dulus)e(c)m (hosen)i(as)h(random,)f(and)g(a)h(random)e Fk(a)e Fj(2)f Fd(N)2863 126 y Fi(\003)2863 181 y Fh(n)2916 159 y Fo(,)33 b(with)e(Jacobi)h(sym)m(b)s(ol)150 271 y(1,)f(then)f(y)m(ou)h(can't)g (tell)f(whether)f Fk(a)i Fo(has)f(a)h(square)f(ro)s(ot)35 b(mo)s(d)30 b Fk(n)g Fo(or)g(not.)291 384 y(Using)g(our)g(notation,)g (the)h(assumption)e(is)g(as)i(follo)m(ws:)291 497 y(Let)j Fk(J)507 511 y Ff(1)547 497 y Fo(\()p Fk(n)p Fo(\))g(denote)h(the)f (selection)f(of)h(a)h(random)e(elemen)m(t)h(of)g Fd(Z)2570 464 y Fi(\003)2570 520 y Fh(n)2638 497 y Fo(mo)s(d)55 b Fk(n)33 b Fo(with)g(Jacobi)h(Sym)m(b)s(ol)150 610 y(1.)291 723 y(Let)29 b Fk(QC)7 b Fo(\()p Fk(a;)15 b(n)p Fo(\))28 b(denote)g(the)h(quadratic)e(c)m(haracter)j(of)e Fk(a)g Fo(mo)s(dulo)f Fk(n)p Fo(.)39 b(That)28 b(is)f(to)i(sa)m(y)-8 b(,)30 b Fk(QC)7 b Fo(\()p Fk(a;)15 b(n)p Fo(\))25 b(=)150 836 y(\\square")31 b(if)e Fk(a)i Fo(is)e(a)i(square)f(and)g Fk(QC)7 b Fo(\()p Fk(a;)15 b(n)p Fo(\))25 b(=)g(\\non-square")31 b(otherwise.)150 999 y Fj(8)p Fk(ppt)p Fj(f)p Fk(A)439 1014 y Fh(k)481 999 y Fj(g9)f Fo(negligible)e(function)h Fk(f)40 b Fo(suc)m(h)30 b(that)h Fj(8)p Fk(k)s Fo(:)291 1137 y Fk(P)13 b(r)s Fo([)p Fk(p)25 b Fj( )g Fk(P)13 b(R)q(I)7 b(M)j(E)5 b(S)1032 1152 y Fh(k)1075 1137 y Fo(;)15 b Fk(q)28 b Fj( )d Fk(P)13 b(R)q(I)7 b(M)j(E)5 b(S)1714 1152 y Fh(k)1757 1137 y Fo(;)15 b Fk(a)26 b Fj( )f Fk(J)2037 1151 y Ff(1)2077 1137 y Fo(\()p Fk(p)20 b Fj(\003)h Fk(q)s Fo(\);)15 b Fk(b)2402 1104 y Fi(0)2451 1137 y Fj( )25 b Fk(A)2635 1152 y Fh(k)2678 1137 y Fo(\()p Fk(n;)15 b(a)p Fo(\))316 1249 y(:)26 b Fk(b)f Fo(=)g Fk(QC)7 b Fo(\()p Fk(a;)15 b(n)p Fo(\)])26 b Fj(\024)1041 1214 y Ff(1)p 1041 1229 36 4 v 1041 1281 a(2)1106 1249 y Fo(+)20 b Fk(f)10 b Fo(\()p Fk(k)s Fo(\))150 1387 y(By)31 b(the)f(Chinese)f(Remainder)g(Theorem)h(w)m(e)h(kno)m(w)g(that)g(there) f(exist)g Fk(a)2697 1401 y Ff(1)2767 1387 y Fo(and)g Fk(a)2992 1401 y Ff(2)3062 1387 y Fo(suc)m(h)g(that)1674 1591 y Fk(a)25 b Fo(=)g Fk(a)1891 1605 y Ff(1)1956 1591 y Fo(mo)s(d)50 b Fk(p)1675 1796 y(a)25 b Fo(=)g Fk(a)1892 1810 y Ff(2)1957 1796 y Fo(mo)s(d)49 b Fk(q)150 2186 y Fl(4)135 b(Useful)46 b(Num)l(b)t(er-Theoretic)f(F)-11 b(acts)150 2389 y Fo(The)24 b(term)h(\\square")h(can)f(b)s(e)g(used)f (in)m(terc)m(hangeably)h(with)e(the)j(term)f(\\quadratic)f(residue")g (through-)150 2502 y(out.)150 2615 y Fp(F)-9 b(act)44 b(0)g(\(F)-9 b(ermat's)42 b(little)h(theorem\):)55 b Fo(If)38 b Fk(p)f Fo(is)h(prime,)g(and)g Fk(a)g Fo(is)f(relativ)m(ely)h (prime)e(to)j Fk(p)p Fo(,)h(then)150 2728 y Fk(a)198 2695 y Fh(p)263 2728 y Fj(\021)25 b Fk(a)g Fo(mo)s(d)g Fk(p)p Fo(.)150 2840 y Fp(F)-9 b(act)38 b(1:)45 b Fd(Z)574 2808 y Fi(\003)574 2863 y Fh(p)642 2840 y Fo(is)32 b(cyclic,)h(i.e.)48 b(there)33 b(exists)f(an)h(elemen)m(t)g Fk(g)g Fj(2)c Fd(Z)2364 2808 y Fi(\003)2364 2863 y Fh(p)2432 2840 y Fo(suc)m(h)j(that)h(for)g(all)f(elemen)m(ts)h Fk(a)c Fj(2)g Fd(Z)3715 2808 y Fi(\003)3715 2863 y Fh(p)150 2953 y Fo(there)36 b(exists)g(a)h(unique)d(n)m(um)m(b)s(er)h(0)g Fj(\024)f Fk(u)h Fj(\024)g Fk(p)24 b Fj(\000)f Fo(1)37 b(suc)m(h)f(that)g Fk(a)f Fj(\021)g Fk(g)2637 2920 y Fh(u)2708 2953 y Fo(mo)s(d)24 b Fk(p)p Fo(.)58 b(Suc)m(h)35 b Fk(g)40 b Fo(is)35 b(called)h(a)150 3066 y Fm(gener)-5 b(ator)37 b Fo(mo)s(d)t Fk(p)p Fo(.)150 3229 y Fp(F)-9 b(act)45 b(2)g(\(The)f(Chinese)g(Remainder)g(Theorem\):)56 b Fo(Let)40 b Fk(m)2486 3243 y Ff(1)2525 3229 y Fk(;)15 b(:)g(:)g(:)i(;)e(m)2807 3244 y Fh(k)2888 3229 y Fo(b)s(e)39 b(pairwise)e(relativ)m(ely)150 3342 y(prime)30 b(in)m(tegers.)45 b(That)31 b(is,)h Fk(g)s(cd)p Fo(\()p Fk(m)1379 3356 y Fh(i)1408 3342 y Fk(;)15 b(m)1528 3356 y Fh(j)1565 3342 y Fo(\))28 b(=)f(1)32 b(for)f(all)g(1)d Fj(\024)f Fk(i)g Fj(\024)g Fk(j)33 b Fj(\024)27 b Fk(k)s Fo(.)45 b(Let)32 b(the)g(v)-5 b(alues)31 b Fk(a)3328 3356 y Fh(i)3384 3342 y Fj(2)26 b Fd(Z)3536 3356 y Fh(m)3598 3366 y Fc(i)3657 3342 y Fo(b)s(e)150 3455 y(giv)m(en)h(for)g(1)f Fj(\024)f Fk(i)h Fj(\024)f Fk(k)s Fo(.)40 b(Let)27 b Fk(m)e Fo(=)1315 3387 y Fe(Q)1416 3455 y Fk(m)1496 3469 y Fh(i)1524 3455 y Fo(.)40 b(There)27 b(exists)g(a)h(unique)d Fk(a)g Fj(2)g Fd(Z)2686 3469 y Fh(m)2776 3455 y Fo(suc)m(h)i(that)h Fk(a)d Fj(\021)g Fk(a)3389 3469 y Fh(i)3443 3455 y Fo(mo)s(d)f Fk(m)3722 3469 y Fh(i)150 3568 y Fo(for)39 b(all)f Fk(i)p Fo(.)66 b(F)-8 b(urthermore)39 b(there)g(is)f(an)h Fk(O)s Fo(\()p Fk(k)s(`)1762 3535 y Ff(2)1802 3568 y Fo(\))g(time)g(algorithm) f(to)h(compute)h Fk(a)f Fo(giv)m(en)g Fk(a)3393 3582 y Ff(1)3432 3568 y Fk(;)15 b(:)g(:)g(:)i(;)e(a)3682 3583 y Fh(k)3725 3568 y Fo(,)150 3681 y Fk(m)230 3695 y Ff(1)269 3681 y Fk(;)g(:)g(:)g(:)i(;)e(m)551 3696 y Fh(k)594 3681 y Fo(,)30 b(where)g Fk(`)25 b Fo(=)g(max\()p Fj(j)p Fk(m)1380 3695 y Fh(i)1409 3681 y Fj(j)p Fo(\).)150 3794 y(W)-8 b(e)37 b(w)m(on't)f(giv)m(e)h(a)f(pro)s(of)f(of)h(F)-8 b(acts)37 b(0,)h(1)e(and)g(2;)j(see,)f(for)d(example,)i(Goldw)m (asser-Bellare's)f(lecture)150 3906 y(notes)31 b(for)f(reference.)150 4019 y Fp(F)-9 b(act)41 b(3:)51 b Fo(If)35 b Fk(p)f(>)g Fo(2)i(is)e(prime,)i(and)f Fk(g)k Fo(is)c(a)h(generator)i(mo)s(d)q Fk(p)p Fo(,)f(then)e Fk(g)k Fo(cannot)d(b)s(e)f(a)h(square)h(mo)s(d)q Fk(p)p Fo(.)150 4132 y(That)30 b(is,)g(for)g(all)f Fk(h)d Fj(2)f Fd(Z)988 4099 y Fi(\003)988 4155 y Fh(p)1023 4132 y Fo(,)31 b(it)f(is)f(not)i(the)f(case)i(that)f Fk(h)2021 4099 y Ff(2)2086 4132 y Fj(\021)25 b Fk(g)j Fo(mo)s(d)d Fk(p)p Fo(.)291 4245 y Fm(Pr)-5 b(o)g(of:)38 b Fo(Supp)s(ose)22 b Fj(9)p Fk(h)h Fo(suc)m(h)g(that)h Fk(h)1483 4212 y Ff(2)1548 4245 y Fo(=)h Fk(g)s Fo(,)g(then)e Fk(h)1992 4212 y Fh(p)p Fi(\000)p Ff(1)2148 4245 y Fo(=)i(1\()20 b(mo)s(d)43 b Fk(p)p Fo(\))24 b(b)m(y)f(F)-8 b(ermats)24 b(Little)f(Theorem,)150 4389 y(and)38 b(so)h(\()p Fk(h)542 4356 y Ff(2)582 4389 y Fo(\))628 4308 y Fc(p)p Fb(\000)p Fa(1)p 628 4324 111 3 v 668 4365 a(2)792 4389 y Fo(=)g(1\()o(mo)s(d)g Fk(p)p Fo(\).)66 b(Consequen)m(tly)38 b Fk(g)1995 4322 y Fc(p)p Fb(\000)p Fa(1)p 1995 4338 V 2035 4379 a(2)2159 4389 y Fo(=)h(1\()o(mo)s(d)g Fk(p)p Fo(\))g(but)f(this)g(is)g(a)h(con)m (tradiction)150 4502 y(b)s(ecause)45 b Fk(g)k Fo(is)44 b(a)i(generator.)87 b(\(Exercise:)70 b(Wh)m(y)46 b(can't)g(it)f(b)s(e)g (that)g(for)h(some)f(1)51 b Fj(\024)f Fk(u)g(<)g(p)29 b Fj(\000)h Fo(1,)150 4614 y Fk(g)196 4581 y Fh(u)267 4614 y Fj(\021)25 b Fo(1)h(mo)s(d)e Fk(n)30 b Fo(if)f Fk(g)34 b Fo(is)c(a)g(generator?\))150 4727 y Fp(F)-9 b(act)36 b(4:)44 b Fo(Let)32 b Fk(p)c(>)f Fo(2)32 b(b)s(e)f(a)h(prime.) 44 b(Let)32 b Fk(g)j Fo(b)s(e)d(a)g(generator)37 b(mo)s(d)s Fk(p)p Fo(.)45 b(Let)32 b Fk(a)c Fj(\021)f Fk(g)2952 4694 y Fh(u)3023 4727 y Fo(mo)s(d)d Fk(p)32 b Fo(b)s(e)f(giv)m(en.)45 b Fk(a)150 4840 y Fo(is)29 b(a)i(square)f(if)g(and)f(only)h(if)f Fk(u)i Fo(is)e(ev)m(en.)291 4953 y Fm(Pr)-5 b(o)g(of:)57 b Fo(Assume)37 b Fk(u)h Fo(is)f(ev)m(en,)k(then)c Fk(a)h Fo(=)g Fk(g)1816 4920 y Ff(2)p Fh(v)1930 4953 y Fo(=)g Fk(g)2085 4920 y Fh(v)2126 4918 y Ff(2)2191 4953 y Fo(mo)s(d)62 b Fk(p)38 b Fo(and)f Fk(a)h Fo(is)f(ob)m(viously)f(a)i(quadratic)150 5066 y(residue.)1889 5416 y(3-3)p eop %%Page: 4 4 4 3 bop 291 159 a Fo(Assume)41 b Fk(a)f Fo(is)g(a)i(quadratic)e (residue.)71 b(Supp)s(ose)39 b(that)j Fk(u)f Fo(is)f(o)s(dd,)i Fk(u)h Fo(=)g(2)p Fk(v)31 b Fo(+)c(1.)73 b(Since)40 b Fk(a)h Fo(is)f(a)150 271 y(quadratic)30 b(residue,)f Fk(g)937 238 y Ff(2)p Fh(v)r Ff(+1)1129 271 y Fo(=)c Fk(b)1264 238 y Ff(2)1329 271 y Fo(mo)s(d)50 b Fk(p)30 b Fo(for)g(some)h Fk(b)p Fo(.)1647 530 y Fk(g)e Fo(=)1847 468 y Fk(b)1886 435 y Ff(2)p 1825 509 123 4 v 1825 592 a Fk(g)1871 566 y Ff(2)p Fh(v)1983 530 y Fo(mo)s(d)49 b Fk(p)1645 813 y Fo(=)25 b(\()1811 752 y Fk(b)p 1786 792 88 4 v 1786 876 a(g)1832 849 y Fh(v)1884 813 y Fo(\))1919 715 y Ff(2)1984 813 y Fo(mo)s(d)50 b Fk(p)291 1019 y Fo(but)30 b(then)g Fk(g)k Fo(w)m(ould)29 b(b)s(e)h(a)g(quadratic)g (residue)f(whic)m(h)g(con)m(tradicts)i(F)-8 b(act)32 b(3.)150 1132 y Fp(F)-9 b(act)30 b(5:)38 b Fo(Let)27 b Fk(p)e(>)g Fo(2)h(b)s(e)g(a)g(prime.)38 b(Exactly)26 b(one)g(half)f(of)i(the)f(elemen)m(ts)g(of)g Fd(Z)2824 1099 y Fi(\003)2824 1154 y Fh(p)2886 1132 y Fo(are)g(quadratic)g (residues)150 1245 y(mo)s(d)t Fk(p)p Fo(.)291 1358 y Fm(Pr)-5 b(o)g(of:)42 b Fo(This)29 b(is)g(simply)f(a)j(corollary)e(to)i (F)-8 b(act)33 b(4.)150 1470 y Fp(F)-9 b(act)40 b(6:)49 b Fo(Let)35 b Fk(p)e(>)f Fo(2)j(b)s(e)f(a)h(prime.)52 b(Let)36 b Fk(a)e Fo(b)s(e)g(a)h(square)i(mo)s(d)q Fk(p)p Fo(.)54 b(Then)34 b Fk(a)g Fo(has)h(exactly)g(t)m(w)m(o)h(square)150 1583 y(ro)s(ots)g(mo)s(d)t Fk(p)p Fo(.)291 1696 y Fm(Pr)-5 b(o)g(of:)50 b Fo(First,)36 b(let)e(us)g(sho)m(w)g(that)i Fk(a)e Fo(has)g(at)i(least)f(t)m(w)m(o)g(square)g(ro)s(ots:)49 b(By)35 b(F)-8 b(act)36 b(4,)h Fk(a)32 b Fj(\021)g Fk(g)3530 1663 y Ff(2)p Fh(v)3641 1696 y Fo(for)150 1809 y(some)26 b(generator)h Fk(g)j Fo(and)25 b(some)h(n)m(um)m(b)s(er)f(1)g Fj(\024)g Fk(v)k Fj(\024)c Fo(\()p Fk(p)11 b Fj(\000)g Fo(1\))p Fk(=)p Fo(2.)41 b(Consider)24 b Fk(g)2734 1776 y Fh(v)2801 1809 y Fo(and)h Fk(g)3019 1776 y Ff(\()p Fh(p)p Fi(\000)p Ff(1\))p Fh(=)p Ff(2+)p Fh(v)3367 1809 y Fo(.)39 b(Observ)m(e)150 1922 y(that)e(they)f(m)m(ust)g(b)s(e)g (distinct.)57 b(Also)36 b(observ)m(e)g(that)h(they)g(are)f(b)s(oth)g (square)g(ro)s(ots)g(of)h Fk(a)p Fo(.)58 b(No)m(w,)38 b(w)m(e)150 2035 y(m)m(ust)c(sho)m(w)h(that)f(no)h(n)m(um)m(b)s(er)e (has)h(more)g(than)h(t)m(w)m(o)g(square)f(ro)s(ots)h(mo)s(dulo)e Fk(p)p Fo(.)52 b(F)-8 b(or)35 b(that,)h(w)m(e)f(do)g(a)150 2148 y(coun)m(ting)i(argumen)m(t:)54 b(b)m(y)36 b(F)-8 b(act)39 b(5,)g(there)e(are)g(\()p Fk(p)25 b Fj(\000)f Fo(1\))p Fk(=)p Fo(2)38 b(quadratic)f(residues.)58 b(W)-8 b(e)38 b(just)e(sa)m(w)h(that)150 2261 y(eac)m(h)c(of)f(them)g(has)f (at)i(least)f(2)g(square)f(ro)s(ots.)45 b(That)32 b(brings)e(us)h(to)i Fk(p)20 b Fj(\000)h Fo(1)32 b(elemem)m(ts)h(accoun)m(ted)g(for.)150 2374 y(There)d(is)f(no)i(more)f(ro)s(om)g(for)g(an)m(y)h(other)g (elemen)m(t)g(to)g(ha)m(v)m(e)g(an)m(y)g(other)g(square)f(ro)s(ot.)150 2487 y Fp(F)-9 b(act)40 b(7:)49 b Fo(Let)36 b Fk(p)c(>)g Fo(2)k(b)s(e)e(a)h(prime.)52 b(Then)34 b(1)h(is)f(a)h(square)i(mo)s(d)q Fk(p)e Fo(and)f(its)g(only)g(square)g(ro)s(ots)h(are)h(1)150 2600 y(and)30 b Fj(\000)p Fo(1)25 b(mo)s(d)g Fk(p)p Fo(.)291 2712 y Fm(Pr)-5 b(o)g(of:)42 b Fo(F)-8 b(act)32 b(7)f(is)e(simply)f(a)j (corollary)f(of)g(F)-8 b(act)33 b(6.)150 2843 y Fp(F)-9 b(act)35 b(8:)41 b Fo(Let)30 b Fk(p)g Fo(b)s(e)g(a)h(prime.)39 b(Then)1471 2742 y Fe(\020)1535 2807 y Fh(ab)p 1535 2822 68 4 v 1551 2874 a(p)1612 2742 y Fe(\021)1692 2843 y Fo(=)1788 2742 y Fe(\020)1852 2807 y Fh(a)p 1852 2822 38 4 v 1853 2874 a(p)1900 2742 y Fe(\021)15 b(\020)2036 2807 y Fh(b)p 2033 2822 36 4 v 2033 2874 a(p)2079 2742 y Fe(\021)2133 2843 y Fo(.)291 3015 y Fm(Pr)-5 b(o)g(of:)47 b Fo(Let)33 b Fk(g)k Fo(b)s(e)32 b(a)i(generator.)49 b(Let)34 b Fk(a)29 b Fj(\021)g Fk(g)1862 2982 y Fh(u)1933 3015 y Fo(mo)s(d)c Fk(p)p Fo(,)33 b Fk(b)d Fj(\021)f Fk(g)2452 2982 y Fh(v)2519 3015 y Fo(mo)s(d)24 b Fk(p)p Fo(.)48 b(By)34 b(F)-8 b(act)35 b(4,)3293 2914 y Fe(\020)3357 2979 y Fh(ab)p 3357 2994 68 4 v 3373 3046 a(p)3435 2914 y Fe(\021)3518 3015 y Fo(=)30 b(1)j(if)150 3187 y(and)g(only)h(if)f Fk(u)22 b Fo(+)h Fk(v)37 b Fo(is)c(ev)m(en,)j(otherwise)d(it's)h Fj(\000)p Fo(1.)52 b(On)33 b(the)h(other)g(hand,)2773 3086 y Fe(\020)2837 3151 y Fh(a)p 2837 3166 38 4 v 2838 3218 a(p)2884 3086 y Fe(\021)16 b(\020)3021 3151 y Fh(b)p 3018 3166 36 4 v 3018 3218 a(p)3063 3086 y Fe(\021)3149 3187 y Fo(=)31 b(1)k(also)f(if)f(and)150 3317 y(only)c(if)h Fk(u)20 b Fo(+)g Fk(v)34 b Fo(is)29 b(ev)m(en,)i(otherwise)f(it's)g Fj(\000)p Fo(1.)41 b(F)-8 b(act)32 b(6)f(follo)m(ws.)150 3430 y Fp(F)-9 b(act)35 b(9:)41 b Fo(The)29 b(Legendre)i(Sym)m(b)s(ol)e (is)g(computed)h(b)m(y)g(the)h(follo)m(wing)e(algorithm:)291 3580 y(T)-8 b(est)31 b Fk(p)p Fj(j)p Fk(a)p Fo(,)g(if)e Fk(p)h Fo(do)s(es)g(divide)e Fk(a)j Fo(output)f(0,)h(otherwise)f (output)f Fk(a)2538 3513 y Fc(p)p Fb(\000)p Fa(1)p 2539 3529 111 3 v 2578 3570 a(2)2688 3580 y Fo(mo)s(d)50 b Fk(p)291 3693 y Fm(Pr)-5 b(o)g(of:)42 b Fo(Case)31 b(1:)41 b Fk(a)30 b Fo(is)f(a)i(square.)41 b(By)30 b(F)-8 b(act)33 b(4)d Fj(9)p Fk(v)k Fo(suc)m(h)c(that)h Fk(a)25 b Fo(=)g Fk(g)2634 3660 y Ff(2)p Fh(v)2736 3693 y Fo(mo)s(d)50 b Fk(p)1208 3927 y(a)1266 3859 y Fc(p)p Fb(\000)p Fa(1)p 1266 3874 V 1306 3916 a(2)1416 3927 y Fo(=)25 b(\()p Fk(g)1593 3889 y Ff(2)p Fh(v)1670 3927 y Fo(\))1715 3842 y Fc(p)p Fb(\000)p Fa(1)p 1715 3858 V 1755 3899 a(2)1865 3927 y Fo(=)g Fk(g)2007 3889 y Fh(v)r Ff(\()p Fh(p)p Fi(\000)p Ff(1\))2255 3927 y Fo(=)g(1)g(mo)s(d)50 b Fk(p)150 4131 y Fo(Th)m(us)29 b(w)m(e)i(ha)m(v)m(e)h(sho)m(wn)d(that)i(if)f Fk(a)g Fo(is)g(a)g(square,)h(the)f(Legendre)h(Sym)m(b)s(ol)d(is)i (computed)g(correctly)-8 b(.)291 4244 y(Case)38 b(2:)57 b Fk(a)38 b Fo(is)f(not)h(a)g(square.)63 b(Then)37 b(b)m(y)h(F)-8 b(act)40 b(4,)h Fk(a)d Fj(\021)f Fk(g)2365 4211 y Ff(2)p Fh(v)r Ff(+1)2557 4244 y Fo(mo)s(d)25 b Fk(p)p Fo(.)63 b(Therefore,)40 b Fk(a)3386 4211 y Ff(\()p Fh(p)p Fi(\000)p Ff(1\))p Fh(=)p Ff(2)3679 4244 y Fj(\021)150 4357 y Fo(\()p Fk(g)231 4324 y Fh(v)273 4357 y Fo(\))308 4324 y Fh(p)p Fi(\000)p Ff(1)438 4357 y Fk(g)484 4324 y Ff(\()p Fh(p)p Fi(\000)p Ff(1\))p Fh(=)p Ff(2)765 4357 y Fo(mo)s(d)25 b Fk(p)p Fo(.)40 b(By)29 b(F)-8 b(ermat's)30 b(little)d(theorem,)j(w)m (e)f(kno)m(w)g(that)g(\()p Fk(g)2828 4324 y Fh(v)2870 4357 y Fo(\))2905 4324 y Fh(p)p Fi(\000)p Ff(1)3060 4357 y Fj(\021)c Fo(1.)41 b(What)29 b(ab)s(out)150 4470 y Fk(g)196 4437 y Ff(\()p Fh(p)p Fi(\000)p Ff(1\))p Fh(=)p Ff(2)477 4470 y Fo(mo)s(d)c Fk(p)p Fo(?)58 b(First,)37 b(observ)m(e)g(that)g(it)f(is)f(a)i(square)f(ro)s(ot)h(of)f(1,)j(since) c Fk(g)2872 4437 y Fh(p)p Fi(\000)p Ff(1)3038 4470 y Fj(\021)g Fo(1.)59 b(No)m(w,)39 b(notice)150 4583 y(that)d(since)f Fk(g)k Fo(is)34 b(a)i(generator,)i(it)d(cannot)h(b)s(e)f(equal)g(to)h (1.)56 b(1)36 b(has)f(only)f(t)m(w)m(o)j(square)e(ro)s(ots,)i(and)e(b)m (y)150 4696 y(F)-8 b(act)32 b(7)f(they)g(are)f(1)h(and)f Fj(\000)p Fo(1.)41 b(W)-8 b(e'v)m(e)32 b(ruled)d(out)i(1,)g(so)f(it)g (m)m(ust)g(b)s(e)g Fj(\000)p Fo(1.)150 4809 y Fp(F)-9 b(act)40 b(10:)49 b Fk(a)35 b Fo(is)f(a)h(quadratic)f(residue)h(mo)s(d) f Fk(n)e Fo(=)h Fk(p)1998 4823 y Ff(1)2037 4809 y Fk(p)2083 4823 y Ff(2)2157 4809 y Fo(if)g(and)h(only)g(if)g Fk(a)2765 4823 y Ff(1)2839 4809 y Fo(is)g(a)h(square)h(mo)s(d)r Fk(p)3530 4823 y Ff(1)3604 4809 y Fo(and)150 4922 y Fk(a)198 4936 y Ff(2)268 4922 y Fo(is)29 b(a)i(square)k(mo)s(d)t Fk(p)951 4936 y Ff(2)991 4922 y Fo(,)30 b(where)g Fk(p)1355 4936 y Ff(1)1419 4922 y Fk(>)25 b Fo(2)31 b(and)f Fk(p)1814 4936 y Ff(2)1878 4922 y Fk(>)25 b Fo(2)31 b(are)g(primes.)291 5035 y Fm(Pr)-5 b(o)g(of:)43 b Fo(Let)31 b(us)g(\014rst)f(sho)m(w)h (the)g Fj(\))g Fo(direction.)41 b(Assume)30 b Fk(a)h Fo(is)f(a)h(quadratic)g(residue.)40 b(W)-8 b(e)33 b(need)d(to)150 5147 y(sho)m(w)g(that)h Fk(a)620 5161 y Ff(1)690 5147 y Fo(and)f Fk(a)915 5161 y Ff(2)985 5147 y Fo(are)g(also)h(quadratic)e (residues.)1889 5416 y(3-4)p eop %%Page: 5 5 5 4 bop 291 159 a Fj(9)p Fk(b)30 b Fo(suc)m(h)g(that)h Fk(b)852 126 y Ff(2)917 159 y Fo(=)25 b Fk(a)g Fo(mo)s(d)55 b Fk(n)p Fo(.)150 271 y(Let)31 b Fk(b)352 285 y Ff(1)417 271 y Fo(=)25 b Fk(b)g Fo(mo)s(d)50 b Fk(p)848 285 y Ff(1)917 271 y Fo(and)30 b Fk(b)p Fo(2)c(=)e Fk(b)i Fo(mo)s(d)49 b Fk(p)1634 285 y Ff(2)1674 271 y Fo(.)1454 476 y Fk(b)1493 438 y Ff(2)1493 498 y(1)1558 476 y Fo(=)25 b Fk(b)1693 438 y Ff(2)1758 476 y Fo(=)g Fk(a)g Fo(=)g Fk(a)2071 490 y Ff(1)2136 476 y Fo(mo)s(d)49 b Fk(p)2406 490 y Ff(1)150 680 y Fo(.)42 b(Therefore,)31 b Fk(a)703 694 y Ff(1)773 680 y Fo(is)f(a)i(square.)42 b(Similarly)27 b(for)j Fk(b)1830 694 y Ff(2)1870 680 y Fo(,)h(whic)m(h)f(pro)m(v)m(es) h(that)h Fk(a)2714 694 y Ff(2)2784 680 y Fo(is)e(a)h(quadratic)f (residue)g(as)150 793 y(w)m(ell.)150 956 y(Assume)g(that)h Fk(a)732 970 y Ff(1)802 956 y Fo(and)e Fk(a)1026 970 y Ff(2)1096 956 y Fo(are)i(quadratic)f(residues.)150 1093 y Fj(9)p Fk(b)240 1107 y Ff(1)279 1093 y Fk(;)15 b(b)358 1107 y Ff(2)437 1093 y Fo(suc)m(h)39 b(that)h Fk(a)905 1107 y Ff(1)984 1093 y Fo(=)g Fk(b)1134 1060 y Ff(2)1134 1118 y(1)1198 1093 y Fo(mo)s(d)65 b Fk(p)39 b Fo(and)f Fk(a)1756 1107 y Ff(2)1836 1093 y Fo(=)h Fk(b)1985 1060 y Ff(2)1985 1118 y(2)2050 1093 y Fo(mo)s(d)64 b Fk(q)s Fo(.)j(Then)38 b(b)m(y)h(the)g(Chinese)f(Remainder)150 1206 y(Theorem,)30 b Fj(9)p Fk(b)g Fo(suc)m(h)g(that)h Fk(b)25 b Fo(=)g Fk(b)1280 1220 y Ff(1)1345 1206 y Fo(mo)s(d)55 b Fk(p)30 b Fo(and)g Fk(b)25 b Fo(=)g Fk(b)2027 1220 y Ff(2)2092 1206 y Fo(mo)s(d)49 b Fk(q)s Fo(.)1659 1432 y Fk(b)1698 1395 y Ff(2)1762 1432 y Fo(=)25 b Fk(a)1906 1446 y Ff(1)1971 1432 y Fo(mo)s(d)50 b Fk(p)1659 1599 y(b)1698 1561 y Ff(2)1762 1599 y Fo(=)25 b Fk(a)1906 1613 y Ff(2)1971 1599 y Fo(mo)s(d)50 b Fk(p)291 1766 y Fo(Therefore)30 b(since)g Fk(a)g Fo(is)g(unique,)e Fk(b)1455 1733 y Ff(2)1520 1766 y Fo(=)d Fk(a)g Fo(mo)s(d)50 b Fk(n)30 b Fo(b)m(y)g(the)h(Chinese)e(Remainder)g(Theorem.)150 1879 y Fp(F)-9 b(act)36 b(11:)43 b Fo(Let)32 b Fk(n)f Fo(b)s(e)g(an)g(RSA)g(mo)s(dulus)e(\(i.e.,)j Fk(n)26 b Fo(=)h Fk(p)2077 1893 y Ff(1)2116 1879 y Fk(p)2162 1893 y Ff(2)2201 1879 y Fo(,)32 b(where)f Fk(p)2568 1893 y Ff(1)2634 1879 y Fk(>)c Fo(2)32 b(and)f Fk(p)3033 1893 y Ff(2)3099 1879 y Fk(>)26 b Fo(2)32 b(are)g(primes\).)150 1992 y(Exactly)f(one)f(quarter)g(of)h(the)g(elemen)m(ts)f(of)h Fd(Z)1773 1959 y Fi(\003)1773 2014 y Fh(n)1846 1992 y Fo(are)f(quadratic)g(residues.)291 2105 y Fm(Pr)-5 b(o)g(of:)42 b Fo(Easy)30 b(to)i(see)e(b)m(y)h(com)m(bining)e(F)-8 b(act)32 b(10)f(with)e(F)-8 b(act)32 b(5.)150 2217 y Fp(F)-9 b(act)36 b(12:)43 b Fo(Let)31 b Fk(n)c Fo(=)f Fk(pq)34 b Fo(b)s(e)c(an)h(RSA)g(mo)s(dulus.)41 b(Exactly)31 b(half)f(of)i(the)f(elemen)m(ts)h(of)f Fd(Z)3212 2184 y Fi(\003)3212 2240 y Fh(n)3286 2217 y Fo(ha)m(v)m(e)h(Jacobi)150 2330 y(Sym)m(b)s(ol)d(1.)291 2443 y Fm(Pr)-5 b(o)g(of:)43 b Fo(This)29 b(is)g(obtained)i(b)m(y)f(a)h(coun)m(ting)g(argumen)m(t.) 42 b(Consider)29 b Fk(a)d Fj(2)g Fd(Z)2869 2410 y Fi(\003)2869 2466 y Fh(p)2904 2443 y Fo(,)31 b Fk(b)26 b Fj(2)f Fd(Z)3176 2410 y Fi(\003)3176 2466 y Fh(q)3211 2443 y Fo(.)42 b(By)31 b(the)g(Chi-)150 2556 y(nese)c(Remainder)f(Theorem,)i(there)g(is)e(a)i (unique)d(elemen)m(t)j Fk(c)f Fo(that)h(corresp)s(onds)e(to)i(them.)40 b(Moreo)m(v)m(er,)150 2687 y(b)m(y)29 b(de\014nition,)704 2613 y Fe(\000)761 2651 y Fh(c)p 755 2666 43 4 v 755 2718 a(n)808 2613 y Fe(\001)875 2687 y Fo(=)971 2586 y Fe(\020)1035 2651 y Fh(a)p 1035 2666 38 4 v 1036 2718 a(p)1083 2586 y Fe(\021)15 b(\020)1218 2651 y Fh(b)p 1216 2666 34 4 v 1216 2718 a(q)1260 2586 y Fe(\021)1315 2687 y Fo(.)40 b(There)29 b(are)h(four)e(p)s(ossibilities.)36 b(Case)30 b(1:)2832 2586 y Fe(\020)2897 2651 y Fh(a)p 2897 2666 38 4 v 2898 2718 a(p)2944 2586 y Fe(\021)3024 2687 y Fo(=)25 b(1,)3220 2586 y Fe(\020)3286 2651 y Fh(b)p 3284 2666 34 4 v 3284 2718 a(q)3328 2586 y Fe(\021)3407 2687 y Fo(=)g(1,)31 b(and)150 2859 y(so)263 2785 y Fe(\000)321 2823 y Fh(c)p 315 2838 43 4 v 315 2890 a(n)368 2785 y Fe(\001)437 2859 y Fo(=)c(1.)46 b(Case)32 b(2:)984 2758 y Fe(\020)1049 2823 y Fh(a)p 1049 2838 38 4 v 1050 2890 a(p)1096 2758 y Fe(\021)1178 2859 y Fo(=)c(1,)1379 2758 y Fe(\020)1446 2823 y Fh(b)p 1444 2838 34 4 v 1444 2890 a(q)1488 2758 y Fe(\021)1570 2859 y Fo(=)f Fj(\000)p Fo(1,)33 b(and)e(so)2133 2785 y Fe(\000)2191 2823 y Fh(c)p 2185 2838 43 4 v 2185 2890 a(n)2237 2785 y Fe(\001)2307 2859 y Fo(=)c Fj(\000)p Fo(1.)46 b(Case)32 b(3:)2925 2758 y Fe(\020)2989 2823 y Fh(a)p 2989 2838 38 4 v 2990 2890 a(p)3037 2758 y 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y(securit)m(y)e(of)h(the)f(encrypted)g(bit,)h(our)f (algorithm)f Fq(A)i Fv(will)d(break)i(the)h(QR)f(assumption,)g(i.e.)52 b(it)34 b(will,)150 4095 y(with)29 b(probabilit)m(y)f(non-negligibly)f (b)s(etter)k(than)f(1)p Fq(=)p Fv(2,)i(determine)d(the)i(quadratic)f(c) m(haracter)i(of)e Fq(a)p Fv(.)3646 4062 y Fn(3)p 150 4157 1440 4 v 254 4210 a Fk(2)288 4242 y Fj(W)-6 b(e)31 b(could)h(sho)n(w)g(instead)g(that)f(existence)h(of)g(Ev)n(e)f(con)n (tradicts)h(some)f(other)h(assumption.)51 b(F)-6 b(or)32 b(example,)g(a)g(re-)150 4334 y(duction)h(that)g(could)h(solv)n(e)g (circuit)f(satis\014abilit)n(y)-6 b(,)37 b(w)n(ould)d(do)f(nicely)-6 b(.)57 b(Ho)n(w)n(ev)n(er,)36 b(it)d(is)h(not)g(kno)n(wn)f(whether)g (this)g(is)150 4425 y(p)r(ossible;)26 b(actually)e(sho)n(wing)g(a)g (reduction)f(from)g(secure)h(encryption)e(to)i(SA)-6 b(T)22 b(is)i(b)r(eliev)n(ed)g(to)f(b)r(e)g(nearly)h(as)g(c)n (hallenging)150 4516 y(as)i(pro)n(ving)g(that)f Fi(P)32 b Fh(6)p Fj(=)21 b Fi(N)8 b(P)j Fj(.)35 b(So)26 b(instead,)g(w)n(e)g (fo)r(cus)h(on)e(hardness)h(assumptions)f(that)h(ha)n(v)n(e)f(nice)h (prop)r(erties.)254 4576 y Fk(3)288 4608 y Fj(Our)i(reduction)h(m)n (ust)e(b)r(e)h(v)n(ery)g(careful)i(with)e(the)h(results)g(returned)e(b) n(y)h(Ev)n(e.)43 b(It)28 b(ma)n(y)f(b)r(e)i(that)f(Ev)n(e)g(op)r (erates)i(in)150 4699 y(a)h(w)n(a)n(y)g(that)g(is)g(least)h(fa)n(v)n (orable)g(to)f(us)f(in)h(an)n(y)f(\(and)h(ev)n(ery)f(situation\).)50 b(Th)n(us,)32 b(our)f(reduction)g(m)n(ust)e(b)r(e)i(sure)g(to)g(b)r(e) 150 4790 y(correct)26 b(ev)n(en)f(in)h(this)g(case.)227 4882 y(An)19 b(example)h(of)h(a)g(wrong)h(approac)n(h:)32 b(the)20 b(reduction)g(receiv)n(es)h(\()p Fi(n;)14 b(a)p Fj(\))20 b(and)g(w)n(an)n(ts)h(to)f(kno)n(w)h(whether)f Fi(a)g Fj(is)h(a)g(quadratic)150 4973 y(residue.)41 b(Creates)29 b Fi(P)11 b(K)30 b Fj(=)25 b(\()p Fi(n;)13 b(u)p Fj(\),)29 b(where)f Fi(u)g Fj(is)g(just)g(some)g(random)e(thing.)41 b(Then)28 b(it)g(giv)n(es)g(\()p Fi(a;)13 b Fj(\()p Fi(n;)g(u)p Fj(\)\))28 b(to)g(Ev)n(e.)40 b(If)28 b(Ev)n(e)150 5064 y(sa)n(ys)f(\\0")g(w)n(e)g(sa)n(y)g(\\square,")g(otherwise)h(w)n(e)f (sa)n(y)f(\\non-square.")37 b(The)27 b(problem)f(is)h(lik)n(e)f(this:) 36 b(Supp)r(ose)26 b(Ev)n(e)g(is)h(sneaky)1889 5416 y Fv(4-3)p eop %%Page: 4 4 4 3 bop 291 159 a Fv(Assume,)29 b(for)g(con)m(tradiction,)h(that)g (there)f(exists)g(an)g(adv)m(ersary)h Fq(E)5 b(v)s(e)30 b Fv(suc)m(h)f(that)h(there)f(exists)g(an)150 271 y(in)m(v)m(erse-p)s (olynomial)f Fq(p)p Fv(\()p Fq(k)s Fv(\))j(suc)m(h)f(that)h(for)f (in\014nitely)d(man)m(y)k Fq(k)s Fv(,)377 459 y(Pr[)15 b(\()p Fq(P)e(K)q(;)i(S)5 b(K)i Fv(\))27 b Fp( )e Fq(G)p Fv(\(1)1213 426 y Fo(k)1257 459 y Fv(\);)515 572 y Fq(b)g Fp( )h(f)p Fv(0)p Fq(;)15 b Fv(1)p Fp(g)p Fv(;)515 685 y Fq(c)26 b Fp( )f Fq(E)5 b Fv(\()p Fq(P)13 b(K)q(;)i(b)p Fv(\);)515 798 y Fq(b)554 765 y Fm(0)603 798 y Fp( )25 b Fq(E)5 b(v)s(e)p Fv(\()p Fq(P)13 b(K)q(;)i(c)p Fv(\))28 b(:)117 b Fq(b)1387 765 y Fm(0)1435 798 y Fv(=)25 b Fq(b)p Fv(])h Fp(\025)f Fv(1)p Fq(=)p Fv(2)c(+)f Fq(p)p Fv(\()p Fq(k)s Fv(\))150 985 y(Consider)35 b(the)i(follo)m(wing)f(algorithm:)53 b(on)37 b(input)e(\()p Fq(n;)15 b(a)p Fv(\),)40 b(set)d Fq(P)13 b(K)43 b Fv(=)36 b(\()p Fq(n;)15 b(a)p Fv(\).)61 b(Then,)38 b(c)m(ho)s(ose)g Fq(b)p Fv(,)p Fq(r)i Fv(at)150 1098 y(random)33 b(and)g(let)g Fq(c)e Fv(=)f Fq(r)1016 1065 y Fn(2)1055 1098 y Fq(a)1103 1065 y Fo(b)1163 1098 y Fv(mo)s(d)25 b Fq(n)p Fv(.)49 b(Obtain)33 b Fq(b)1843 1065 y Fm(0)1897 1098 y Fv(=)d(Ev)m(e)q(\()p Fq(P)13 b(K)q(;)i(c)p Fv(\).)52 b(If)33 b Fq(b)2654 1065 y Fm(0)2708 1098 y Fv(=)d Fq(b)p Fv(,)35 b(output)e(\\Non-square")150 1211 y(\(or)e(1\))g(Otherwise,)e(output)h(\\Square")h(\(or)f(0\).)150 1324 y Fw(Claim)39 b(1:)51 b Fv(if)34 b Fq(a)h Fv(is)g(a)h(non-square,) g(then)f(the)g(probabilit)m(y)e(that)j Fq(b)2521 1291 y Fm(0)2578 1324 y Fv(=)d Fq(b)j Fv(in)e(this)g(exp)s(erimen)m(t)g(is)h (the)150 1437 y(same)c(as)f(the)h(probabilit)m(y)d(that)j(Ev)m(e)g (guesses)g(the)f(encrypted)g(bit)f(correctly)-8 b(.)42 b(More)31 b(precisely)-8 b(,)377 1625 y(Pr[)25 b Fq(n)g Fp( )g Fq(R)q(S)5 b(AM)1008 1640 y Fo(k)1051 1625 y Fv(;)525 1738 y Fq(a)26 b Fp( )f Fq(QR)q Fv(\()p Fq(n)p Fv(\);)525 1850 y Fq(b)h Fp( )f(f)p Fv(0)p Fq(;)15 b Fv(1)p Fp(g)p Fv(;)525 1963 y Fq(r)28 b Fp( )d Fl(Z)775 1930 y Fm(\003)775 1986 y Fo(n)818 1963 y Fv(;)525 2076 y Fq(c)h Fp( )f Fq(r)750 2043 y Fn(2)789 2076 y Fq(a)837 2043 y Fo(b)871 2076 y Fv(;)525 2189 y Fq(b)564 2156 y Fm(0)613 2189 y Fp( )g Fq(E)5 b(v)s(e)p Fv(\(\()p Fq(n;)15 b(a)p Fv(\))p Fq(;)g(c)p Fv(\))28 b(:)117 b Fq(b)1461 2156 y Fm(0)1509 2189 y Fv(=)25 b Fq(b)p Fv(])h Fp(\025)f Fv(1)p Fq(=)p Fv(2)c(+)f Fq(p)p Fv(\()p Fq(k)s Fv(\))150 2377 y Ft(Pr)-5 b(o)g(of:)44 b Fv(The)32 b(claim)e(follo)m(ws)h(b)s(ecause)h(in)e(this) h(case)i(w)m(e)f(are)g(cop)m(ying)g(the)g(GM)g(cryptosystem)g(set-up) 150 2490 y(exactly)-8 b(.)46 b(The)31 b(\014rst)g(t)m(w)m(o)j(steps)d (comprise)g Fq(G)p Fv(,)i(the)f(next)g(step)g(is)f(the)h(selection)f (of)h(the)g(message,)i(and)150 2603 y(the)e(next)h(t)m(w)m(o)g(steps)g (are)f(the)h(encryption)e Fq(E)5 b Fv(.)47 b(This)30 b(is)h(exactly)i(what)f(w)m(e)h(ha)m(v)m(e)h(assumed)d(that)i Fq(E)5 b(v)s(e)150 2716 y Fv(can)31 b(do.)3241 b Fd(\003)150 2829 y Fw(Claim)33 b(2:)41 b Fv(if)29 b Fq(a)i Fv(is)e(a)i(square,)f (then)g(the)h(probabilit)m(y)d(that)j Fq(b)2283 2796 y Fm(0)2331 2829 y Fv(=)25 b Fq(b)31 b Fv(is)e(exactly)i(1)p Fq(=)p Fv(2.)42 b(More)31 b(precisely:)377 3016 y Fp(8)p Fq(P)13 b(P)g(T)g(F)g Fp(f)p Fq(E)5 b(v)s(e)913 3031 y Fo(k)957 3016 y Fp(g)521 3129 y Fv(Pr[)25 b Fq(n)g Fp( )g Fq(R)q(S)5 b(AM)1152 3144 y Fo(k)1195 3129 y Fv(;)669 3242 y Fq(a)25 b Fp( )h Fq(QR)q Fv(\()p Fq(n)p Fv(\);)669 3355 y Fq(b)f Fp( )h(f)p Fv(0)p Fq(;)15 b Fv(1)p Fp(g)p Fv(;)669 3468 y Fq(r)28 b Fp( )d Fl(Z)919 3435 y Fm(\003)919 3490 y Fo(n)962 3468 y Fv(;)669 3581 y Fq(c)h Fp( )f Fq(r)894 3548 y Fn(2)933 3581 y Fq(a)981 3548 y Fo(b)1015 3581 y Fv(;)669 3694 y Fq(b)708 3661 y Fm(0)757 3694 y Fp( )g Fq(E)5 b(v)s(e)1034 3709 y Fo(k)1078 3694 y Fv(\(\()p Fq(n;)15 b(a)p Fv(\))p Fq(;)g(c)p Fv(\))27 b(:)116 b Fq(b)1647 3661 y Fm(0)1696 3694 y Fv(=)25 b Fq(b)p Fv(])g(=)g(1)p Fq(=)p Fv(2)150 3881 y Ft(Pr)-5 b(o)g(of)40 b(\(intuition\):)53 b Fv(Supp)s(ose)34 b Fq(a)i Fv(is)g(a)g(quadratic)g(residue.)57 b(Then,)37 b(no)f(matter)i(what)e Fq(b)g Fv(is,)h Fq(c)g Fv(is)e(dis-)150 3994 y(tributed)26 b(in)g(the)i(same)g(w)m(a)m(y)-8 b(,)30 b(i.e.)39 b Fq(c)28 b Fv(is)f(alw)m(a)m(ys)h(a)f(random)g(quadratic)g (residue.)38 b(Th)m(us)27 b Fq(b)g Fv(is)g(distributed)150 4107 y(indep)s(enden)m(tly)h(on)i(\()p Fq(n;)15 b(a;)g(c)p Fv(\).)44 b(Therefore,)31 b(w)m(e)g(can)g(view)f(Ev)m(e's)i(output)e Fq(b)2752 4074 y Fm(0)2806 4107 y Fv(as)h(\014xed)f(b)s(efore)h Fq(b)f Fv(is)g(ev)m(en)150 4220 y(decided.)40 b(Therefore,)30 b(the)h(probabilit)m(y)c(that)k Fq(b)g Fv(will)c(come)32 b(out)e(equal)g(to)h Fq(b)2794 4187 y Fm(0)2848 4220 y Fv(is)e(exactly)j(1)p Fq(=)p Fv(2.)150 4333 y Ft(Pr)-5 b(o)g(of)40 b(\(formal\):)55 b Fv(Supp)s(ose)35 b Fq(a)i Fv(is)f(a)h(quadratic)f(residue)g(and)g Fq(n)g Fv(as)h(giv)m(en.)60 b(Then,)38 b Fq(E)5 b Fv(\(\()p Fq(n;)15 b(a)p Fv(\))p Fq(;)g Fv(0\))40 b(and)150 4446 y Fq(E)5 b Fv(\(\()p Fq(n;)15 b(a)p Fv(\))p Fq(;)g Fv(1\))40 b(ha)m(v)m(e)e(distributions)c (iden)m(tical)i(to)i(that)g(of)f(a)h(randomly)e(selected)h(quadratic)g (residue.)150 4559 y(Consider)25 b(a)j(quadratic)f(residue)e Fq(r)s Fv(.)39 b(Observ)m(e)27 b(\014rst)g(that)h Fp(j)p Fq(QR)q Fp(j)d Fv(=)g Fq(')p Fv(\()p Fq(n)p Fv(\))p Fq(=)p Fv(4)i(=)e(\()p Fq(p)14 b Fp(\000)g Fv(1\)\()p Fq(q)j Fp(\000)d Fv(1\))p Fq(=)p Fv(4.)41 b(Th)m(us,)p 150 4620 1440 4 v 150 4706 a Fj(and)26 b(she)g(can)g(really)h(tell)g(QRs)e(from) h(QNRs.)35 b(So)26 b(here's)h(what)f(she)h(do)r(es:)36 b(if)26 b(\()p Fi(n;)14 b(u)p Fj(\))26 b(is)h(a)f(v)l(alid)g(PK)g (\(i.e.,)i Fi(u)e Fj(is)h(a)f(QNR\),)150 4797 y(then)33 b(she)i(giv)n(es)f(the)g(righ)n(t)g(answ)n(er.)61 b(Otherwise,)37 b(she)d(sneakily)h(do)r(es)f(this:)52 b(if)2613 4737 y Fc(\000)2659 4766 y Ff(u)p 2658 4780 39 4 v 2658 4822 a(n)2706 4737 y Fc(\001)2777 4797 y Fj(=)35 b Fh(\000)p Fj(1)f(she)h(alw)n(a)n(ys)g(outputs)e(0.)150 4888 y(Otherwise,)c(if)f Fi(u)f Fj(is)h(a)g(square,)g(she)g(giv)n(es)g(the)f Fb(wr)l(ong)h Fj(answ)n(er.)40 b(Then)28 b(our)f(reduction)g(can)h(only)f(w)n(ork)h (if)g(it)f(can)h(pic)n(k)f Fi(u)150 4980 y Fj(correctly!)35 b(But)26 b Fi(u)g Fj(w)n(as)h(pic)n(k)n(ed)e(at)g(random,)h(so)g(with)g (prob)g(1/2)g(it)g(is)h(go)r(o)r(d,)g(and)e(w)n(e)h(are)h(righ)n(t,)f (and)f(with)h(prob)g(1/2)h(it)150 5071 y(is)f(bad)g(and)f(w)n(e)h(are)g (wrong,)h(so)f(w)n(e)g(are)h(really)f(not)g(doing)g(an)n(y)f(b)r(etter) g(than)g(1/2.)227 5162 y(So)g(w)n(e)i(ha)n(v)n(e)e(to)g(outsmart)h (sneaky)f(Ev)n(e)g(so)h(she)g(has)g(no)g(clue.)1889 5416 y Fv(4-4)p eop %%Page: 5 5 5 4 bop 150 159 a Fv(for)36 b(a)h(quadratic)f(residue)f(c)m(hosen)i (uniformly)d(at)j(random)f(from)g(the)h(space)g(of)g(quadratic)f (residues)150 271 y(mo)s(dulo)29 b Fq(n)p Fv(,)h(probabilit)m(y)e(of)i (selecting)h Fq(r)h Fv(is)e Fp(j)p Fq(QR)q Fp(j)1881 238 y Fm(\000)p Fn(1)1975 271 y Fv(,)h(or)f(4)p Fq(=')p Fv(\()p Fq(n)p Fv(\).)291 384 y(No)m(w,)k(let)e(us)g(consider)f Fq(\032)1181 398 y Fn(1)1249 384 y Fv(=)d(Pr)o([)p Fq(r)1514 351 y Fm(0)1566 384 y Fp( )g Fq(E)5 b Fv(\(\()p Fq(n;)15 b(a)p Fv(\))p Fq(;)g Fv(1\))31 b(:)e Fq(r)2254 351 y Fm(0)2305 384 y Fv(=)g Fq(r)s Fv(].)46 b(This)30 b(means)i(w)m(e)h(tak) m(e)h(a)f(random)150 497 y Fq(x)202 464 y Fn(2)270 497 y Fv(and)28 b(transform)f(it)h(b)m(y)h(m)m(ultiplication)c(b)m(y)j Fq(a)p Fv(.)40 b(Since)28 b Fq(a)g Fv(is)g(a)g(quadratic)g(residue)f (\(b)m(y)i(assumption\),)150 610 y(the)i(pro)s(duct)e(is)g(also)h(a)h (quadratic)f(residue.)39 b(W)-8 b(e)32 b(sho)m(w)e(that)h(this)e(is)h (a)g(bijectiv)m(e)h(map.)150 798 y Fw(one-to-one)46 b Fv(W)-8 b(e)31 b(need)f Fq(a)1091 765 y Fm(\000)p Fn(1)1216 798 y Fv(to)h(exist;)g(but)e(since)h(this)f(is)h(a)g(subset)g(of)h Fl(Z)2749 765 y Fm(\003)2749 820 y Fo(n)2791 798 y Fv(,)g(it)f(do)s (es.)150 985 y Fw(on)m(to)46 b Fv(W)-8 b(e)40 b(m)m(ust)e(sho)m(w)h (that)g(giv)m(en)g Fq(y)1527 952 y Fn(2)1566 985 y Fv(,)i(there)e(is)e Fq(x)2024 952 y Fn(2)2102 985 y Fv(suc)m(h)h(that)i Fq(y)2569 952 y Fn(2)2647 985 y Fv(=)f Fq(ax)2857 952 y Fn(2)2896 985 y Fv(.)66 b(Let)39 b Fq(\013)3216 952 y Fn(2)3295 985 y Fv(=)f Fq(a)p Fv(.)66 b(Then)377 1098 y(\()p Fq(y)s(=\013)p Fv(\))598 1065 y Fn(2)664 1098 y Fv(=)25 b Fq(x)812 1065 y Fn(2)852 1098 y Fv(.)150 1286 y(Th)m(us,)30 b Fq(\032)453 1300 y Fn(1)522 1286 y Fv(is)g(the)g(same)h(as)g(the)f(probabilit)m(y)e (of)j(selecting)f(an)m(y)h(quadratic)e(residue,)g(or)i(4)p Fq(=')p Fv(\()p Fq(n)p Fv(\).)291 1399 y(It)g(is)e(clear)h(that)150 1512 y Fq(r)s(ho)290 1526 y Fn(0)355 1512 y Fv(=)25 b Fq(P)13 b(r)s Fv([)p Fq(r)635 1479 y Fm(0)682 1512 y Fp( )26 b Fq(E)5 b Fv(\(\()p Fq(n;)15 b(a)p Fv(\))p Fq(;)g Fv(0\))28 b(:)d Fq(r)1361 1479 y Fm(0)1409 1512 y Fv(=)g Fq(r)s Fv(])30 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Fl(0)26 b Fe(\024)f Ff(i)g(<)g(l)r Fj(,)797 461 y Fl(Pr[\()p Ff(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))26 b Fe( )g Ff(G)p Fl(\(1)1618 423 y Fd(k)1662 461 y Fl(\);)15 b Ff(b)26 b Fe( )f(f)p Fl(0)p Ff(;)15 b Fl(1)p Fe(g)p Fl(;)484 599 y Ff(c)26 b Fe( )f Ff(E)5 b Fl(\()p Ff(P)13 b(K)q(;)i(m)1041 561 y Fd(i)p Fc(+)p Fd(b)1156 599 y Fl(\);)g Ff(b)1270 561 y Fb(0)1319 599 y Fe( )25 b Ff(A)1503 614 y Fd(k)1546 599 y Fl(\()p Ff(P)13 b(K)q(;)i(c;)g(m)1929 613 y Fc(0)1970 599 y Ff(;)g(m)2090 613 y Fc(1)2130 599 y Fl(\))83 b(:)h Ff(b)25 b Fl(=)g Ff(b)2556 561 y Fb(0)2579 599 y Fl(])h Fe(\025)f Fl(1)p Ff(=)p Fl(2)c(+)f(1)p Ff(=)p Fl(\()p Ff(l)j Fe(\003)e Ff(p)p Fl(\()p Ff(k)s Fl(\)\))150 788 y Fj(Mor)-5 b(e)g(over,)34 b(given)e Ff(A)883 803 y Fd(k)925 788 y Fj(,)h Ff(m)1066 802 y Fc(0)1138 788 y Fj(and)g Ff(m)1394 802 y Fc(1)1433 788 y Fj(,)g(one)g(c)-5 b(an)33 b(e\016ciently)g(determine)g Ff(i)p Fj(.)291 986 y Fl(Our)28 b(reduction)h Ff(R)h Fl(receiv)m(es)g(as)f(input)f(a)i(public)c(k)m(ey) k Ff(P)13 b(K)2329 953 y Fb(0)2382 986 y Fl(from)29 b(the)g (cryptosystem)h(\()p Ff(G)3413 953 y Fb(0)3437 986 y Ff(;)15 b(E)3549 953 y Fb(0)3573 986 y Ff(;)g(D)3691 953 y Fb(0)3715 986 y Fl(\))150 1099 y(and)32 b(a)h(c)m(hallenge)f (ciphertext)h Ff(c)1274 1066 y Fb(0)1297 1099 y Fl(.)47 b(Our)32 b(goal)h(is)e(to)j(determine)d(whether)h Ff(c)h Fl(is)e(an)i(encryption)e(of)i(0)g(or)150 1212 y(an)d(encryption)g(of)g (1.)291 1325 y(No)m(w,)43 b(w)m(e)d(create)h(the)f(c)m(hallenge)g (ciphertext)f(for)h(our)f(adv)m(ersary)h Ff(A)2773 1340 y Fd(k)2816 1325 y Fl(,)i(as)e(follo)m(ws:)58 b(the)40 b(\014rst)f Ff(i)150 1437 y Fl(blo)s(c)m(ks)30 b(of)h Ff(c)h Fl(will)c(b)s(e)i(encryptions)g(of)h(the)g(\014rst)f Ff(i)h Fl(bits)f(of)h Ff(m)2259 1451 y Fc(1)2298 1437 y Fl(.)43 b(The)30 b Ff(i)21 b Fl(+)f(1st)32 b(blo)s(c)m(k)e(will)e(b)s (e)j(the)g(input)150 1550 y(ciphertext)i Ff(c)624 1517 y Fb(0)648 1550 y Fl(.)49 b(The)33 b(last)g Ff(l)24 b Fe(\000)e Ff(i)g Fe(\000)g Fl(1)34 b(blo)s(c)m(ks)f(will)d(b)s(e)j(the) g(encryptions)g(of)g(the)g(last)h Ff(l)24 b Fe(\000)e Ff(i)g Fe(\000)g Fl(1)34 b(bits)e(of)150 1663 y Ff(m)230 1677 y Fc(0)269 1663 y Fl(.)48 b(Note)34 b(that,)g(in)d(case)j Ff(c)1131 1630 y Fb(0)1187 1663 y Fl(is)d(the)i(encryption)f(of)g(0,)i Ff(c)f Fl(will)d(b)s(e)i(the)h(encryption)e(of)i Ff(m)3274 1630 y Fd(i)3274 1691 y(k)3317 1663 y Fl(;)h(otherwise)150 1776 y Ff(c)g Fl(will)c(b)s(e)j(the)h(encryption)e(of)h Ff(m)1325 1738 y Fd(i)p Fc(+1)1325 1806 y Fd(k)1443 1776 y Fl(.)50 b(Our)32 b(reduction)g(will)f(simply)g(output)i(whatev)m(er)h Ff(A)3345 1791 y Fd(k)3421 1776 y Fl(outputs.)150 1889 y(Let)d(us)f(estimate)h(the)f(success)h(probabilit)m(y)d(of)i(our)g (reduction)f Ff(R)2463 1904 y Fd(k)2506 1889 y Fl(:)285 2191 y(Pr[\()p Ff(P)13 b(K)598 2154 y Fb(0)621 2191 y Ff(;)i(S)5 b(K)806 2154 y Fb(0)830 2191 y Fl(\))25 b Fe( )g Ff(G)1077 2154 y Fb(0)1101 2191 y Fl(;)15 b Ff(b)26 b Fe( )f(f)p Fl(0)p Ff(;)15 b Fl(1)p Fe(g)p Fl(;)g Ff(c)1621 2154 y Fb(0)1672 2191 y Fe( )26 b Ff(E)1861 2154 y Fb(0)1884 2191 y Fl(\()p Ff(P)13 b(K)2074 2154 y Fb(0)2098 2191 y Ff(;)i(b)p Fl(\);)g Ff(b)2291 2154 y Fb(0)2340 2191 y Fe( )25 b Ff(R)2525 2206 y Fd(k)2568 2191 y Fl(\()p Ff(P)13 b(K)2758 2154 y Fb(0)2782 2191 y Ff(;)i(c)2861 2154 y Fb(0)2885 2191 y Fl(\))25 b(:)h Ff(b)f Fl(=)g Ff(b)3195 2154 y Fb(0)3218 2191 y Fl(])83 b(=)1291 2329 y(Pr)o([\()p Ff(P)13 b(K)1603 2292 y Fb(0)1627 2329 y Ff(;)i(S)5 b(K)1812 2292 y Fb(0)1835 2329 y Fl(\))26 b Fe( )f Ff(G)2083 2292 y Fb(0)2107 2329 y Fl(;)15 b Ff(b)26 b Fe( )f(f)p Fl(0)p Ff(;)15 b Fl(1)p Fe(g)p Fl(;)g Ff(c)2627 2292 y Fb(0)2678 2329 y Fe( )25 b Ff(E)2866 2292 y Fb(0)2890 2329 y Fl(\()p Ff(P)13 b(K)3080 2292 y Fb(0)3103 2329 y Ff(;)i(b)p Fl(\);)150 2467 y Ff(c)26 b Fe( )f Fl(\()p Ff(E)438 2430 y Fb(0)462 2467 y Fl(\()p Ff(P)13 b(K)652 2430 y Fb(0)675 2467 y Ff(;)i(m)795 2430 y Fc(1)795 2490 y Fd(k)r(;)p Fc(0)893 2467 y Fl(\))p Ff(;)g(:)g(:)g(:)i(;)e(E)1202 2430 y Fb(0)1226 2467 y Fl(\()p Ff(P)e(K)1416 2430 y Fb(0)1439 2467 y Ff(;)i(m)1559 2430 y Fd(i)1559 2490 y(k)r(;)p Fc(0)1657 2467 y Fl(\))p Ff(;)g(c)1771 2430 y Fb(0)1795 2467 y Ff(;)g(E)1907 2430 y Fb(0)1931 2467 y Fl(\()p Ff(P)e(K)2121 2430 y Fb(0)2145 2467 y Ff(;)i(m)2265 2429 y Fd(i)p Fc(+2)2265 2496 y Fd(k)r(;)p Fc(1)2383 2467 y Fl(\))p Ff(;)g(:)g(:)g(:)i(;)e(E)2692 2430 y Fb(0)2716 2467 y Fl(\()p Ff(P)e(K)2906 2430 y Fb(0)2930 2467 y Ff(;)i(m)3050 2430 y Fd(l)3050 2490 y(k)r(;)p Fc(1)3148 2467 y Fl(\)\);)2277 2617 y Ff(b)2316 2579 y Fb(0)2364 2617 y Fe( )26 b Ff(A)2549 2632 y Fd(k)2591 2617 y Fl(\()p Ff(P)13 b(K)2781 2579 y Fb(0)2805 2617 y Ff(;)i(c)p Fl(\))26 b(:)g Ff(b)f Fl(=)g Ff(b)3195 2579 y Fb(0)3218 2617 y Fl(])83 b(=)523 2755 y(Pr[\()p Ff(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))27 b Fe( )e Ff(G)p Fl(;)15 b Ff(b)26 b Fe( )f(f)p Fl(0)p Ff(;)15 b Fl(1)p Fe(g)p Fl(;)g Ff(c)29 b Fe( )c Ff(E)5 b Fl(\()p Ff(m)2116 2717 y Fd(i)p Fc(+)p Fd(b)2116 2784 y(k)2229 2755 y Fl(\);)15 b Ff(b)2343 2717 y Fb(0)2393 2755 y Fe( )25 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3604 y Ff(c)26 b Fe( )f Ff(E)5 b Fl(\()p Ff(P)13 b(K)q(;)i(m)1226 3619 y Fd(b)1262 3604 y Fl(\);)669 3717 y Ff(b)708 3684 y Fb(0)757 3717 y Fe( )25 b Fl(Ev)m(e)q(\()p Ff(P)13 b(K)q(;)i(c)p Fl(\))118 b(:)e Ff(b)26 b Fl(=)f Ff(b)1778 3684 y Fb(0)1801 3717 y Fl(])g Fe(\024)g Fl(1)p Ff(=)p Fl(2)d(+)e Ff(f)10 b Fl(\()p Ff(k)s Fl(\))291 3910 y(Ho)m(w)m(ev)m(er,)33 b(if)c(the)i(de\014nition)d(do)s(es)i(not)g (hold)f(for)i(\()p Ff(G)2141 3877 y Fb(0)2165 3910 y Ff(;)15 b(E)2277 3877 y Fb(0)2301 3910 y Ff(;)g(D)2419 3877 y Fb(0)2442 3910 y Fl(\),)31 b(then:)377 4103 y Fe(9)p Ff(P)13 b(P)g(T)g(F)g Fe(f)p Ff(E)5 b(v)s(e)913 4118 y Fd(k)957 4103 y Fe(g)15 b(9)p Ff(m)1148 4117 y Fc(0)1187 4103 y Ff(;)g(m)1307 4117 y Fc(1)1372 4103 y Fe(2)25 b(f)p Fl(0)p Ff(;)15 b Fl(1)p Fe(g)1678 4070 y Fd(l)1706 4103 y Fe(9)p Ff(f)10 b Fl(\()p Ff(k)s Fl(\))25 b Fe(2)g Fl(negl)521 4216 y(Pr[)g(\()p Ff(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))27 b Fe( )e Ff(G)p Fl(\(1)1367 4183 y Fd(k)1411 4216 y Fl(\);)669 4329 y Ff(b)g Fe( )h(f)p Fl(0)p Ff(;)15 b Fl(1)p Fe(g)p Fl(;)669 4442 y Ff(c)26 b Fe( )f Ff(E)5 b Fl(\()p Ff(P)13 b(K)q(;)i(m)1226 4457 y Fd(b)1262 4442 y Fl(\)or)p Ff(P)e(K)32 b Fl(=)25 b Ff(m)1734 4457 y Fd(b)1768 4442 y Fl(;)669 4555 y Ff(b)708 4522 y Fb(0)757 4555 y Fe( )g Fl(Ev)m(e)q(\()p Ff(P)13 b(K)q(;)i(c)p Fl(\))118 b(:)e Ff(b)26 b Fl(=)f Ff(b)1778 4522 y Fb(0)1801 4555 y Fl(])g Ff(>)g Fl(1)p Ff(=)p Fl(2)d(+)e(1)p Ff(=pol)r(y)291 4748 y Fl(Th)m(us,)30 b(\()p Ff(G)653 4715 y Fb(0)677 4748 y Ff(;)15 b(E)789 4715 y Fb(0)813 4748 y Ff(;)g(D)931 4715 y Fb(0)954 4748 y Fl(\))31 b(can)g(only)e(b)s(ecome)i(insecure)e (if)g Fe(9)p Ff(m)2275 4762 y Fc(0)2314 4748 y Ff(;)15 b(m)2434 4762 y Fc(1)2504 4748 y Fl(suc)m(h)30 b(that:)521 4941 y(Pr[)25 b(\()p Ff(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))27 b Fe( )e Ff(G)p Fl(\(1)1367 4908 y Fd(k)1411 4941 y Fl(\);)669 5054 y Ff(b)g Fe( )h(f)p Fl(0)p Ff(;)15 b Fl(1)p Fe(g)p Fl(;)669 5167 y Ff(P)e(K)32 b Fl(=)25 b Ff(m)1025 5182 y Fd(b)1059 5167 y Fl(])h Fe(\025)f Fl(1)p Ff(=pol)r(y)1889 5416 y Fl(7-4)p eop %%Page: 5 5 5 4 bop 291 159 a Fl(If)29 b(this)g(w)m(ere)h(p)s(ossible,)e(then)i(an) f Ff(E)5 b(v)s(e)31 b Fl(equipp)s(ed)c(with)i Ff(S)5 b(K)36 b Fl(corresp)s(onding)28 b(to)j Ff(m)3202 174 y Fd(b)3265 159 y Fl(w)m(ould)e(break)150 271 y(3-pass)35 b(securit)m(y)-8 b(.)56 b(This)33 b(is)h(a)i(con)m(tradiction,)g(since) f(w)m(e)g(ha)m(v)m(e)i(assumed)d(3-pass)h(securit)m(y)-8 b(.)56 b(Th)m(us,)35 b(b)m(y)150 384 y(con)m(tradiction,)c(w)m(e)f(ha)m (v)m(e)i(sho)m(wn)e(that)h(3-pass)f(securit)m(y)h(do)s(es)f(not)g (imply)e(1-pass)j(securit)m(y)-8 b(.)150 576 y Fm(Theorem)34 b(7)46 b Fj(1-bit)32 b(GM-se)-5 b(curity)32 b Fe(\))h Fj(one-p)-5 b(ass)34 b(GM-Se)-5 b(curity.)41 b(\(If)33 b(you)g(encrypt)g(bit-by-bit.\))150 767 y Fm(Sk)m(etc)m(h)j(of)g(Pro)s (of)123 b Fl(This)29 b(pro)s(of)i(go)s(es)h(roughly)e(the)h(same)g(as)h (last)f(time)g(\(when)f(w)m(e)i(sho)m(w)m(ed)f(1-bit)150 880 y(securit)m(y)f(implies)e(l-bit)h(securit)m(y\).)291 993 y(Supp)s(ose)40 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Fd(i)p Fc(+2)2046 1681 y(0)2185 1655 y Fe(\016)21 b Ff(:)15 b(:)g(:)21 b Fe(\016)g Ff(m)2523 1607 y Fb(j)p Fd(m)2605 1616 y Fa(1)2639 1607 y Fb(j)2523 1681 y Fc(0)2663 1655 y Fl(\))559 1768 y Ff(b)598 1735 y Fb(0)647 1768 y Fe( )k Ff(E)5 b(v)s(e)924 1783 y Fd(k)967 1768 y Fl(\()p Ff(\013;)15 b(c)1139 1735 y Fb(0)1164 1768 y Fl(\))559 1881 y(Return)30 b Ff(b)908 1848 y Fb(0)291 2072 y Fl(Note)i(that)f(this)e(is)g(exacrly) i(ho)m(w)f(w)m(e)h(brok)m(e)g(the)f(1-bit)h(securit)m(y)f(sc)m(heme)h (from)f(last)g(time.)p 3514 2072 59 59 v 150 2441 a Fi(4)135 b(Seman)l(tic)46 b(securit)l(y)150 2644 y Fl(In)m(tuition)23 b(tells)g(us)h(that)h(the)f(righ)m(t)g(w)m(a)m(y)i(to)f(de\014ne)e (securit)m(y)i(is:)36 b(\\Whatev)m(er)27 b(can)d(b)s(e)g(computed)g(ab) s(out)150 2756 y(the)33 b(message)h(from)e(the)h(ciphertext,)g(can)g(b) s(e)f(computed)h(without)f(the)g(ciphertext.")48 b(Let)34 b(us)e(try)g(to)150 2869 y(capture)f(this)e(in)m(tuition.)291 2982 y(What)38 b(if)d(Ev)m(e)j(has)f(ev)m(en)g(greater)h(p)s(o)m(w)m (er)f(o)m(v)m(er)h(Bob?)60 b(Let)38 b(us)e(generalize)h(this)f (de\014nition)e(ev)m(en)150 3095 y(further.)61 b(Supp)s(ose)35 b(that)k(Ev)m(e,)h(up)s(on)c(examining)g(Alice's)h(public)e(k)m(ey)-8 b(,)40 b(decides)d(that)h(the)g(message)150 3208 y(should)23 b(b)s(e)g(selected)i(according)g(to)g(some)g(e\016cien)m(tly)f (samplable)f(distribution)e Ff(M)10 b Fl(.)39 b(F)-8 b(urther,)25 b(supp)s(ose)150 3321 y(that)j(Ev)m(e's)g(goal)f(is)g(not) g(to)h(determine)e(whic)m(h)g(sp)s(eci\014c)g(message)i(w)m(as)g(sen)m (t,)g(but)f(rather,)h(to)g(compute)150 3434 y(some)34 b(function)e(of)h(the)h(message,)h(for)e(example,)h(to)g(determine)f (whether)f(this)g(is)h(a)g(lo)m(v)m(e)i(letter)e(or)h(a)150 3547 y(hate)d(letter.)41 b(This)29 b(scenario)h(is)f(summarized)g(in)g (Figure)h(3.)960 4201 y @beginspecial 0 @llx 0 @lly 564 @urx 155 @ury 2376 @rwi @setspecial %%BeginDocument: scen3.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: scen3.fig %%Creator: 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4365 660 l gs col0 s gr gr % arrowhead n 4185 615 m 4365 660 l 4185 705 l col0 s /Times-Roman ff 210.00 scf sf 3450 735 m gs 1 -1 sc (PK) col0 sh gr % Polyline gs clippath 3915 2085 m 3735 2040 l 3915 1995 l 3720 1995 l 3720 2085 l cp clip n 4335 2040 m 3735 2040 l gs col0 s gr gr % arrowhead n 3915 2085 m 3735 2040 l 3915 1995 l col0 s /Times-Roman ff 210.00 scf sf 3600 2100 m gs 1 -1 sc (z) col0 sh gr % Polyline gs clippath 3915 975 m 3735 930 l 3915 885 l 3720 885 l 3720 975 l cp clip n 4335 930 m 3735 930 l gs col0 s gr gr % arrowhead n 3915 975 m 3735 930 l 3915 885 l col0 s /Times-Roman ff 210.00 scf sf 2850 1005 m gs 1 -1 sc (\( , M, R\)) col0 sh gr /Symbol ff 210.00 scf sf 2985 990 m gs 1 -1 sc (a) col0 sh gr /Times-Roman ff 210.00 scf sf 3090 1845 m gs 1 -1 sc (\( ,c\)) col0 sh gr /Symbol ff 210.00 scf sf 3195 1845 m gs 1 -1 sc (a) col0 sh gr /Times-Roman ff 210.00 scf sf 5745 555 m gs 1 -1 sc (Eve) col0 sh gr /Times-Roman ff 210.00 scf sf 6120 675 m gs 1 -1 sc (k) col0 sh gr /Times-Roman ff 210.00 scf sf 5850 420 m gs 1 -1 sc (~) col0 sh gr % Polyline gs clippath 2205 1170 m 2025 1125 l 2205 1080 l 2010 1080 l 2010 1170 l cp clip n 2625 1125 m 2025 1125 l gs col0 s gr gr % arrowhead n 2205 1170 m 2025 1125 l 2205 1080 l col0 s % Polyline gs clippath 4170 1725 m 4350 1770 l 4170 1815 l 4365 1815 l 4365 1725 l cp clip n 3750 1770 m 4350 1770 l gs col0 s gr gr % arrowhead n 4170 1725 m 4350 1770 l 4170 1815 l col0 s % Polyline n 4425 450 m 5505 450 l 5505 2235 l 4425 2235 l cp gs col0 s gr % Polyline n 2700 300 m 6300 300 l 6300 2400 l 2700 2400 l cp gs col0 s gr $F2psEnd rs %%EndDocument @endspecial 1067 4748 a(Figure)f(1:)41 b(Visual)29 b(idea)h(b)s(ehind) d(reduction)2694 4725 y(~)2643 4748 y(Ev)m(e)2790 4763 y Ff(k)291 5054 y Fl(Let's)41 b(estimate)f(the)h(success)f(probabilit)m (y)d(of)2030 5031 y(~)1971 5054 y Fh(E)5 b(v)s(e)2133 5069 y Ff(k)2176 5054 y Fl(:)60 b(Let)40 b Fh(A)g Fl(b)s(e)g(the)g(ev)m (en)m(t)h(that)g Fh(R)q Fl(\()p Fh(m)3482 5069 y Ff(b)3516 5054 y Fh(;)15 b(z)t Fl(\))42 b(=)150 5167 y Fh(T)13 b(R)q(U)d(E)5 b Fl(.)40 b(More)32 b(formally)-8 b(,)1889 5416 y(8-2)p eop %%Page: 3 3 3 2 bop 521 159 a Fh(A)26 b Fl(=\()k(\()p Fh(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))27 b Fg( )e Fh(G)p Fl(\(1)1449 126 y Ff(k)1493 159 y Fl(\);)685 276 y(\()9 b(~)-54 b Fh(\013)q(;)15 b(m)899 290 y Fd(0)939 276 y Fh(;)g(m)1059 290 y Fd(1)1099 276 y Fl(\))25 b Fg( )1326 253 y Fl(~)1275 276 y(Ev)m(e)1423 291 y Ff(k)1466 276 y Fl(\()p Fh(P)13 b(K)7 b Fl(\);)685 389 y Fh(b)26 b Fg( )f(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)p Fl(;)g Fh(c)28 b Fg( )d Fh(E)5 b Fl(\()p Fh(P)13 b(K)q(;)i(m)1685 404 y Ff(b)1721 389 y Fl(\);)685 502 y Fh(z)30 b Fg( )25 b Fl(Ev)m(e)1021 517 y Ff(k)1063 502 y Fl(\()p Fh(\013;)15 b(c)p Fl(\))119 b(:)d Fh(R)q Fl(\()p Fh(m)1715 517 y Ff(b)1749 502 y Fh(;)15 b(z)t Fl(\))32 b(\))291 714 y(Let)f Fh(B)k Fl(b)s(e)30 b(the)g(ev)m(en)m(t)i(that)f Fh(R)q Fl(\()p Fh(m)1457 721 y Fd(\026)1460 739 y Ff(b)1494 714 y Fh(;)15 b(z)t Fl(\))27 b(=)d Fh(T)13 b(R)q(U)d(E)5 b Fl(.)41 b(More)31 b(formally:)521 902 y Fh(B)f Fl(=\()g(\()p Fh(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))27 b Fg( )e Fh(G)p Fl(\(1)1454 869 y Ff(k)1498 902 y Fl(\);)691 1020 y(\()9 b(~)-54 b Fh(\013)q(;)15 b(m)905 1034 y Fd(0)944 1020 y Fh(;)g(m)1064 1034 y Fd(1)1104 1020 y Fl(\))25 b Fg( )1332 997 y Fl(~)1281 1020 y(Ev)m(e)1428 1035 y Ff(k)1471 1020 y Fl(\()p Fh(P)13 b(K)7 b Fl(\);)691 1133 y Fh(b)25 b Fg( )g(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)p Fl(;)g Fh(c)28 b Fg( )d Fh(E)5 b Fl(\()p Fh(P)13 b(K)q(;)i(m)1690 1148 y Ff(b)1726 1133 y Fl(\);)691 1245 y Fh(z)29 b Fg( )c Fl(Ev)m(e)1026 1260 y Ff(k)1069 1245 y Fl(\()p Fh(\013;)15 b(c)p Fl(\))118 b(:)e Fh(R)q Fl(\()p Fh(m)1717 1252 y Fd(\026)1720 1270 y Ff(b)1755 1245 y Fh(;)15 b(z)t Fl(\))31 b(\))291 1446 y(Note)h(that,)f(for)f(our)g(adv)m(ersary)1498 1423 y(~)1447 1446 y(Ev)m(e)1594 1461 y Ff(k)1637 1446 y Fl(,)864 1672 y(Pr[)p Fh(success)p Fl(])25 b(=)g(Pr[)p Fh(A)20 b Fg(^)1748 1649 y Fl(\026)1726 1672 y Fh(B)5 b Fl(])20 b(+)g(1)p Fh(=)p Fl(2\(Pr)q([)p Fh(A)h Fg(^)f Fh(B)5 b Fl(])20 b(+)g(Pr)o([)2756 1649 y(\026)2732 1672 y Fh(A)h Fg(^)2923 1649 y Fl(\026)2902 1672 y Fh(B)t Fl(]\))291 1839 y(whic)m(h)33 b(is)g(the)h(probabilit)m(y)d(either)j (that)g Fh(R)h Fl(correctly)f(accepts)i Fh(z)i Fl(for)c Fh(m)2850 1854 y Ff(b)2918 1839 y Fl(and)f(rejects)i Fh(z)j Fl(for)c Fh(m)3688 1846 y Fd(\026)3691 1863 y Ff(b)3725 1839 y Fl(,)150 1952 y(or)c(that)h(the)g(coin)f(\015ip,)f(b)m (y)h(c)m(hance,)i(is)d Fh(b)p Fl(.)41 b(On)29 b(the)i(other)f(hand,)933 2156 y(Pr[)p Fh(f)10 b(ail)r Fl(])25 b(=)g(Pr[)1512 2133 y(\026)1488 2156 y Fh(A)20 b Fg(^)g Fh(B)5 b Fl(])20 b(+)g(1)p Fh(=)p Fl(2\(Pr)q([)p Fh(A)h Fg(^)f Fh(B)5 b Fl(])20 b(+)g(Pr)o([)2687 2133 y(\026)2663 2156 y Fh(A)h Fg(^)2854 2133 y Fl(\026)2832 2156 y Fh(B)5 b Fl(]\))291 2360 y(whic)m(h)28 b(is)g(the)i(probabilit)m(y)d(either)h(that)i Fh(R)g Fl(incorrectly)e(accepts)j Fh(z)i Fl(for)c Fh(m)2870 2367 y Fd(\026)2873 2384 y Ff(b)2937 2360 y Fl(and)f(rejects)i Fh(z)k Fl(for)29 b Fh(m)3691 2375 y Ff(b)3725 2360 y Fl(,)150 2473 y(or)h(that)h(the)g(coin)f(\015ip,)f(b)m(y)h(c)m(hance,)i (is)1524 2449 y(\026)1527 2473 y Fh(b)e Fl(Therefore,)448 2677 y(Pr[)p Fh(success)p Fl(])20 b Fg(\000)g Fl(Pr[)p Fh(f)10 b(ail)r Fl(])83 b(=)g(Pr)o([)p Fh(A)21 b Fg(^)1869 2654 y Fl(\026)1848 2677 y Fh(B)t Fl(])f Fg(\000)g Fl(Pr[)2204 2654 y(\026)2180 2677 y Fh(A)h Fg(^)e Fh(B)5 b Fl(])1402 2815 y(=)83 b(Pr)o([)p Fh(A)21 b Fg(^)1869 2792 y Fl(\026)1848 2815 y Fh(B)t Fl(])f(+)g(Pr[)p Fh(A)h Fg(^)e Fh(B)5 b Fl(])20 b Fg(\000)g Fl(Pr[)p Fh(A)g Fg(^)g Fh(B)5 b Fl(])20 b Fg(\000)g Fl(Pr)o([)3208 2792 y(\026)3183 2815 y Fh(A)h Fg(^)f Fh(B)5 b Fl(])1402 2953 y(=)83 b(Pr)o([)p Fh(A)p Fl(])21 b Fg(\000)f Fl(Pr)o([)p Fh(B)5 b Fl(])p Fh(:)291 3157 y Fl(So,)31 b(let)f(us)g(estimate)h(Pr)o([)p Fh(A)p Fl(])g(and)f(Pr)o([)p Fh(B)5 b Fl(].)521 3370 y(Pr[)p Fh(A)p Fl(])26 b(=)f(Pr)o([)h(\()p Fh(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))26 b Fg( )f Fh(G)p Fl(\(1)1704 3337 y Ff(k)1748 3370 y Fl(\);)1007 3487 y(\()9 b(~)-54 b Fh(\013;)15 b(m)1220 3501 y Fd(0)1260 3487 y Fh(;)g(m)1380 3501 y Fd(1)1420 3487 y Fl(\))25 b Fg( )1648 3464 y Fl(~)1596 3487 y(Ev)m(e)1744 3502 y Ff(k)1787 3487 y Fl(\()p Fh(P)13 b(K)7 b Fl(\);)1007 3600 y Fh(b)25 b Fg( )g(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)p Fl(;)g Fh(c)28 b Fg( )d Fh(E)5 b Fl(\()p Fh(P)13 b(K)q(;)i(m)2006 3615 y Ff(b)2042 3600 y Fl(\);)1007 3713 y Fh(z)29 b Fg( )c Fl(Ev)m(e)1342 3728 y Ff(k)1385 3713 y Fl(\()p Fh(\013;)15 b(c)p Fl(\))118 b(:)e Fh(R)q Fl(\()p Fh(m)2036 3728 y Ff(b)2070 3713 y Fh(;)15 b(z)t Fl(\)])27 b(=)521 3826 y(Pr[)e(\()p Fh(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))27 b Fg( )e Fh(G)p Fl(\(1)1367 3793 y Ff(k)1411 3826 y Fl(\);)669 3939 y(\()p Fh(\013;)15 b(M)5 b(;)15 b(R)q Fl(\))27 b Fg( )e Fl(Ev)m(e)1331 3954 y Ff(k)1374 3939 y Fl(\()p Fh(P)13 b(K)7 b Fl(\);)669 4052 y Fh(m)749 4066 y Fd(0)814 4052 y Fg( )25 b Fh(M)10 b Fl(;)15 b Fh(m)1148 4066 y Fd(1)1213 4052 y Fg( )25 b Fh(M)10 b Fl(;)669 4165 y Fh(b)25 b Fg( )h(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)p Fl(;)g Fh(c)28 b Fg( )d Fh(E)5 b Fl(\()p Fh(P)13 b(K)q(;)i(m)1669 4180 y Ff(b)1705 4165 y Fl(\);)669 4278 y Fh(z)30 b Fg( )25 b Fl(Ev)m(e)1005 4293 y Ff(k)1047 4278 y Fl(\()p Fh(\013;)15 b(c)p Fl(\))118 b(:)f Fh(R)q Fl(\()p Fh(m)1699 4293 y Ff(b)1733 4278 y Fh(;)15 b(z)t Fl(\)])27 b(=)521 4391 y(Pr[)e(\()p Fh(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))27 b Fg( )e Fh(G)p Fl(\(1)1367 4358 y Ff(k)1411 4391 y Fl(\);)669 4504 y(\()p Fh(\013;)15 b(M)5 b(;)15 b(R)q Fl(\))27 b Fg( )e Fl(Ev)m(e)1331 4519 y Ff(k)1374 4504 y Fl(\()p Fh(P)13 b(K)7 b Fl(\);)669 4616 y Fh(m)25 b Fg( )g Fh(M)10 b Fl(;)15 b Fh(c)27 b Fg( )e Fh(E)5 b Fl(\()p Fh(P)13 b(K)q(;)i(m)p Fl(\);)669 4729 y Fh(z)30 b Fg( )25 b Fl(Ev)m(e)1005 4744 y Ff(k)1047 4729 y Fl(\()p Fh(\013;)15 b(c)p Fl(\))118 b(:)f Fh(R)q Fl(\()p Fh(m;)15 b(z)t Fl(\)])26 b(=)521 4842 y Fh(p)567 4856 y Fd(0)606 4842 y Fl(\()p Fh(k)s Fl(\).)1889 5416 y(8-3)p eop %%Page: 4 4 4 3 bop 521 159 a Fl(Pr[)p Fh(B)5 b Fl(])25 b(=)g(Pr)o([)h(\()p Fh(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))27 b Fg( )e Fh(G)p Fl(\(1)1710 126 y Ff(k)1754 159 y Fl(\);)1012 276 y(\()9 b(~)-54 b Fh(\013)q(;)15 b(m)1226 290 y Fd(0)1265 276 y Fh(;)g(m)1385 290 y Fd(1)1425 276 y Fl(\))26 b Fg( )1653 253 y Fl(~)1602 276 y(Ev)m(e)1750 291 y Ff(k)1792 276 y Fl(\()p Fh(P)13 b(K)7 b Fl(\);)1012 389 y Fh(b)25 b Fg( )g(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)p Fl(;)g Fh(c)29 b Fg( )c Fh(E)5 b Fl(\()p Fh(P)13 b(K)q(;)i(m)2012 404 y Ff(b)2048 389 y Fl(\);)1012 502 y Fh(z)29 b Fg( )d Fl(Ev)m(e)1347 517 y Ff(k)1390 502 y Fl(\()p Fh(\013;)15 b(c)p Fl(\))118 b(:)f Fh(R)q Fl(\()p Fh(m)2039 509 y Fd(\026)2042 526 y Ff(b)2076 502 y Fh(;)15 b(z)t Fl(\)])26 b(=)521 615 y(Pr[)f(\()p Fh(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))27 b Fg( )e Fh(G)p Fl(\(1)1367 582 y Ff(k)1411 615 y Fl(\);)669 728 y(\()p Fh(\013;)15 b(M)5 b(;)15 b(R)q Fl(\))27 b Fg( )e Fl(Ev)m(e)1331 743 y Ff(k)1374 728 y Fl(\()p Fh(P)13 b(K)7 b Fl(\);)669 841 y Fh(m)749 855 y Fd(0)814 841 y Fg( )25 b Fh(M)10 b Fl(;)15 b Fh(m)1148 855 y Fd(1)1213 841 y Fg( )25 b Fh(M)10 b Fl(;)669 954 y Fh(b)25 b Fg( )h(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)p Fl(;)g Fh(c)28 b Fg( )d Fh(E)5 b Fl(\()p Fh(P)13 b(K)q(;)i(m)1669 969 y Ff(b)1705 954 y Fl(\);)669 1067 y Fh(z)30 b Fg( )25 b Fl(Ev)m(e)1005 1082 y Ff(k)1047 1067 y Fl(\()p Fh(\013;)15 b(c)p Fl(\))118 b(:)f Fh(R)q Fl(\()p Fh(m)1696 1074 y Fd(\026)1699 1091 y Ff(b)1733 1067 y Fh(;)15 b(z)t Fl(\)])27 b(=)521 1180 y(Pr[)e(\()p Fh(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))27 b Fg( )e Fh(G)p Fl(\(1)1367 1147 y Ff(k)1411 1180 y Fl(\);)669 1292 y(\()p Fh(\013;)15 b(M)5 b(;)15 b(R)q Fl(\))27 b Fg( )e Fl(Ev)m(e)1331 1307 y Ff(k)1374 1292 y Fl(\()p Fh(P)13 b(K)7 b Fl(\);)669 1405 y Fh(m)25 b Fg( )g Fh(M)10 b Fl(;)15 b Fh(m)1108 1372 y Fc(0)1157 1405 y Fg( )25 b Fh(M)10 b Fl(;)15 b Fh(c)27 b Fg( )e Fh(E)5 b Fl(\()p Fh(P)13 b(K)q(;)i(m)1969 1372 y Fc(0)1994 1405 y Fl(\);)669 1518 y Fh(z)30 b Fg( )25 b Fl(Ev)m(e)1005 1533 y Ff(k)1047 1518 y Fl(\()p Fh(\013;)15 b(c)p Fl(\))118 b(:)f Fh(R)q Fl(\()p Fh(m;)15 b(z)t Fl(\)])26 b(=)521 1631 y(Pr[)f(\()p Fh(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))27 b Fg( )e Fh(G)p Fl(\(1)1367 1598 y Ff(k)1411 1631 y Fl(\);)669 1744 y(\()p Fh(\013;)15 b(M)5 b(;)15 b(R)q Fl(\))27 b Fg( )e Fl(Ev)m(e)1331 1759 y Ff(k)1374 1744 y Fl(\()p Fh(P)13 b(K)7 b Fl(\))669 1857 y Fh(m)25 b Fg( )g Fh(M)10 b Fl(;)669 1970 y Fh(z)30 b Fg( )25 b Fh(S)913 1985 y Ff(k)955 1970 y Fl(\()p Fh(\013;)15 b(M)5 b(;)15 b(R)q(;)g(P)e(K)7 b Fl(\))118 b(:)f Fh(R)q Fl(\()p Fh(m;)15 b(z)t Fl(\)])26 b(=.)521 2083 y Fh(p)567 2097 y Fd(1)606 2083 y Fl(\()p Fh(S)697 2098 y Ff(k)740 2083 y Fh(;)15 b(k)s Fl(\).)291 2282 y(Note)32 b(that)f(the)f(last)h(equalit)m(y)f(uses)f(the)i (earlier)e(de\014nition)f(of)j Fh(S)2574 2297 y Ff(k)2616 2282 y Fl(.)291 2395 y(Therefore,)f(Pr)o([)p Fh(success)p Fl(])20 b Fg(\000)f Fl(Pr)o([)p Fh(f)10 b(ail)r Fl(])25 b(=)g(Pr[)p Fh(A)p Fl(])19 b Fg(\000)g Fl(Pr[)p Fh(B)5 b Fl(])25 b(=)g Fh(p)2431 2409 y Fd(0)2470 2395 y Fl(\()p Fh(k)s Fl(\))20 b Fg(\000)f Fh(p)2746 2409 y Fd(1)2785 2395 y Fl(\()p Fh(S)2876 2410 y Ff(k)2919 2395 y Fh(;)c(k)s Fl(\))26 b Fg(\025)f Fh(\017)p Fl(\()p Fh(S)3294 2410 y Ff(k)3336 2395 y Fh(;)15 b(k)s Fl(\),)31 b(where)150 2508 y Fh(\017)f Fl(is)g(a)g(non-negligible)e(function.)291 2621 y(This)h(con)m(tradicts)i(GM-securit)m(y)g(of)f(\()p Fh(G;)15 b(E)5 b(;)15 b(D)s Fl(\).)43 b(\(Exercise:)e(wh)m(y?\))150 2862 y Fb(2.2)112 b(Seman)m(tic)37 b(securit)m(y)f(implies)f(GM)i (securit)m(y)150 3033 y Fl(Supp)s(ose)26 b(that)j(our)e(cryptosystem)i (\()p Fh(G;)15 b(E)5 b(;)15 b(D)s Fl(\))30 b(is)d(not)h(GM-secure.)41 b(W)-8 b(e)30 b(wish)c(to)j(sho)m(w)e(that)i(it)f(is)f(not)150 3146 y(seman)m(tically)j(secure)g(either.)291 3259 y(By)44 b(our)f(assumption,)j(there)e(exists)f(an)h(adv)m(ersary)g Fg(f)p Fl(Ev)m(e)2414 3274 y Ff(k)2456 3259 y Fg(g)g Fl(suc)m(h)g(that)g(there)g(exists)f(a)h(non-)150 3372 y(negligible)28 b(function)h Fh(\017)p Fl(\()p Fh(k)s Fl(\),)i(suc)m(h)g(that)521 3571 y(Pr[)25 b(\()p Fh(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))27 b Fg( )e Fh(G)p Fl(\(1)1367 3538 y Ff(k)1411 3571 y Fl(\);)669 3684 y(\()p Fh(\013;)15 b(m)882 3698 y Fd(0)923 3684 y Fh(;)g(m)1043 3698 y Fd(1)1082 3684 y Fl(\))26 b Fg( )f Fl(Ev)m(e)1407 3699 y Ff(k)1450 3684 y Fl(\()p Fh(P)13 b(K)7 b Fl(\))30 b(where)g Fg(j)p Fh(m)2073 3698 y Fd(0)2113 3684 y Fg(j)25 b Fl(=)g Fg(j)p Fh(m)2364 3698 y Fd(1)2404 3684 y Fg(j)p Fl(,)669 3797 y Fh(b)g Fg( )h(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)p Fl(;)669 3910 y Fh(c)26 b Fg( )f Fh(E)5 b Fl(\()p Fh(P)13 b(K)q(;)i(m)1226 3925 y Ff(b)1262 3910 y Fl(\);)669 4023 y Fh(b)708 3990 y Fc(0)757 4023 y Fg( )25 b Fl(Ev)m(e)q(\()p Fh(\013;)15 b(c)p Fl(\))118 b(:)e Fh(b)25 b Fl(=)g Fh(b)1686 3990 y Fc(0)1710 4023 y Fl(])g Fg(\025)g Fl(1)p Fh(=)p Fl(2)d(+)e Fh(\017)p Fl(\()p Fh(k)s Fl(\))291 4235 y(Let)43 b(us)e(construct)i(an) f(adv)m(ersary)g Fg(f)1665 4212 y Fl(~)1613 4235 y(Ev)m(e)1762 4250 y Ff(k)1804 4235 y Fg(g)h Fl(that)g(breaks)f(seman)m(tic)g (securit)m(y)g(of)g(\()p Fh(G;)15 b(E)5 b(;)15 b(D)s Fl(\))45 b(as)150 4348 y(follo)m(ws:)286 4546 y Fg(\017)h Fl(On)30 b(input)e(\()p Fh(P)13 b(K)7 b Fl(\),)1103 4524 y(~)1052 4546 y(Ev)m(e)1200 4561 y Ff(k)1273 4546 y Fl(do)s(es:)480 4730 y Fm({)45 b Fl(Obtain)29 b(\()p Fh(\013;)15 b(m)1098 4744 y Fd(0)1139 4730 y Fh(;)g(m)1259 4744 y Fd(1)1299 4730 y Fl(\))25 b Fg( )g Fl(Ev)m(e)1623 4745 y Ff(k)1666 4730 y Fl(\()p Fh(P)13 b(K)7 b Fl(\).)480 4871 y Fm({)45 b Fl(Let)34 b(~)-54 b Fh(\013)25 b Fl(=)g(\()p Fh(\013;)15 b(m)1126 4885 y Fd(0)1167 4871 y Fh(;)g(m)1287 4885 y Fd(1)1326 4871 y Fl(\).)39 b(Let)25 b Fh(M)35 b Fl(=)25 b Fg(f)p Fh(m)1926 4885 y Fd(0)1966 4871 y Fh(;)15 b(m)2086 4885 y Fd(1)2125 4871 y Fg(g)p Fl(.)39 b(Let)24 b Fh(R)h Fl(b)s(e)e(the)h(equalit)m(y)f(relation.)38 b(Output)577 4984 y(\()9 b(~)-54 b Fh(\013)q(;)15 b(M)5 b(;)15 b(R)q Fl(\).)286 5167 y Fg(\017)46 b Fl(On)30 b(input)e(\()9 b(~)-54 b Fh(\013)q(;)15 b(c)p Fl(\),)32 b(where)39 b(~)-54 b Fh(\013)25 b Fl(=)g(\()p Fh(\013;)15 b(m)1691 5181 y Fd(0)1732 5167 y Fh(;)g(m)1852 5181 y Fd(1)1891 5167 y Fl(\),)2034 5144 y(~)1982 5167 y(Ev)m(e)31 b(do)s(es:)1889 5416 y(8-4)p eop %%Page: 5 5 5 4 bop 480 159 a Fm({)45 b Fl(Obtain)29 b Fh(b)924 126 y Fc(0)973 159 y Fg( )c Fl(Ev)m(e)1237 174 y Ff(k)1279 159 y Fl(\()p Fh(\013;)15 b(c)p Fl(\).)480 302 y Fm({)45 b Fl(Output)30 b Fh(m)981 317 y Ff(b)1011 298 y Fa(0)1037 302 y Fl(.)291 507 y(It)h(is)e(clear)h(that)h(the)g(the)g(v)-5 b(alue)29 b Fh(p)1489 521 y Fd(0)1529 507 y Fl(\()p Fh(k)s Fl(\))i(for)f(this)f(adv)m(ersary)i(is:)521 712 y Fh(p)567 726 y Fd(0)606 712 y Fl(\()p Fh(k)s Fl(\))c(=)d(Pr[)h(\()p Fh(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))27 b Fg( )e Fh(G)p Fl(\(1)1694 679 y Ff(k)1738 712 y Fl(\);)996 825 y(\()p Fh(\013;)15 b(m)1209 839 y Fd(0)1250 825 y Fh(;)g(m)1370 839 y Fd(1)1410 825 y Fl(\))25 b Fg( )g Fl(Ev)m(e)1734 840 y Ff(k)1777 825 y Fl(\()p Fh(P)13 b(K)7 b Fl(\),)996 937 y Fh(b)26 b Fg( )f(f)p Fl(0)p Fh(;)15 b Fl(1)p Fg(g)p Fl(;)996 1050 y Fh(c)26 b Fg( )f Fh(E)5 b Fl(\()p Fh(P)13 b(K)q(;)i(m)1553 1065 y Ff(b)1589 1050 y Fl(\);)996 1163 y Fh(b)1035 1130 y Fc(0)1084 1163 y Fg( )25 b Fl(Ev)m(e)q(\()p Fh(\013;)15 b(c)p Fl(\))118 b(:)e Fh(b)26 b Fl(=)f Fh(b)2014 1130 y Fc(0)2037 1163 y Fl(])g Fg(\025)g Fl(1)p Fh(=)p Fl(2)d(+)e Fh(\017)p Fl(\()p Fh(k)s Fl(\))291 1368 y(On)34 b(the)g(other)h(hand,)f(what)g(is)g(the)g(b)s(est)g(c)m(hances)h(for)f (an)g(algorithm)g(that)h(do)s(es)f(not)g(tak)m(e)i Fh(c)f Fl(as)150 1481 y(input?)43 b(It)32 b(cannot)g(b)s(e)g(more)g(than)f(1)p Fh(=)p Fl(2,)j(since)d(w)m(e)i(ma)m(y)f(think)f(of)h(it)f(as)h(though)g (\014rst)f(the)h(sim)m(ulator)150 1594 y(outputs)22 b(a)h(guess,)i(and) d(then)g(bit)g Fh(b)h Fl(is)f(pic)m(k)m(ed.)38 b(Therefore,)24 b(for)f(all)e(sim)m(ulators)h Fh(S)2931 1609 y Ff(k)2973 1594 y Fl(,)j Fh(p)3069 1608 y Fd(0)3108 1594 y Fl(\()p Fh(k)s Fl(\))5 b Fg(\000)g Fh(p)3355 1608 y Fd(1)3394 1594 y Fl(\()p Fh(S)3485 1609 y Ff(k)3528 1594 y Fh(;)15 b(k)s Fl(\))26 b Fg(\025)150 1707 y Fh(\017)p Fl(\()p Fh(k)s Fl(\),)31 b(whic)m(h)e(con)m(tradicts)i(the)g(seman)m(tic)g (securit)m(y)f(of)g(\()p Fh(G;)15 b(E)5 b(;)15 b(D)s Fl(\).)150 1992 y Fi(3)135 b(Other)45 b(de\014nitions)150 2195 y Fl(Tw)m(en)m(t)m(y)25 b(y)m(ears)f(ago,)i(there)e(w)m(ere)g(no)g (established)e(de\014nitions)f(of)j(securit)m(y)-8 b(.)39 b(But)23 b(p)s(eople)g(had)g(in)m(tuition)150 2308 y(that)35 b(\\seemed)f(righ)m(t.")52 b(The)34 b(in)m(tuition)e(b)s(ehind)f(the)j (RSA)f(cryptosystem)i(is)e(that,)j(without)d(an)m(y)h(a-)150 2421 y(priori)k(information)h(on)h(the)h(message,)j(it)c(is)g(hard)f (to)i(infer)e(it)h(from)g(seeing)g(its)g(encrypted)g(form.)150 2534 y(As)c(w)m(e)g(ha)m(v)m(e)h(stressed)e(b)s(efore,)i(this)d(in)m (tuition)g(is)h(m)m(uc)m(h)g(w)m(eak)m(er)i(than)f(the)g(in)m(tuition)d (b)s(ehind)g(GM-)150 2647 y(securit)m(y)-8 b(.)45 b(Indeed,)31 b(if)f(w)m(e)i(kno)m(w)g(a)g(candidate)f(message)i(sen)m(t)f(using)e (RSA)h(encryption,)g(then)h(w)m(e)g(can)150 2760 y(c)m(hec)m(k)g (whether)e(or)g(not)h(this)e(w)m(as)i(the)f(message)i(sen)m(t.)291 2872 y(Y)-8 b(ao)31 b(had)e(another)h(in)m(tuition,)d(whic)m(h)h(had)h (to)h(do)g(with)e(e\013ectiv)m(eness)j(of)e(data)h(compression.)40 b(His)150 2985 y(in)m(tuition)30 b(is)i(that,)i(if)d(a)i(cryptosystem)g (is)e(secure,)i(then)g(it)f(tak)m(es)h(just)f(as)h(man)m(y)g(bits)e(to) i(transmit)f(a)150 3098 y(message,)37 b(whether)c(I)h(happ)s(en)f(to)i (ha)m(v)m(e)h(this)d(message)i(in)f(encrypted)f(form)h(already)g(or)h (not.)52 b(F)-8 b(rom)150 3211 y(the)27 b(p)s(oin)m(t)e(of)h(view)g(of) g(an)h(information-theorist,)e(this)h(in)m(tuition)e(is)h(v)m(ery)i (natural,)f(and)g(quite)g(akin)f(to)150 3324 y(Shannon's)j(ideas)h(ab)s (out)g(securit)m(y)-8 b(.)40 b(It)30 b(turns)e(out)h(that)h(Y)-8 b(ao's)31 b(de\014nition)c(is)i(equiv)-5 b(alen)m(t)28 b(to)j(seman)m(tic)150 3437 y(securit)m(y)-8 b(.)41 b(This)28 b(w)m(as)j(sho)m(wn)f(b)m(y)g(Micali,)g(Rac)m(k)m(o\013,)j(and)c (Sloan.)291 3550 y(It)45 b(is)g(p)s(ossible)d(that)k(other)f(notions)g (of)g(securit)m(y)g(ma)m(y)h(pro)m(v)m(e)g(useful,)h(ev)m(en)f(if)e (they)h(are)h(not)150 3663 y(comparable)29 b(to)g(existing)f(ones.)41 b(The)28 b(study)g(of)h(useful)e(notions)i(of)g(securit)m(y)g(is)f (on-going,)h(and)g(if)f(y)m(ou)150 3776 y(ha)m(v)m(e)k(an)m(y)e(in)m (teresting)g(ideas)g(related)g(to)h(it,)f(please)h(b)s(e)e(sure)h(to)h (follo)m(w)f(up)f(on)h(them!)150 4061 y Fi(4)135 b(Wh)l(y)44 b(w)l(orry)i(ab)t(out)f(de\014nitions)g(so)g(m)l(uc)l(h?)150 4264 y Fl(W)-8 b(e)33 b(fo)s(cus)f(on)g Fj(pr)-5 b(ovable)34 b Fl(securit)m(y)-8 b(,)33 b(i.e.,)g(if)f(w)m(e)g(claim)g(that)h(a)f (cryptosystem)h(is)e(secure,)i(then)f(w)m(e)h(also)150 4377 y(exhibit)42 b(a)i(pro)s(of)f(of)i(this)d(claim,)47 b(under)42 b(some)j(reasonable)e(computational)h(assumption)e(\(where) 150 4490 y(\\reasonable")d(is)f(in)g(the)h(ey)m(e)h(of)e(the)h(b)s (eholder,)g(of)g(course\).)67 b(Th)m(us)37 b(the)i(need)g(for)f(a)h (de\014nition)e(of)150 4603 y(securit)m(y)f(is)f(natural.)56 b(Whether)36 b(a)g(giv)m(en)g(formal)g(de\014nition)d(satis\014es)j (one's)g(need)f(for)h(securit)m(y)-8 b(,)38 b(or)150 4716 y(comes)g(short)f(of)h(it,)h(is,)g(needless)d(to)j(sa)m(y)-8 b(,)40 b(in)c(the)i(ey)m(e)h(of)e(the)h(b)s(eholder)e(as)h(w)m(ell.)61 b(A)38 b(discussion)d(of)150 4828 y(p)s(ossible)28 b(de\014nitions)g (and)i(their)f(strength)h(is)g(therefore)h(cen)m(tral.)291 4941 y(Wh)m(y)22 b(do)f(w)m(e)h(fo)s(cus)e(so)i(m)m(uc)m(h)f(on)g (these)h(sp)s(eci\014c)e(de\014nitions?)35 b(The)21 b(concepts)h(of)f (seman)m(tic)h(securit)m(y)150 5054 y(and)28 b(indistinguishabil)o(i)o (t)m(y)23 b(are)29 b(cen)m(tral)f(in)f(cryptograph)m(y)-8 b(.)41 b(In)27 b(the)h(end,)h(it's)e(all)g(ab)s(out)h(the)h(bad)e(guys) 150 5167 y(not)k(b)s(eing)e(an)m(y)h(b)s(etter)h(o\013)g(after)f (seeing)h(something)e(than)i(they)f(w)m(ere)h(without)e(seeing)i(it.) 1889 5416 y(8-5)p eop %%Page: 6 6 6 5 bop 291 159 a Fl(Pro)m(ving)35 b(that)g(this)f(v)m(ery)h(p)s(o)m(w) 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y(public-k)m(ey)j (encryption)g(in)g(depth.)150 1882 y Ff(1.1)112 b(F)-9 b(undamen)m(tal)38 b(problems)150 2054 y Fk(Encryption)46 b Fj(\(secure)31 b(transmission)d(of)i(messages\))377 2253 y(-)h(Public)d(k)m(ey)377 2453 y(-)i(Priv)-5 b(ate)30 b(k)m(ey)h(\(Alice)e(and)h(Bob)g(ha)m(v)m(e)h(a)f(secret)h(to)g(b)s (egin)d(with;)h(this)g(t)m(yp)s(e)h(of)g(encryption)f(can)377 2566 y(b)s(e)23 b(m)m(uc)m(h)g(more)g(e\016cien)m(t)g(than)g(public)d (k)m(ey)k(encryption,)g(and)e(it)h(ma)m(y)g(not)h(rely)e(on)h (assumptions)377 2679 y(as)31 b(strongly)-8 b(.\))150 2866 y Fk(Pseudorandomness)46 b Fj(\(e\016cien)m(t)32 b(random)d(generator\))377 3065 y(This)44 b(problem)f(is)h(ab)s(out)h (generating)h(a)f(long)g(binary)e(sequence)j(from)e(a)i(shorter)f (random)377 3178 y(sequence,)35 b(so)f(that)g(the)f(resulting)f(long)h (sequence)h(is)e(indistinguishable)c(from)33 b(a)h(random)e(se-)377 3291 y(quence)f(of)f(the)h(same)g(length.)150 3478 y Fk(Authen)m(tication)45 b Fj(\(v)m(eri\014cation)31 b(of)f(iden)m(tit)m (y)g(of)h(the)f(sender\))377 3678 y(-)37 b(F)-8 b(or)38 b(public)c(k)m(ey)k(encryption,)g(digital)d(signatures)h(are)i(used.)59 b(F)-8 b(or)38 b(secret)g(k)m(ey)g(encryption,)377 3791 y(Message)29 b(Authen)m(tication)d(Co)s(des)h(\(MA)m(C\))h(are)f(used.) 39 b(In)26 b(b)s(oth)g(cases)i(a)f(de\014nition)d(of)j(securit)m(y)377 3904 y(is)j(needed)g(to)h(establish)e(the)h(notion)g(of)h(e\013ectiv)m (eness)g(of)g(the)f(authen)m(tication)h(metho)s(d.)150 4090 y Fk(Secure)k(proto)s(cols)47 b Fj(T)m(yp)s(es)30 b(of)g(secure)h(proto)s(cols:)377 4290 y(-)g(Con)m(tract)g(signing)377 4490 y(-)g(Cryptosystem)f(where)g(ma)5 b(jorit)m(y)30 b(of)h(p)s(eople)e(can)i(decrypt,)f(but)g(no)g(minorit)m(y)f(can.)150 4733 y Ff(1.2)112 b(Building)36 b(blo)s(c)m(ks)h(\(=)g(cryptographic)f (primitiv)m(es\))150 4904 y Fk(T)-9 b(rap)s(do)s(or)35 b(predicates)46 b Fj(T)-8 b(ailor-made)30 b(for)g(public-k)m(ey)f (encryption.)39 b(F)-8 b(or)32 b(example,)e(QRA.)377 5054 y(Predicate)d Fe(P)39 b Fj(is)25 b(trap)s(do)s(or)g(if)g(there)i (are)f(algorithms)f(\()p Fe(G;)15 b(D)s Fj(\))28 b(suc)m(h)e(that)g Fe(G)h Fj(is)e(a)i(k)m(ey)g(generation)377 5167 y(algorithm)c(with)g (the)i(prop)s(ert)m(y)e(that,)k Fe(P)37 b Fj(is)23 b(an)h(encryption)f (algorithm,)i(and)e Fe(D)k Fj(is)c(a)i(decryption)1889 5416 y(9-1)p eop %%Page: 2 2 2 1 bop 377 159 a Fj(algorithm:)66 b(\(1\))44 b Fe(D)i Fj(faithfully)40 b(determines)i(the)i(predicate,)i(i.e.,)h(for)c(all)f (\()p Fe(P)13 b(K)q(;)i(S)5 b(K)i Fj(\))48 b Fd(2)e Fe(G)p Fj(,)377 271 y Fe(D)s Fj(\()p Fe(S)5 b(K)q(;)15 b(P)e Fj(\()p Fe(P)g(K)q(;)i(b)p Fj(\)\))34 b(=)d Fe(b)i Fj(and)g(\(2\))i (the)f(predicate)f(is)g(secure,)i(i.e.)50 b(for)34 b(all)e(PPTF)h Fd(f)p Fe(A)3399 286 y Fc(k)3443 271 y Fd(g)p Fj(,)i(there)377 384 y(exists)30 b(a)h(negligible)d Fe(\027)36 b Fj(suc)m(h)30 b(that)487 634 y(Pr)o([\()p Fe(P)13 b(K)q(;)i(S)5 b(K)i Fj(\))27 b Fd( )e Fe(G)p Fj(\(1)1307 597 y Fc(k)1351 634 y Fj(\);)15 b Fe(b)26 b Fd( )g(f)p Fj(0)p Fe(;)15 b Fj(1)p Fd(g)p Fj(;)g Fe(c)28 b Fd( )d Fe(P)13 b Fj(\()p Fe(P)g(K)q(;)i(b)p Fj(\);)g Fe(b)2499 597 y Fb(0)2550 634 y Fd( )25 b Fe(A)2734 649 y Fc(k)2802 634 y Fj(:)g Fe(b)h Fj(=)f Fe(b)3052 597 y Fb(0)3075 634 y Fj(])g Fd(\024)g Fj(1)p Fe(=)p Fj(2)d(+)e Fe(\027)6 b Fj(\()p Fe(k)s Fj(\))377 866 y(As)28 b(w)m(e)h(sa)m(w)g(in)e(previous)f (lectures,)j(secure)f(public-k)m(ey)f(encryption)g(can)h(b)s(e)g (constructed)g(from)377 979 y(trap)s(do)s(or)i(predicates.)40 b(The)30 b Fe(S)5 b(K)37 b Fj(is)29 b(the)i Fh(tr)-5 b(ap)g(do)g(or)p Fj(.)150 1164 y Fk(T)c(rap)s(do)s(or)35 b(one-w)m(a)m(y)g(p)s(erm)m(utations)44 b Fj(F)-8 b(or)32 b(example,)e(RSA.)377 1313 y(Here,)35 b(a)f(function)e(family)g Fe(a)i Fj(is)e(a)i(function)e(that)i(tak)m(es)h(t)m(w)m(o)g(inputs,)e (a)g(parameter)h Fe(P)13 b(K)7 b Fj(,)35 b(and)377 1426 y(another)c(input)d Fe(x)p Fj(.)377 1575 y(A)22 b(function)e(family)g Fe(p)i Fj(is)f(a)h(trap)s(do)s(or)e(p)s(erm)m(utation)h(if)g(there)g (are)i(e\016cien)m(t)f(algorithms)e(\()p Fe(G;)15 b(S;)g(D)s Fj(\))377 1688 y(suc)m(h)29 b(that)g(\(1\))h Fe(G)f Fj(is)f(a)h(k)m(ey) g(generation)g(algorithm;)g(\(2\))h Fe(S)j Fj(is)28 b(an)h(algorithm)f (that)h(samples)f(the)377 1801 y(input)23 b(domain;)i(\(3\))h Fe(D)s Fj(\()p Fe(S)5 b(K)q(;)15 b(p)p Fj(\()p Fe(P)e(K)q(;)i(x)p Fj(\)\))28 b(=)d Fe(x)g Fj(for)f(all)f(\()p Fe(P)13 b(K)q(;)i(S)5 b(K)i Fj(\))27 b Fd(2)e Fe(G)p Fj(,)h(for)e(all)g Fe(x)h Fd(2)g Fe(S)5 b Fj(\()p Fe(P)13 b(K)7 b Fj(\))25 b(and)377 1914 y(\(4\))i(the)f(p)s(erm)m(utation)e(is)h(hard)f(to)j(in)m(v)m (ert,)g(i.e.:)38 b(for)25 b(all)g(PPTF)g Fd(f)p Fe(A)2691 1929 y Fc(k)2734 1914 y Fd(g)p Fj(,)i(there)f(exists)f(a)h(negligible) 377 2027 y Fe(\027)36 b Fj(suc)m(h)30 b(that)729 2259 y(Pr)o([\()p Fe(P)13 b(K)q(;)i(S)5 b(K)i Fj(\))27 b Fd( )f Fe(G)p Fj(\(1)1550 2221 y Fc(k)1594 2259 y Fj(\);)15 b Fe(x)25 b Fd( )h Fe(S)5 b Fj(\()p Fe(P)13 b(K)7 b Fj(\);)15 b Fe(x)2241 2221 y Fb(0)2290 2259 y Fd( )25 b Fe(A)2474 2274 y Fc(k)2517 2259 y Fj(\()p Fe(p)p Fj(\()p Fe(x)p Fj(\)\))h(:)g Fe(x)f Fj(=)g Fe(x)3057 2221 y Fb(0)3080 2259 y Fj(])h Fd(\024)f Fe(\027)6 b Fj(\()p Fe(k)s Fj(\))150 2491 y Fk(One-w)m(a)m(y)34 b(p)s(erm)m(utation)g(family)44 b Fj(F)-8 b(or)31 b(example,)g(exp)s(onen)m(tiation)e(and)h(DES,)g(AES) 377 2640 y(This)41 b(is)g(a)h(relaxation)g(of)g(trap)s(do)s(or)f(one-w) m(a)m(y)j(p)s(erm)m(utations,)g(in)d(that)h(w)m(e)h(do)f(not)g(require) 377 2753 y(algorithm)30 b Fe(D)j Fj(to)e(b)s(e)f(e\016cien)m(t.)41 b(In)29 b(other)i(w)m(ords:)377 2902 y(A)g(function)e(family)g Fe(p)i Fj(is)e(a)i(one-w)m(a)m(y)h(p)s(erm)m(utation)e(if)g(there)h (are)g(e\016cien)m(t)g(algorithms)e(\()p Fe(G;)15 b(S)5 b Fj(\),)377 3015 y(and)41 b(a)g(not)g(necessarily)f(e\016cien)m(t)i (algorithm)e Fe(D)k Fj(suc)m(h)d(that)g(AS)g(ABO)m(VE:)h(\(1\))g Fe(G)g Fj(is)e(a)h(k)m(ey)377 3128 y(generation)48 b(algorithm;)55 b(\(2\))48 b Fe(S)k Fj(is)46 b(an)h(algorithm)g(that)g(samples)g(the)g (input)e(domain;)55 b(\(3\))377 3240 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b(example,)f(m)m(ultiplication,)e(exp)s(onen)m(tiation,)i(DES,) g(AES)377 4080 y(This)c(is)h(a)i(relaxation)e(of)i(one-w)m(a)m(y)g(p)s (erm)m(utation)e(family)g(in)f(that)j(w)m(e)f(do)g(not)g(require)f (that)i(the)377 4192 y(trasformation)h(should)f(b)s(e)h(in)m(v)m (ertible)e(\(1-1\).)377 4341 y(A)g(function)f(family)g Fe(f)37 b Fj(is)27 b(one-w)m(a)m(y)j(if)d(there)h(are)h(e\016cien)m(t)f (algorithms)f(\()p Fe(G;)15 b(S)5 b Fj(\))30 b(suc)m(h)e(that)g(\(1\))i Fe(G)377 4454 y Fj(is)i(a)h(k)m(ey)h(generation)f(algorithm;)g(\(2\))h Fe(S)j Fj(is)32 b(an)g(algorithm)g(that)h(samples)f(the)h(input)e (domain;)377 4567 y(\(3\))h Fe(f)39 b Fj(is)30 b(hard)f(to)i(in)m(v)m (ert,)g(i.e.:)41 b(for)30 b(all)f(PPTF)h Fd(f)p Fe(A)2148 4582 y Fc(k)2191 4567 y Fd(g)p Fj(,)h(there)g(exists)f(a)h(negligible)d Fe(\027)36 b Fj(suc)m(h)30 b(that)600 4822 y(Pr)o([\()p Fe(P)13 b(K)q(;)i(S)5 b(K)i Fj(\))27 b Fd( )e Fe(G)p Fj(\(1)1420 4785 y Fc(k)1464 4822 y Fj(\);)15 b Fe(x)26 b Fd( )f Fe(S)5 b Fj(\()p 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Fn(\()p Fi(k)s Fn(\))33 b Fl(such)g(that)321 1784 y Fn(Pr[)p Fi(P)13 b(K)32 b Fh( )25 b Fi(G)p Fn(\(1)891 1747 y Ff(k)935 1784 y Fn(\);)15 b Fi(x)26 b Fh( )f Fi(S)5 b Fn(\()p Fi(P)13 b(K)7 b Fn(\);)15 b Fi(y)29 b Fh( )c Fi(f)1765 1798 y Ff(P)10 b(K)1887 1784 y Fn(\()p Fi(x)p Fn(\);)15 b Fi(x)2101 1747 y Fg(0)2151 1784 y Fh( )25 b Fi(A)2335 1799 y Ff(k)2378 1784 y Fn(\()p Fi(P)13 b(K)q(;)i(y)s Fn(\))27 b(:)e Fi(f)10 b Fn(\()p Fi(x)p Fn(\))26 b(=)f Fi(f)10 b Fn(\()p Fi(x)3203 1747 y Fg(0)3225 1784 y Fn(\)])26 b Fh(\024)f Fi(\027)6 b Fn(\()p Fi(k)s Fn(\))150 1983 y Fl(and)34 b(for)f(al)5 b(l)33 b Fi(P)13 b(K)32 b Fh(2)25 b Fi(G)p Fl(,)33 b(for)g(al)5 b(l)33 b Fi(x)25 b Fh(2)g Fi(S)5 b Fn(\()p Fi(P)13 b(K)7 b Fn(\))p Fl(,)33 b Fi(f)1830 1997 y Ff(P)10 b(K)1985 1983 y Fl(is)32 b(a)h(function.)150 2190 y Fo(De\014nition)i(2)46 b Fl(A)31 b(family)i(of)f(one-way)h (functions)g Fn(\()p Fi(G;)15 b(S;)g(p)p Fn(\))33 b Fl(is)f(a)g(family) h(of)g(one-way)f(p)-5 b(ermutations)150 2303 y(\(O)n(WP)32 b(family\))h(if)f(for)i(al)5 b(l)33 b Fi(P)13 b(K)32 b Fh(2)25 b Fi(G)p Fl(,)32 b Fi(p)1560 2317 y Ff(P)10 b(K)1716 2303 y Fl(is)32 b(a)h(p)-5 b(ermutation.)150 2510 y Fo(De\014nition)35 b(3)46 b Fl(A)35 b(family)h(of)f(one-way)h(p) -5 b(ermutations)38 b Fn(\()p Fi(G;)15 b(S;)g(p)p Fn(\))36 b Fl(is)f(tr)-5 b(ap)g(do)g(or)40 b(\(TDP)34 b(family\))i(if)f(\(1\)) 150 2623 y(the)42 b(output)h(of)f Fi(G)g Fl(c)-5 b(onsists)43 b(of)f(two)h(values,)h(denote)-5 b(d)44 b Fn(\()p Fi(P)13 b(K)q(;)i(S)5 b(K)i Fn(\))p Fl(;)47 b(and)c(\(2\))f(ther)-5 b(e)43 b(is)f(an)g(e\016cient)150 2736 y(algorithm)35 b Fi(D)g Fl(such)e(that)h(for)f(al)5 b(l)33 b Fi(k)s Fl(,)543 2935 y Fn(Pr[\()p Fi(P)13 b(K)q(;)i(S)5 b(K)i Fn(\))27 b Fh( )e Fi(G)p Fn(\(1)1364 2897 y Ff(k)1408 2935 y Fn(\);)15 b Fi(x)26 b Fh( )f Fi(S)5 b Fn(\()p Fi(P)13 b(K)7 b Fn(\);)15 b Fi(x)2055 2897 y Fg(0)2104 2935 y Fh( )26 b Fi(D)s Fn(\()p Fi(S)5 b(K)q(;)15 b(p)2559 2949 y Ff(P)10 b(K)2682 2935 y Fn(\()p Fi(x)p Fn(\)\))27 b(:)e Fi(x)g Fn(=)g Fi(x)3141 2897 y Fg(0)3165 2935 y Fn(])g(=)g(1)150 3142 y Fo(De\014nition)35 b(4)46 b Fl(A)37 b(family)g(of)h(one-way)g(functions)f(\(r)-5 b(esp.,)39 b(p)-5 b(ermutations\))40 b Fn(\()p Fi(G;)15 b(S;)g(f)10 b Fn(\))39 b Fl(is)e(a)g(one-way)150 3255 y(function)c(\(r)-5 b(esp.,)33 b(p)-5 b(ermutation\))35 b(if)e Fi(G)g Fl(is)f(a)h (deterministic)h(algorithm.)150 3498 y Fj(1.2)112 b(Hardcore)38 b(bit)150 3669 y Fn(Motiv)-5 b(ation:)55 b(Ho)m(w)37 b(to)h(encrypt)f(using)f(a)h(trap)s(do)s(or)f(p)s(erm)m(utation)g (family)-8 b(,)38 b(suc)m(h)f(as)g(RSA.)g(Ho)m(w)h(to)150 3782 y(construct)31 b(pseudorandom)e(sequences.)150 3989 y Fo(De\014nition)35 b(5)46 b Fl(L)-5 b(et)41 b Fn(\()p Fi(G;)15 b(S;)g(f)10 b Fn(\))43 b Fl(b)-5 b(e)41 b(a)g(O)n(WF)g (family.)68 b(L)-5 b(et)42 b Fi(b)2286 4008 y Fe(\()p Fg(\016)p Fe(\))2380 3989 y Fn(\()p Fh(\016)p Fn(\))h Fl(b)-5 b(e)41 b(a)g(Bo)-5 b(ole)g(an)44 b(function.)67 b Fi(b)41 b Fl(is)h(a)150 4102 y(har)-5 b(dc)g(or)g(e)36 b(bit)c(for)h Fn(\()p Fi(G;)15 b(S;)g(f)10 b Fn(\))34 b Fl(if)257 4285 y(1.)46 b Fi(b)416 4299 y Ff(P)10 b(K)539 4285 y Fn(\()p Fi(x)p Fn(\))36 b Fl(is)f(appr)-5 b(oximately)39 b(a)d(bit)f(of)h(information)h(ab)-5 b(out)37 b Fi(x)p Fl(.)50 b(Mor)-5 b(e)35 b(pr)-5 b(e)g(cisely,)38 b(ther)-5 b(e)36 b(exists)g(a)377 4398 y(ne)-5 b(gligible)33 b(function)f Fi(\026)h Fl(such)f(that)i(for)f(al)5 b(l)34 b Fi(P)13 b(K)32 b Fh(2)24 b Fi(G)p Fn(\(1)2297 4365 y Ff(k)2341 4398 y Fn(\))1282 4597 y Fh(j)15 b Fn(Pr[)p Fi(x)26 b Fh( )f Fi(S)5 b Fn(\()p Fi(P)13 b(K)7 b Fn(\);)15 b Fi(b)2004 4611 y Ff(P)10 b(K)2127 4597 y Fn(\()p Fi(x)p Fn(\)])22 b Fh(\000)e Fn(1)p Fi(=)p Fn(2)p Fh(j)26 b(\024)f Fi(\026)p Fn(\()p Fi(k)s Fn(\))257 4833 y Fl(2.)46 b(It)30 b(is)f(har)-5 b(d)31 b(to)f(c)-5 b(ompute)31 b Fi(b)1268 4847 y Ff(P)10 b(K)1391 4833 y Fn(\()p Fi(x)p Fn(\))30 b Fl(fr)-5 b(om)30 b(the)g(values)g Fi(P)13 b(K)36 b Fl(and)30 b Fi(f)2573 4847 y Ff(P)10 b(K)2695 4833 y Fn(\()p Fi(x)p Fn(\))p Fl(.)42 b(Mor)-5 b(e)29 b(pr)-5 b(e)g(cisely,)31 b(for)f(al)5 b(l)377 4946 y(PPTF)33 b Fh(f)p Fi(A)771 4961 y Ff(k)814 4946 y Fh(g)g Fl(ther)-5 b(e)33 b(exists)g(a)g(ne)-5 b(gligible)33 b(function)f Fi(\027)6 b Fn(\()p Fi(k)s Fn(\))33 b Fl(such)g(that)471 5144 y Fn(Pr[)p Fi(P)13 b(K)32 b Fh( )25 b Fi(G)p Fn(\(1)1041 5107 y Ff(k)1085 5144 y Fn(\);)15 b Fi(x)26 b Fh( )f Fi(S)5 b Fn(\()p Fi(P)13 b(K)7 b Fn(\);)15 b Fi(b)1719 5107 y Fg(0)1769 5144 y Fh( )25 b Fi(A)1953 5159 y Ff(k)1996 5144 y Fn(\()p Fi(P)13 b(K)q(;)i(f)2265 5158 y Ff(P)10 b(K)2388 5144 y Fn(\()p Fi(x)p Fn(\)\))27 b(:)e Fi(b)2661 5107 y Fg(0)2710 5144 y Fn(=)g Fi(b)2845 5158 y Ff(P)10 b(K)2968 5144 y Fn(\()p Fi(x)p Fn(\)])26 b Fh(\024)f Fn(1)p Fi(=)p Fn(2)c(+)f Fi(\027)6 b Fn(\()p Fi(k)s Fn(\))1867 5416 y(10-1)p eop %%Page: 2 2 2 1 bop 291 159 a Fn(Note)26 b(that)g(although)e(a)h(hardcore)g(bit)f (is)g(de\014ned)g(for)g(O)m(WF)i(families,)e(this)g(concept)i(mak)m(es) g(m)m(uc)m(h)150 271 y(more)31 b(sense)f(when)f(w)m(e)i(consider)e(O)m (WP)i(families.)291 384 y(Giv)m(en)f(a)h(hardcore)f(bit,)g(w)m(e)h(can) g(build:)286 568 y Fh(\017)46 b Fn(Secure)31 b(PK)f(cryptosystem:)42 b(giv)m(en)30 b Fi(P)13 b(K)7 b Fn(,)31 b(to)g(encrypt)g Fi(m)25 b Fh(2)g Fn(0)p Fi(;)15 b Fn(1)r(,)31 b(pic)m(k)f Fi(x)g Fn(suc)m(h)h(that)g Fi(b)3408 582 y Ff(P)10 b(K)3531 568 y Fn(\()p Fi(x)p Fn(\))26 b(=)377 681 y Fi(m)p Fn(,)31 b(send)e Fi(f)765 695 y Ff(P)10 b(K)888 681 y Fn(\()p Fi(x)p Fn(\).)286 867 y Fh(\017)46 b Fn(Psudorandom)30 b(generator)i(\(con)m(v)m(ert)h(short)d(string)g(to)i(long)f(string\))f (\(see)i(next)f(lecture.\))43 b(This)377 980 y(allo)m(ws)30 b(us)g(to)h(increase)f(complexit)m(y)g(of)h Fi(P)13 b(K)37 b Fn(cryptosystem.)150 1266 y Fk(2)135 b(The)44 b(Goldreic)l(h-Levin)i (theorem)150 1468 y Fo(Theorem)34 b(6)46 b Fl(F)-7 b(or)36 b(any)g(O)n(WP)e(family,)j(for)f(al)5 b(l)36 b(but)f(a)g(ne)-5 b(gligible)36 b(fr)-5 b(action)36 b(of)g Fi(r)d Fh(2)c(f)p Fn(0)p Fi(;)15 b Fn(1)p Fh(g)3350 1436 y Ff(`)3386 1468 y Fl(,)35 b Fi(b)3488 1482 y Ff(r)3526 1468 y Fn(\()p Fi(x)p Fn(\))c Fh(\021)150 1581 y Fi(x:r)d Fh(\021)392 1513 y Fd(L)493 1540 y Ff(`)493 1608 y(i)p Fe(=1)626 1581 y Fi(x)678 1595 y Ff(i)707 1581 y Fi(r)748 1595 y Ff(i)808 1581 y Fl(is)33 b(har)-5 b(dc)g(or)g(e.)291 1789 y Fn(This)22 b(is)g(a)i(rather)f(tec)m(hnical)h(pro)s(of.)37 b(The)23 b(reason)h(that)g(I)f(w)m(an)m(t)h(to)g(sho)m(w)f(it)g(to)h(y) m(ou)g(is)f(that)h(I)f(b)s(eliev)m(e)150 1902 y(it)32 b(is)g(v)m(ery)h(imp)s(ortan)m(t.)47 b(In)32 b(some)h(sense,)h(this)e (theorem)h(sa)m(ys)g(that)g(if)f(there)h(is)f(an)m(y)h(hardness)e(in)h (the)150 2015 y(w)m(orld,)g(then)h(w)m(e)g(can)f(put)g(it)g(to)i(go)s (o)s(d)e(use!)47 b(Without)33 b(this)e(theorem,)j(one)f(can)g(imagine)e (that)i(all)f(of)150 2128 y(crypto)c(w)m(ould)e(break)h(do)m(wn)g(once) h(some)g(sp)s(eci\014c)e(assumption)g(gets)j(brok)m(en.)39 b(But)28 b(with)e(it,)i(w)m(e)g(kno)m(w)150 2241 y(that)k(w)m(e)f(are)g (OK)g(as)g(long)g(as)g(one-w)m(a)m(y)h(\(or,)g(for)f(public-k)m(ey)e (encryption,)i(trap)s(do)s(or\))f(p)s(erm)m(utations)150 2354 y(exist.)291 2467 y(Another)h(reason)h(wh)m(y)f(I)g(w)m(an)m(t)h (to)g(sho)m(w)f(it)g(to)h(y)m(ou)f(is)g(that)g(I)g(think)f(it)h(is)f (rather)h(ingen)m(uous)g(and)150 2580 y(I)f(hop)s(e)g(y)m(ou)h(will)c (b)s(e)j(impressed.)150 2692 y Fo(Pro)s(of)123 b Fn(It)32 b(is)f(su\016cien)m(t)g(to)h(sho)m(w)g(that,)h(if)d Fi(p)1795 2659 y Fg(0)1795 2719 y Ff(P)10 b(K)1918 2692 y Fn(\()p Fh(\001)p Fn(\))33 b(is)d(a)i(one-w)m(a)m(y)i(p)s(erm)m(utation)c (family)g(of)i Fi(S)3529 2706 y Ff(P)10 b(K)3679 2692 y Fh(\022)150 2805 y(f)p Fn(0)p Fi(;)15 b Fn(1)p Fh(g)370 2772 y Ff(`)p Fe(\()p Ff(P)10 b(K)5 b Fe(\))609 2805 y Fn(then)29 b Fi(p)861 2819 y Ff(P)10 b(K)984 2805 y Fn(\()p Fh(\001)p Fi(;)15 b Fh(\001)p Fn(\))32 b(where)d(the)h(second)g (argumen)m(t)h(is)e(randomly)f(c)m(hosen)i(from)g Fh(f)p Fn(0)p Fi(;)15 b Fn(1)p Fh(g)3516 2772 y Ff(`)p Fe(\()p Ff(P)10 b(K)5 b Fe(\))3725 2805 y Fn(,)150 2918 y(and)30 b Fi(p)373 2932 y Ff(P)10 b(K)495 2918 y Fn(\()p Fi(x;)15 b(r)s Fn(\))26 b(=)g Fi(p)870 2885 y Fg(0)870 2945 y Ff(P)10 b(K)992 2918 y Fn(\()p Fi(x)p Fn(\))21 b Fh(\016)g Fi(r)33 b Fn(is)c(a)i(one-w)m(a)m(y)h(p)s(erm)m(utation)e(with)f (hardcore)h(bit)g Fi(b)p Fn(\()p Fi(x;)15 b(r)s Fn(\))26 b(=)f Fi(x:r)s Fn(.)40 b(It)31 b(is)150 3031 y(clear)h(that)i Fi(p)e Fn(is)f(a)i(O)m(WP)g(family)-8 b(.)46 b(It)32 b(is)g(also)g(clear)h(that)g Fi(b)f Fn(giv)m(es)h(one)g(bit)e(of)i (information)d(ab)s(out)j(its)150 3144 y(input.)291 3257 y(Supp)s(ose)c(for)h(con)m(tradiction)g(that)h Fi(b)p Fn(\()p Fi(x;)15 b(r)s Fn(\))31 b(is)e(not)i(hardcore)f(for)g Fi(p)p Fn(.)291 3370 y(Then)e(there)h(exists)g(a)g(predictor)f Fh(f)p Fi(A)1583 3385 y Ff(k)1626 3370 y Fh(g)i Fn(and)e(a)h(p)s (olynomial)d Fi(a)p Fn(\()p Fi(k)s Fn(\))k(suc)m(h)f(that)g(for)g (in\014nitely)d(man)m(y)150 3483 y Fi(k)s Fn('s)150 3682 y Fi(p)f Fn(=)g(Pr)o([)p Fi(P)13 b(K)33 b Fh( )25 b Fi(G)p Fn(\(1)887 3645 y Ff(k)931 3682 y Fn(\);)15 b Fi(x)26 b Fh( )f Fi(S)5 b Fn(\()p Fi(P)13 b(K)7 b Fn(\);)15 b Fi(r)28 b Fh( )e(f)p Fn(0)p Fi(;)15 b Fn(1)p Fh(g)1932 3645 y Ff(`)p Fe(\()p Ff(P)10 b(K)5 b Fe(\))2140 3682 y Fn(;)15 b Fi(b)2219 3645 y Fg(0)2268 3682 y Fn(=)25 b Fi(A)2432 3697 y Ff(k)2475 3682 y Fn(\()p Fi(P)13 b(K)q(;)i(p)2745 3696 y Ff(P)10 b(K)2869 3682 y Fn(\()p Fi(x;)15 b(r)s Fn(\))26 b(:)g Fi(b)3191 3645 y Fg(0)3239 3682 y Fn(=)f Fi(x:r)s Fn(])g Fh(\025)g Fn(1)p Fi(=)p Fn(2+1)p Fi(=a)p Fn(\()p Fi(k)s Fn(\))291 3882 y(Consider)34 b(a)i(v)-5 b(alue)35 b Fi(k)k Fn(for)c(whic)m(h)f(the)i(statemen)m(t)h(ab)s(o)m(v) m(e)g(is)e(true.)56 b(It)36 b(is)e(easy)j(to)f(see)g(that)g(for)g(a)150 3995 y(p)s(olynomial)28 b(fraction)i(of)g(all)g Fi(P)13 b(K)32 b Fh(2)24 b Fi(G)p Fn(\(1)1602 3962 y Ff(k)1646 3995 y Fn(\),)31 b(call)f(them)g(the)h("go)s(o)s(d")g(PK's,)159 4217 y Fi(p)205 4232 y Ff(g)r(ood)374 4217 y Fn(=)25 b(Pr[)p Fi(x)g Fh( )g Fi(S)5 b Fn(\()p Fi(P)13 b(K)7 b Fn(\);)15 b Fi(r)29 b Fh( )c(f)p Fn(0)p Fi(;)15 b Fn(1)p Fh(g)1518 4179 y Ff(`)p Fe(\()p Ff(P)10 b(K)5 b Fe(\))1727 4217 y Fn(;)15 b Fi(b)1806 4179 y Fg(0)1855 4217 y Fn(=)25 b Fi(A)2019 4232 y Ff(k)2062 4217 y Fn(\()p Fi(P)13 b(K)q(;)i(p)2332 4231 y Ff(P)10 b(K)2456 4217 y Fn(\()p Fi(x;)15 b(r)s Fn(\))25 b(:)h Fi(b)2777 4179 y Fg(0)2826 4217 y Fn(=)e Fi(x:r)s Fn(])i Fh(\025)f Fn(1)p Fi(=)p Fn(2)c(+)f(1)p Fi(=)p Fn(2)p Fi(a)p Fn(\()p Fi(k)s Fn(\))291 4416 y(Wh)m(y?)41 b(Let)31 b Fi(g)e Fn(=)c(Pr)o([)p Fi(P)13 b(K)32 b Fh( )25 b Fi(G)p Fn(\(1)1464 4383 y Ff(k)1508 4416 y Fn(\))h(:)f Fi(P)13 b(K)37 b Fn(is)30 b(go)s(o)s(d].)41 b(Supp)s(ose)28 b Fi(g)h(<)c Fn(1)p Fi(=a)p Fn(\()p Fi(k)s Fn(\).)43 b(Then)1096 4616 y Fi(p)83 b Fn(=)g Fi(p)1425 4631 y Ff(g)r(ood)1589 4616 y Fh(\003)21 b Fi(g)j Fn(+)c Fi(p)1859 4631 y Ff(bad)1987 4616 y Fh(\003)g Fn(\(1)h Fh(\000)f Fi(g)s Fn(\))1225 4754 y Fh(\024)83 b Fn(1)20 b Fh(\003)h Fi(g)j Fn(+)c(\(1)p Fi(=)p Fn(2)i(+)e(1)p Fi(=)p Fn(2)p Fi(a)p Fn(\()p Fi(k)s Fn(\)\)\(1)j Fh(\000)d Fi(g)s Fn(\))1225 4892 y(=)83 b(1)p Fi(=)p Fn(2)21 b(+)f(1)p Fi(=)p Fn(2)p Fi(a)p Fn(\()p Fi(k)s Fn(\))j(+)d Fi(g)s Fn(\(1)p Fi(=)p Fn(2)i Fh(\000)e Fn(1)p Fi(=)p Fn(2)p Fi(a)p Fn(\()p Fi(k)s Fn(\)\))1225 5029 y Fi(<)83 b Fn(1)p Fi(=)p Fn(2)21 b(+)f(1)p Fi(=)p Fn(2)p Fi(a)p Fn(\()p Fi(k)s Fn(\))j(+)d(1)p Fi(=)p Fn(2)p Fi(a)p Fn(\()p Fi(k)s Fn(\))i Fh(\000)e Fn(1)p Fi(=)p Fn(2)p Fi(a)2642 4992 y Fe(2)2683 5029 y Fn(\()p Fi(k)s Fn(\))1225 5167 y Fi(<)83 b Fn(1)p Fi(=)p Fn(2)21 b(+)f(1)p Fi(=a)p Fn(\()p Fi(k)s Fn(\))1867 5416 y(10-2)p eop %%Page: 3 3 3 2 bop 291 159 a Fn(But)31 b(w)m(e)g(kno)m(w)f(that)h Fi(p)25 b(>)g Fn(1)p Fi(=)p Fn(2)c(+)f(1)p Fi(=a)p Fn(\()p Fi(k)s Fn(\),)33 b(so)d(this)g(is)f(a)i(con)m(tradiction.)291 271 y(By)39 b(the)g(same)g(tok)m(en,)j(for)c(all)g(go)s(o)s(d)g(PK's,)j (for)d(a)h(p)s(olynomial)d(fraction)i(of)h Fi(x)g Fh(2)f Fi(S)5 b Fn(\()p Fi(P)13 b(K)7 b Fn(\))39 b(\(call)150 384 y(these)31 b Fi(x)p Fn('s)f("go)s(o)s(d"\))547 586 y(Pr)o([)p Fi(r)e Fh( )d(f)p Fn(0)p Fi(;)15 b Fn(1)p Fh(g)1074 548 y Ff(`)p Fe(\()p Ff(P)10 b(K)5 b Fe(\))1283 586 y Fn(;)15 b Fi(b)1362 548 y Fg(0)1411 586 y Fh( )25 b Fi(A)1595 601 y Ff(k)1638 586 y Fn(\()p Fi(P)13 b(K)q(;)i(p)1908 600 y Ff(P)10 b(K)2032 586 y Fn(\()p Fi(x;)15 b(r)s Fn(\)\))26 b(:)g Fi(b)2389 548 y Fg(0)2437 586 y Fn(=)f Fi(x:r)s Fn(])h Fh(\025)f Fn(1)p Fi(=)p Fn(2)c(+)f(1)p Fi(=)p Fn(4)p Fi(a)p Fn(\()p Fi(k)s Fn(\))291 787 y(Supp)s(ose)42 b(that)i Fi(P)13 b(K)50 b Fn(and)43 b Fi(x)g Fn(are)h(go)s(o)s(d,)i (and)d(that)h(w)m(e)g(are)g(giv)m(en)f Fi(y)50 b Fn(=)d Fi(p)3003 754 y Fg(0)3003 814 y Ff(P)10 b(K)3125 787 y Fn(\()p Fi(x)p Fn(\).)81 b(Since)42 b(this)150 900 y(o)s(ccurs)28 b(with)e(non-negligible)g(probabilit)m(y)-8 b(,)26 b(it)i(is)f(enough)h(to)g(sho)m(w)g(ho)m(w)g(to)h(in)m(v)m(ert)f Fi(p)3112 914 y Ff(P)10 b(K)3262 900 y Fn(when)27 b(w)m(e)i(are)150 1013 y(restricted)h(to)h(this)e(situation.)291 1126 y(Let)i(us)f(sho)m 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b(do)g(w)m(e)g(get)g(that?)150 2029 y(Observing)f(that)j Fi(x:e)910 2043 y Ff(i)974 2029 y Fn(=)e Fi(x:)p Fn(\()p Fi(r)27 b Fh(\010)d Fn(\()p Fi(r)j Fh(\010)d Fi(e)1595 2043 y Ff(i)1623 2029 y Fn(\)\))36 b(=)e(\()p Fi(x:r)s Fn(\))25 b Fh(\010)f Fn(\()p Fi(x:)p Fn(\()p Fi(r)j Fh(\010)d Fi(e)2497 2043 y Ff(i)2526 2029 y Fn(\)\),)38 b(w)m(e)f(will)d(do)i(it)f(b)m(y)i(taking)f(the)150 2142 y(ma)5 b(jorit)m(y)30 b(of)h(\()p Fi(x:r)781 2109 y Ff(j)818 2142 y Fn(\))20 b Fh(\010)g Fn(\()p Fi(x:)p Fn(\()p Fi(r)1155 2109 y Ff(j)1212 2142 y Fh(\010)g Fi(e)1345 2156 y Ff(i)1374 2142 y Fn(\)\).)291 2255 y(Ho)m(w)43 b(do)f(w)m(e)h(get)g(the)g(v)-5 b(alues)41 b Fi(x:r)1529 2222 y Ff(j)1608 2255 y Fn(for)h(a)g(bunc)m(h)f(of)i(totally)f(random)g Fi(r)2936 2222 y Ff(j)2972 2255 y Fn('s?)76 b(The)42 b(answ)m(er)g(is:)150 2368 y(w)m(e)36 b(w)m(on't.)55 b(If)35 b(our)g Fi(r)895 2335 y Ff(j)931 2368 y Fn('s)g(aren't)h (totally)f(random)g(but)f(are)i Fl(p)-5 b(airwise)38 b(indep)-5 b(endent)p Fn(.)57 b(W)-8 b(e)37 b(don't)e(ha)m(v)m(e)150 2481 y(to)j(compute)f Fi(r)687 2448 y Ff(j)723 2481 y Fn('s,)i(but)e(can)g(just)g(guess)g(them)g(correctly)h(with)d(go)s(o)s (d)i(probabilit)m(y)-8 b(.)59 b(This)36 b(will)e(also)150 2594 y(b)s(e)h(su\016cien)m(t,)i(as)f(it)f(turns)g(out,)i(for)f(the)g (ma)5 b(jorit)m(y)36 b(of)g(the)g(tests)g(to)h(giv)m(e)f(us)f(the)h (righ)m(t)f(answ)m(er)h(for)150 2706 y Fi(x:)p Fn(\()p Fi(r)306 2673 y Ff(j)363 2706 y Fh(\010)20 b Fi(e)496 2720 y Ff(i)524 2706 y Fn(\).)291 2819 y(\(Recall)34 b(that)g(random)f(v)-5 b(ariables)32 b Fi(X)41 b Fn(and)33 b Fi(Y)53 b Fn(are)34 b(pairwise)d(indep)s(enden)m(t)h(if)g(for)i(an)m (y)f(v)-5 b(alue)33 b Fi(y)k Fn(of)150 2932 y(the)c(random)f(v)-5 b(ariable)32 b Fi(Y)20 b Fn(,)34 b(Pr)o([)p Fi(X)7 b Fh(j)p Fi(Y)51 b Fn(=)29 b Fi(y)s Fn(])h(=)f(Pr)o([)p Fi(X)7 b Fn(])34 b(and)e(for)h(an)m(y)g(v)-5 b(alue)33 b Fi(x)g Fn(of)g(the)g(random)f(v)-5 b(ariable)150 3045 y Fi(X)7 b Fn(,)31 b(Pr)o([)p Fi(Y)21 b Fh(j)p Fi(X)33 b Fn(=)25 b 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Ff(J)3679 3643 y Fn(=)377 3688 y Fd(L)478 3783 y Ff(l)q Fg(2)p Ff(J)611 3756 y Fi(x:s)731 3723 y Ff(l)757 3756 y Fn(\),)31 b(and)f(so)g(our)g(guess)h(for)f Fi(x:r)1797 3723 y Ff(J)1876 3756 y Fn(is)f Fi(\032)2014 3723 y Ff(J)2088 3756 y Fn(=)2184 3688 y Fd(L)2285 3783 y Ff(l)q Fg(2)p Ff(J)2418 3756 y Fi(\033)2473 3723 y Ff(l)2500 3756 y Fn(\).)291 3941 y(It)42 b(is)e(easy)i(to)g(see)g(that)g(\(1\))h(the)e (v)-5 b(alues)41 b Fi(r)1853 3908 y Ff(J)1943 3941 y Fn(are)h(pairwise)d(indep)s(enden)m(t,)j(and)f(\(2\))h(IF)g(for)f(all) 150 4054 y(1)27 b Fh(\024)f Fi(l)j Fh(\024)d Fi(u)p Fn(,)32 b Fi(\033)636 4021 y Ff(l)693 4054 y Fn(is)e(the)h(righ)m(t)g(guess)g (for)g Fi(x:s)1662 4021 y Ff(l)1719 4054 y Fn(\(call)g(this)f(ev)m(en)m (t)j Fi(A)p Fn(\),)f(THEN)f(for)f(all)h Fi(J)k Fh(\022)26 b Fn([)p Fi(u)p Fn(],)33 b Fi(\032)3452 4021 y Ff(J)3532 4054 y Fn(is)d(the)150 4167 y(righ)m(t)g(guess)g(for)g Fi(x:r)868 4134 y Ff(J)917 4167 y Fn(.)291 4280 y(No)m(w)42 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Fg(A)2361 1493 y Fd(k)2404 1478 y Ff(g)h Fj(ther)-5 b(e)33 b(exists)f(a)h(ne)-5 b(gligible)32 b(function)150 1590 y Fg(\027)6 b Fl(\()p Fg(k)s Fl(\))33 b Fj(such)g(that)321 1795 y Fl(Pr[)p Fg(P)13 b(K)32 b Ff( )25 b Fg(G)p Fl(\(1)891 1757 y Fd(k)935 1795 y Fl(\);)15 b Fg(x)26 b Ff( )f Fg(S)5 b Fl(\()p Fg(P)13 b(K)7 b Fl(\);)15 b Fg(y)29 b Ff( )c Fg(f)1765 1809 y Fd(P)10 b(K)1887 1795 y Fl(\()p Fg(x)p Fl(\);)15 b Fg(x)2101 1757 y Fe(0)2151 1795 y Ff( )25 b Fg(A)2335 1810 y Fd(k)2378 1795 y Fl(\()p Fg(P)13 b(K)q(;)i(y)s Fl(\))27 b(:)e Fg(f)10 b Fl(\()p Fg(x)p Fl(\))26 b(=)f Fg(f)10 b Fl(\()p Fg(x)3203 1757 y Fe(0)3225 1795 y Fl(\)])26 b Ff(\024)f Fg(\027)6 b Fl(\()p Fg(k)s Fl(\))150 1999 y Fj(and)34 b(for)f(al)5 b(l)33 b Fg(P)13 b(K)32 b Ff(2)25 b Fg(G)p Fj(,)33 b(for)g(al)5 b(l)33 b Fg(x)25 b Ff(2)g Fg(S)5 b Fl(\()p Fg(P)13 b(K)7 b Fl(\))p Fj(,)33 b Fg(f)1830 2013 y Fd(P)10 b(K)1985 1999 y Fj(is)32 b(a)h(function.)150 2211 y Fm(De\014nition)i(2)46 b Fj(A)31 b(family)i(of)f(one-way)h (functions)g Fl(\()p Fg(G;)15 b(S;)g(p)p Fl(\))33 b Fj(is)f(a)g(family) h(of)g(one-way)f(p)-5 b(ermutations)150 2324 y(\(O)n(WP)32 b(family\))h(if)f(for)i(al)5 b(l)33 b Fg(P)13 b(K)32 b Ff(2)25 b Fg(G)p Fj(,)32 b Fg(p)1560 2338 y Fd(P)10 b(K)1716 2324 y Fj(is)32 b(a)h(p)-5 b(ermutation.)150 2537 y Fm(De\014nition)35 b(3)46 b Fj(A)35 b(family)h(of)f(one-way)h(p) -5 b(ermutations)38 b Fl(\()p Fg(G;)15 b(S;)g(p)p Fl(\))36 b Fj(is)f(tr)-5 b(ap)g(do)g(or)40 b(\(TDP)34 b(family\))i(if)f(\(1\)) 150 2650 y(the)42 b(output)h(of)f Fg(G)g Fj(c)-5 b(onsists)43 b(of)f(two)h(values,)h(denote)-5 b(d)44 b Fl(\()p Fg(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))p Fj(;)47 b(and)c(\(2\))f(ther)-5 b(e)43 b(is)f(an)g(e\016cient)150 2763 y(algorithm)35 b Fg(D)g Fj(such)e(that)h(for)f(al)5 b(l)33 b Fg(k)s Fj(,)561 2967 y Fl(Pr[\()p Fg(P)13 b(K)q(;)i(S)5 b(K)i Fl(\))27 b Ff( )e Fg(G)p Fl(\(1)1382 2929 y Fd(k)1426 2967 y Fl(\);)15 b Fg(x)26 b Ff( )f Fg(S)5 b Fl(\()p Fg(P)13 b(K)7 b Fl(\);)15 b Fg(x)2073 2929 y Fe(0)2122 2967 y Ff( )25 b Fg(D)s Fl(\()p Fg(S)5 b(K)q(;)15 b(p)2576 2981 y Fd(P)10 b(K)2700 2967 y Fl(\()p Fg(x)p Fl(\))26 b(:)g Fg(x)f Fl(=)g Fg(x)3124 2929 y Fe(0)3147 2967 y Fl(])g(=)g(1)150 3180 y Fm(De\014nition)35 b(4)46 b Fj(A)37 b(family)g(of)h(one-way)g(functions)f(\(r)-5 b(esp.,)39 b(p)-5 b(ermutations\))40 b Fl(\()p Fg(G;)15 b(S;)g(f)10 b Fl(\))39 b Fj(is)e(a)g(one-way)150 3292 y(function)c(\(r)-5 b(esp.,)33 b(p)-5 b(ermutation\))35 b(if)e Fg(G)g Fj(is)f(a)h (deterministic)h(algorithm.)150 3505 y Fl(A)c(TDP)h(family)e(is)g (conceiv)-5 b(able)30 b(but)g(is)f(a)i(TDP)f(\(with)g(no)g(family\))f (conceiv)-5 b(able?)150 3618 y(No.)41 b(F)-8 b(or)31 b(eac)m(h)h Fg(k)s Fl(,)f(there)f(is)g(just)g(one)g Fg(f)1518 3633 y Fd(k)1560 3618 y Fl(.)41 b(So)30 b(if)g(there)g(exists)g(trap)s (do)s(or,)g Fg(S)2780 3633 y Fd(k)2822 3618 y Fl(,)h(for)f Fg(f)3062 3633 y Fd(k)3104 3618 y Fl(,)h(then)f Fg(A)3435 3633 y Fd(k)3508 3618 y Fl(migh)m(t)150 3731 y(kno)m(w)g Fg(S)443 3746 y Fd(k)486 3731 y Fl(.)150 3974 y Fh(1.2)112 b(Hardcore)38 b(bit)150 4146 y Fl(Motiv)-5 b(ation:)55 b(Ho)m(w)37 b(to)h(encrypt)f(using)f(a)h(trap)s(do)s(or)f(p)s(erm)m (utation)g(family)-8 b(,)38 b(suc)m(h)f(as)g(RSA.)g(Ho)m(w)h(to)150 4259 y(construct)31 b(pseudorandom)e(sequences.)150 4471 y Fm(De\014nition)35 b(5)46 b Fj(L)-5 b(et)41 b Fl(\()p Fg(G;)15 b(S;)g(f)10 b Fl(\))43 b Fj(b)-5 b(e)41 b(a)g(O)n(WF)g (family.)68 b(L)-5 b(et)42 b Fg(b)2286 4490 y Fc(\()p Fe(\016)p Fc(\))2380 4471 y Fl(\()p Ff(\016)p Fl(\))h Fj(b)-5 b(e)41 b(a)g(Bo)-5 b(ole)g(an)44 b(function.)67 b Fg(b)41 b Fj(is)h(a)150 4584 y(har)-5 b(dc)g(or)g(e)36 b(bit)c(for)h Fl(\()p Fg(G;)15 b(S;)g(f)10 b Fl(\))34 b Fj(if)257 4772 y(1.)46 b Fg(b)416 4786 y Fd(P)10 b(K)539 4772 y Fl(\()p Fg(x)p Fl(\))36 b Fj(is)f(appr)-5 b(oximately)39 b(a)d(bit)f(of)h(information)h(ab)-5 b(out)37 b Fg(x)p Fj(.)50 b(Mor)-5 b(e)35 b(pr)-5 b(e)g(cisely,)38 b(ther)-5 b(e)36 b(exists)g(a)377 4885 y(ne)-5 b(gligible)33 b(function)f Fg(\026)h Fj(such)f(that)i(for)f(al)5 b(l)34 b Fg(P)13 b(K)32 b Ff(2)24 b Fg(G)p Fl(\(1)2297 4852 y Fd(k)2341 4885 y Fl(\))1282 5089 y Ff(j)15 b Fl(Pr[)p Fg(x)26 b Ff( )f Fg(S)5 b Fl(\()p Fg(P)13 b(K)7 b Fl(\);)15 b Fg(b)2004 5103 y Fd(P)10 b(K)2127 5089 y Fl(\()p Fg(x)p Fl(\)])22 b Ff(\000)e Fl(1)p Fg(=)p Fl(2)p Ff(j)26 b(\024)f Fg(\026)p Fl(\()p Fg(k)s Fl(\))1867 5416 y(11-1)p eop %%Page: 2 2 2 1 bop 257 159 a Fj(2.)46 b(It)30 b(is)f(har)-5 b(d)31 b(to)f(c)-5 b(ompute)31 b Fg(b)1268 173 y Fd(P)10 b(K)1391 159 y Fl(\()p Fg(x)p Fl(\))30 b Fj(fr)-5 b(om)30 b(the)g(values)g Fg(P)13 b(K)36 b Fj(and)30 b Fg(f)2573 173 y Fd(P)10 b(K)2695 159 y Fl(\()p Fg(x)p Fl(\))p Fj(.)42 b(Mor)-5 b(e)29 b(pr)-5 b(e)g(cisely,)31 b(for)f(al)5 b(l)377 271 y(PPTF)33 b Ff(f)p Fg(A)771 286 y Fd(k)814 271 y Ff(g)g Fj(ther)-5 b(e)33 b(exists)g(a)g(ne)-5 b(gligible)33 b(function)f Fg(\027)6 b Fl(\()p Fg(k)s Fl(\))33 b Fj(such)g(that)471 476 y Fl(Pr[)p Fg(P)13 b(K)32 b Ff( )25 b Fg(G)p Fl(\(1)1041 438 y Fd(k)1085 476 y Fl(\);)15 b Fg(x)26 b Ff( )f Fg(S)5 b Fl(\()p Fg(P)13 b(K)7 b Fl(\);)15 b Fg(b)1719 438 y Fe(0)1769 476 y Ff( )25 b Fg(A)1953 491 y Fd(k)1996 476 y Fl(\()p Fg(P)13 b(K)q(;)i(f)2265 490 y Fd(P)10 b(K)2388 476 y Fl(\()p Fg(x)p Fl(\)\))27 b(:)e Fg(b)2661 438 y Fe(0)2710 476 y Fl(=)g Fg(b)2845 490 y Fd(P)10 b(K)2968 476 y Fl(\()p Fg(x)p Fl(\)])26 b Ff(\024)f Fl(1)p Fg(=)p Fl(2)c(+)f Fg(\027)6 b Fl(\()p Fg(k)s Fl(\))291 717 y(Note)26 b(that)g(although)e(a)h(hardcore)g(bit)f(is)g(de\014ned)g(for)g(O)m(WF) i(families,)e(this)g(concept)i(mak)m(es)g(m)m(uc)m(h)150 830 y(more)31 b(sense)f(when)f(w)m(e)i(consider)e(O)m(WP)i(families.) 291 943 y(Note)h(that)f(actually)f(the)g(non-trivialit)m(y)f(condition) g(follo)m(ws)g(from)h(the)h(hardness)e(condition.)150 1230 y Fi(2)135 b(Construcing)45 b(a)g(one-bit)g(PR)l(G)g(from)h(a)f (sp)t(ecial)g(O)l(WP)150 1432 y Fl(Recall)30 b(the)h(de\014nition)d(of) i(a)h(pseudorandom)e(generator:)150 1645 y Fm(De\014nition)35 b(6)46 b Fj(L)-5 b(et)38 b Fg(l)f(<)e(m)p Fj(,)k(and)g(let)g Fg(G)f Fj(is)g(a)h(deterministic)g(algorithm)h(which,)g(given)e(a)g(se) -5 b(quenc)g(e)150 1758 y(of)39 b Fg(l)i Fj(bits,)f(outputs)g(a)g(se)-5 b(quenc)g(e)38 b(of)i Fg(m)e Fj(bits.)61 b Fg(G)40 b Fj(is)f(a)g(pseudor)-5 b(andom)42 b(gener)-5 b(ator)41 b(\(PR)n(G\))e(if)g(for)g(al)5 b(l)150 1871 y Fg(P)13 b(P)g(T)g(F)45 b(A)529 1886 y Fd(l)588 1871 y Fj(ther)-5 b(e)34 b(is)e(a)h(ne)-5 b(gligible)32 b(function)h Fg(\027)6 b Fl(\()p Fg(l)r Fl(\))33 b Fj(such)g(that)251 2075 y Fl(Pr[)p Fg(x)25 b Ff( )g(f)p Fl(0)p Fg(;)15 b Fl(1)p Ff(g)787 2037 y Fd(l)816 2075 y Fl(;)g Fg(y)901 2089 y Fc(0)965 2075 y Ff( )26 b Fg(G)p Fl(\()p Fg(x)p Fl(\);)15 b Fg(y)1360 2089 y Fc(1)1425 2075 y Ff( )25 b(f)p Fl(0)p Fg(;)15 b Fl(1)p Ff(g)1761 2037 y Fd(m)1830 2075 y Fl(;)g Fg(b)26 b Ff( )f(f)p Fl(0)p Fg(;)15 b Fl(1)p Ff(g)p Fl(;)g Fg(b)2350 2037 y Fe(0)2401 2075 y Fl(=)25 b Fg(A)2565 2090 y Fd(l)2591 2075 y Fl(\()p Fg(y)2671 2090 y Fd(b)2706 2075 y Fl(\))58 b(:)g Fg(b)2921 2037 y Fe(0)2969 2075 y Fl(=)25 b Fg(b)p Fl(])h Ff(\024)f Fl(1)p Fg(=)p Fl(2)c(+)f Fg(\027)6 b Fl(\()p Fg(l)r Fl(\))150 2288 y Fm(Theorem)34 b(7)46 b Fj(Supp)-5 b(ose)27 b Fg(p)1077 2303 y Fd(k)1171 2288 y Fl(:)52 b Ff(f)p Fl(0)p Fg(;)15 b Fl(1)p Ff(g)1468 2255 y Fd(k)1538 2288 y Ff(7!)25 b(f)p Fl(0)p Fg(;)15 b Fl(1)p Ff(g)1874 2255 y Fd(k)1945 2288 y Fj(is)26 b(a)h(one-way)g(p) -5 b(ermutation.)42 b(Supp)-5 b(ose)28 b(that)f Fg(B)3591 2303 y Fd(k)3634 2288 y Fl(\()p Ff(\016)p Fl(\))150 2400 y Fj(is)33 b(har)-5 b(dc)g(or)g(e)35 b(for)f Fg(p)804 2415 y Fd(k)846 2400 y Fl(\()p Ff(\016)p Fl(\))p Fj(.)43 b(Then)33 b Fg(G)p Fl(\()p Fg(x)p Fl(\))26 b(=)f Fg(p)1630 2419 y Fe(j)p Fd(x)p Fe(j)1713 2400 y Fl(\()p Fg(x)p Fl(\))c Ff(\016)f Fg(B)1990 2419 y Fe(j)p Fd(x)p Fe(j)2073 2400 y Fl(\()p Fg(x)p Fl(\))34 b Fj(is)e(a)h(pseudor)-5 b(andom)36 b(gener)-5 b(ator.)150 2613 y Fm(Pro)s(of)126 b Fl(Supp)s(ose)32 b(that)i Fg(G)g Fl(is)f(not)h(a)h(PR)m(G.)f(Then)f (there)h(is)f(a)i(distinguisher)30 b Fg(A)3050 2628 y Fd(k)3126 2613 y Fl(that)35 b(can)f(distin-)150 2726 y(guish)i Fg(G)p Fl(\()p Fg(x)p Fl(\))j(from)e(a)g(random)g Fg(y)s Fl(.)62 b(W)-8 b(e)39 b(m)m(ust)e(use)g(this)f(distinguisher)e (to)k(tell)f(us)g Fg(B)5 b Fl(\()p Fg(x)p Fl(\))37 b(from)g Fg(p)p Fl(\()p Fg(x)p Fl(\),)150 2839 y(con)m(tradicting)30 b(the)h(supp)s(osition)c(that)k Fg(B)k Fl(is)29 b(hard-core)i(for)f Fg(p)p Fl(.)150 3051 y(De\014ne)h(the)f(reduction)f Fg(R)i Fl(as)g(follo)m(ws:)150 3164 y(On)f(input)e Fg(z)h Fl(=)c Fg(p)p Fl(\()p Fg(x)p Fl(\))377 3277 y(1.)41 b(Run)30 b Fg(A)755 3292 y Fd(k)797 3277 y Fl(\()p Fg(z)25 b Ff(\016)c Fl(0\))26 b(=)f Fg(b)1206 3291 y Fc(0)377 3390 y Fl(2.)41 b(Run)30 b Fg(A)755 3405 y Fd(k)797 3390 y Fl(\()p Fg(z)25 b Ff(\016)c Fl(1\))26 b(=)f Fg(b)1206 3404 y Fc(1)377 3503 y Fl(3.)41 b(In)30 b(the)h(case)g(where)f Fg(b)1253 3517 y Fc(0)1318 3503 y Ff(6)p Fl(=)25 b Fg(b)1453 3517 y Fc(1)1492 3503 y Fl(,)31 b(output)f Fg(a)g Fl(if)f Fg(b)2046 3517 y Fd(a)2113 3503 y Fl(=)c(0.)377 3616 y(4.)41 b(Otherwise)29 b(output)h(the)h(result)e(of)i(a)g(coin)e (\015ip.)150 3829 y(Pr)o([)p Fg(x)p Ff( )d(f)p Fl(0)p Fg(;)15 b Fl(1)p Ff(g)661 3796 y Fd(k)706 3829 y Fl(;)g Fg(b)25 b Ff( )h Fg(R)q Fl(\()p Fg(p)p Fl(\()p Fg(x)p Fl(\)\))g(:)f Fg(b)g Fl(=)g Fg(B)5 b Fl(\()p Fg(x)p Fl(\)])26 b(=)325 3942 y Fg(P)13 b(r)s Fl([)p Fg(x)25 b Ff( )g(f)p Fl(0)p Fg(;)15 b Fl(1)p Ff(g)878 3909 y Fd(k)948 3942 y Fl(:)25 b Fg(A)1066 3957 y Fd(k)1109 3942 y Fl(\()p Fg(p)p Fl(\()p Fg(x)p Fl(\))c Ff(\016)g Fg(B)5 b Fl(\()p Fg(x)p Fl(\)\))26 b(=)f(0)20 b Ff(^)g Fg(A)1966 3957 y Fd(k)2009 3942 y Fl(\()p Fg(p)p Fl(\()p Fg(x)p Fl(\))h Ff(\016)2320 3919 y Fl(\026)2299 3942 y Fg(B)t Fl(\()p Fg(x)p Fl(\))26 b(=)f(1]+)335 4019 y Fc(1)p 335 4034 36 4 v 335 4086 a(2)400 4055 y Ff(\003)c Fg(P)13 b(r)s Fl([)p Fg(x)25 b Ff( )g(f)p Fl(0)p Fg(;)15 b Fl(1)p Ff(g)1019 4022 y Fd(k)1089 4055 y Fl(:)26 b Fg(A)1208 4070 y Fd(k)1250 4055 y Fl(\()p Fg(p)p Fl(\()p Fg(x)p Fl(\))21 b Ff(\016)g Fg(B)5 b Fl(\()p Fg(x)p Fl(\)\))26 b(=)f Fg(A)1961 4070 y Fd(k)2004 4055 y Fl(\()p Fg(p)p Fl(\()p Fg(x)p Fl(\))c Ff(\016)2315 4032 y Fl(\026)2293 4055 y Fg(B)5 b Fl(\()p Fg(x)p Fl(\)\)])150 4267 y(Let)31 b Fg(P)371 4281 y Fd(G)456 4267 y Fl(=)25 b Fg(P)13 b(r)s Fl([)p Fg(x)744 4281 y Fc(0)808 4267 y Ff( )25 b(f)p Fl(0)p Fg(;)15 b Fl(1)p Ff(g)1144 4234 y Fd(k)1214 4267 y Fl(:)26 b Fg(A)1333 4282 y Fd(k)1376 4267 y Fl(\()p Fg(p)p Fl(\()p Fg(x)1544 4281 y Fc(0)1584 4267 y Fl(\))20 b Ff(\016)h Fg(B)5 b Fl(\()p Fg(x)1866 4281 y Fc(0)1905 4267 y Fl(\)\))26 b(=)f(0])31 b(and)313 4380 y Fg(P)371 4394 y Fd(r)434 4380 y Fl(=)25 b Fg(P)13 b(r)s Fl([)p Fg(x)722 4394 y Fc(0)787 4380 y Ff( )25 b(f)p Fl(0)p Fg(;)15 b Fl(1)p Ff(g)1123 4347 y Fd(k)1193 4380 y Fl(:)26 b Fg(A)1312 4395 y Fd(k)1354 4380 y Fl(\()p Fg(p)p Fl(\()p Fg(x)1522 4394 y Fc(0)1562 4380 y Fl(\))21 b Ff(\016)g Fg(r)s Fl(\))k(=)g(0])31 b(where)f Fg(r)i Fl(is)e(a)h(random)e(bit.)150 4593 y(Because)j Fg(A)566 4608 y Fd(k)639 4593 y 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%%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: Letter %%EndSetup %%Page: 1 1 1 0 bop 157 71 3586 4 v 157 574 4 504 v 246 164 a Fo(6.875J/18.425J)38 b(Cryptograph)m(y)d(and)g(Cryptanalysis)500 b Fn(Octob)s(er)30 b(22,)i(2001)1660 371 y Fm(Lecture)38 b(12)246 531 y Fl(Instructor:)43 b(A)n(nna)33 b(Lysyanskaya)1043 b(Scrib)-5 b(e)g(d)33 b(by:)42 b(Jelena)32 b(Sp)-5 b(asojevic)p 3739 574 V 157 577 3586 4 v 150 987 a Fk(1)135 b(Recap)45 b(of)g(de\014nitions)150 1193 y Fj(1.1)112 b(One-w)m(a)m(y)39 b(functions)e(and)i(one-w)m(a)m(y)f(and)g(trap)s(do)s(or)g(p)s(erm)m (utations)150 1365 y Fo(De\014nition)d(1)46 b Fl(A)32 b(set)h(of)g(e\016cient)f(pr)-5 b(ob)g(abilistic)35 b(algorithms)g Fn(\()p Fi(G)p Fn(\()p Fh(\001)p Fn(\))p Fi(;)15 b(S)5 b Fn(\()p Fh(\001)p Fn(\))p Fi(;)15 b(f)2860 1379 y Fg(\001)2887 1365 y Fn(\()p Fh(\001)p Fn(\)\))p Fl(,)33 b(sp)-5 b(e)g(ci\014es)34 b(a)f(family)150 1478 y(of)f(one-way)h(functions)g(\(O)n(WF)f(family\)) h(if)f(for)g(al)5 b(l)33 b(PPTF)f Fh(f)p Fi(A)2361 1493 y Ff(k)2404 1478 y Fh(g)h Fl(ther)-5 b(e)33 b(exists)f(a)h(ne)-5 b(gligible)32 b(function)150 1590 y Fi(\027)6 b Fn(\()p Fi(k)s Fn(\))33 b Fl(such)g(that)321 1795 y Fn(Pr[)p Fi(P)13 b(K)32 b Fh( )25 b Fi(G)p Fn(\(1)891 1757 y Ff(k)935 1795 y Fn(\);)15 b Fi(x)26 b Fh( )f Fi(S)5 b Fn(\()p Fi(P)13 b(K)7 b Fn(\);)15 b Fi(y)29 b Fh( )c Fi(f)1765 1809 y Ff(P)10 b(K)1887 1795 y Fn(\()p Fi(x)p Fn(\);)15 b Fi(x)2101 1757 y Fg(0)2151 1795 y Fh( )25 b Fi(A)2335 1810 y Ff(k)2378 1795 y Fn(\()p Fi(P)13 b(K)q(;)i(y)s Fn(\))27 b(:)e Fi(f)10 b Fn(\()p Fi(x)p Fn(\))26 b(=)f Fi(f)10 b Fn(\()p Fi(x)3203 1757 y Fg(0)3225 1795 y Fn(\)])26 b Fh(\024)f Fi(\027)6 b Fn(\()p Fi(k)s Fn(\))150 1999 y Fl(and)34 b(for)f(al)5 b(l)33 b Fi(P)13 b(K)32 b Fh(2)25 b Fi(G)p Fl(,)33 b(for)g(al)5 b(l)33 b Fi(x)25 b Fh(2)g Fi(S)5 b Fn(\()p Fi(P)13 b(K)7 b Fn(\))p Fl(,)33 b Fi(f)1830 2013 y Ff(P)10 b(K)1985 1999 y Fl(is)32 b(a)h(function.)150 2211 y Fo(De\014nition)i(2)46 b Fl(A)31 b(family)i(of)f(one-way)h (functions)g Fn(\()p Fi(G;)15 b(S;)g(p)p Fn(\))33 b Fl(is)f(a)g(family) h(of)g(one-way)f(p)-5 b(ermutations)150 2324 y(\(O)n(WP)32 b(family\))h(if)f(for)i(al)5 b(l)33 b Fi(P)13 b(K)32 b Fh(2)25 b Fi(G)p Fl(,)32 b Fi(p)1560 2338 y Ff(P)10 b(K)1716 2324 y Fl(is)32 b(a)h(p)-5 b(ermutation.)150 2537 y Fo(De\014nition)35 b(3)46 b Fl(A)35 b(family)h(of)f(one-way)h(p) -5 b(ermutations)38 b Fn(\()p Fi(G;)15 b(S;)g(p)p Fn(\))36 b Fl(is)f(tr)-5 b(ap)g(do)g(or)40 b(\(TDP)34 b(family\))i(if)f(\(1\)) 150 2650 y(the)42 b(output)h(of)f Fi(G)g Fl(c)-5 b(onsists)43 b(of)f(two)h(values,)h(denote)-5 b(d)44 b Fn(\()p Fi(P)13 b(K)q(;)i(S)5 b(K)i Fn(\))p Fl(;)47 b(and)c(\(2\))f(ther)-5 b(e)43 b(is)f(an)g(e\016cient)150 2763 y(algorithm)35 b Fi(D)g Fl(such)e(that)h(for)f(al)5 b(l)33 b Fi(k)s Fl(,)561 2967 y Fn(Pr[\()p Fi(P)13 b(K)q(;)i(S)5 b(K)i Fn(\))27 b Fh( )e Fi(G)p Fn(\(1)1382 2929 y Ff(k)1426 2967 y Fn(\);)15 b Fi(x)26 b Fh( )f Fi(S)5 b Fn(\()p Fi(P)13 b(K)7 b Fn(\);)15 b Fi(x)2073 2929 y Fg(0)2122 2967 y Fh( )25 b Fi(D)s Fn(\()p Fi(S)5 b(K)q(;)15 b(p)2576 2981 y Ff(P)10 b(K)2700 2967 y Fn(\()p Fi(x)p Fn(\))26 b(:)g Fi(x)f Fn(=)g Fi(x)3124 2929 y Fg(0)3147 2967 y Fn(])g(=)g(1)150 3180 y Fo(De\014nition)35 b(4)46 b Fl(A)37 b(family)g(of)h(one-way)g(functions)f(\(r)-5 b(esp.,)39 b(p)-5 b(ermutations\))40 b Fn(\()p Fi(G;)15 b(S;)g(f)10 b Fn(\))39 b Fl(is)e(a)g(one-way)150 3292 y(function)c(\(r)-5 b(esp.,)33 b(p)-5 b(ermutation\))35 b(if)e Fi(G)g Fl(is)f(a)h (deterministic)h(algorithm.)150 3536 y Fj(1.2)112 b(Hardcore)38 b(bit)150 3707 y Fo(De\014nition)d(5)46 b Fl(L)-5 b(et)41 b Fn(\()p Fi(G;)15 b(S;)g(f)10 b Fn(\))43 b Fl(b)-5 b(e)41 b(a)g(O)n(WF)g(family.)68 b(L)-5 b(et)42 b Fi(b)2286 3726 y Fe(\()p Fg(\016)p Fe(\))2380 3707 y Fn(\()p Fh(\016)p Fn(\))h Fl(b)-5 b(e)41 b(a)g(Bo)-5 b(ole)g(an)44 b(function.)67 b Fi(b)41 b Fl(is)h(a)150 3820 y(har)-5 b(dc)g(or)g(e)36 b(bit)d(for)h Fn(\()p Fi(G;)15 b(S;)g(f)10 b Fn(\))34 b Fl(if)e(it)h(is)g(har)-5 b(d)35 b(to)f(c)-5 b(ompute)34 b Fi(b)2150 3834 y Ff(P)10 b(K)2273 3820 y Fn(\()p Fi(x)p Fn(\))34 b Fl(fr)-5 b(om)34 b(the)f(values)g Fi(P)13 b(K)40 b Fl(and)34 b Fi(f)3477 3834 y Ff(P)10 b(K)3599 3820 y Fn(\()p Fi(x)p Fn(\))p Fl(.)150 3933 y(Mor)-5 b(e)33 b(pr)-5 b(e)g(cisely,)34 b(for)f(al)5 b(l)33 b(PPTF)f Fh(f)p Fi(A)1448 3948 y Ff(k)1492 3933 y Fh(g)h Fl(ther)-5 b(e)33 b(exists)g(a)g(ne)-5 b(gligible)32 b(function)h Fi(\027)6 b Fn(\()p Fi(k)s Fn(\))33 b Fl(such)g(that)358 4138 y Fn(Pr)o([)p Fi(P)13 b(K)32 b Fh( )26 b Fi(G)p Fn(\(1)928 4100 y Ff(k)972 4138 y Fn(\);)15 b Fi(x)26 b Fh( )f Fi(S)5 b Fn(\()p Fi(P)13 b(K)7 b Fn(\);)15 b Fi(b)1606 4100 y Fg(0)1655 4138 y Fh( )25 b Fi(A)1839 4153 y Ff(k)1882 4138 y Fn(\()p Fi(P)13 b(K)q(;)i(f)2151 4152 y Ff(P)10 b(K)2275 4138 y Fn(\()p Fi(x)p Fn(\)\))26 b(:)g Fi(b)2548 4100 y Fg(0)2596 4138 y Fn(=)f Fi(b)2731 4152 y Ff(P)10 b(K)2854 4138 y Fn(\()p Fi(x)p Fn(\)])26 b Fh(\024)f Fn(1)p Fi(=)p Fn(2)d(+)e Fi(\027)6 b Fn(\()p Fi(k)s Fn(\))150 4424 y Fk(2)135 b(F)-11 b(actoring)45 b(and)g(RSA-based)f(O)l(WP)h(families)i(and)d(O)l(WFs)150 4627 y Fn(\(This)29 b(is)g(con)m(tin)m(ued)h(from)g(previous)f (lecture\))291 4740 y(Last)d(time)e(w)m(e)i(sa)m(w)f(a)h(few)f (candidates)f(based)h(on)g(DL-t)m(yp)s(e)h(assumption.)37 b(No)m(w)26 b(let)f(us)f(talk)h(ab)s(out)150 4853 y(factoring)31 b(and)e(RSA.)291 4966 y(RSA)h(is)g(a)h(O)m(WP)g(family)-8 b(.)40 b(RSA)31 b(is)e(a)i(rather)g(strong)g(assumption.)40 b(Let)31 b(me)f(describ)s(e)f(the)i(gener-)150 5079 y(ation)f (algorithm.)1867 5416 y(12-1)p eop %%Page: 2 2 2 1 bop 291 159 a Fn(On)24 b(input)e(1)718 126 y Ff(k)785 159 y Fn(generate)k(t)m(w)m(o)g Fi(k)s Fn(-bit)e(primes)e Fi(p)p Fn(,)k Fi(q)h Fn(and)d(set)g Fi(n)h Fh( )g Fi(pq)s Fn(,)h(\014nd)d Fi(l)j Fn(suc)m(h)e(that)h(\()p Fi(l)r(;)15 b(\036)p Fn(\()p Fi(n)p Fn(\)\))26 b(=)f(1.)150 271 y(Output)k Fi(p)519 290 y Fe(\()p Ff(n;l)q Fe(\))662 271 y Fn(\()p Fi(x)p Fn(\))d(=)f Fi(x)958 238 y Ff(l)1014 271 y Fn(mo)s(d)30 b Fi(n)p Fn(.)291 384 y(W)-8 b(e)32 b(kno)m(w)e(ho)m(w)g(to)i(in)m(v)m (ert)e(it)g(from)g(p)s(ervious)e(lectures.)291 497 y(LSB)i(is)g (hardcore)g(for)g(RSA.)g(This)f(result)g(is)g(due)h(to)h(Alexi,)f (Chor,)g(Goldreic)m(h)f(and)h(Sc)m(hnorr.)291 610 y(But)g(some)g(ma)m (y)g(sa)m(y:)41 b(I)29 b(can)h(b)s(eliev)m(e)e(factoring)i(is)f(hard,)g (but)f(I)i(am)f(sk)m(eptical)h(ab)s(out)f(RSA.)g(No)m(w)150 723 y(w)m(e)d(in)m(tro)s(duce)f(an)h(O)m(WP)g(that)g(is)f(based)h(on)f (w)m(eak)m(er)i(assumption,)f(whic)m(h)e(is)h(that)h(it)g(is)f(hard)f (to)j(factor)150 836 y(Blum)k(in)m(tegers.)48 b(Let)33 b(me)g(de\014ne)f(Blum)f(in)m(tegers.)47 b(Generation)33 b(algorithm)f(is)f(as)i(same)g(as)g(for)f(RSA)150 949 y(with)c Fi(l)f Fn(=)e(2)30 b(and)f(with)f(addition)f(w)m(e)j(pic)m(k)f (up)f(n)h(to)h(b)s(e)e(Blum)h(in)m(teger,)h(i.e.)40 b(w)m(e)30 b(pic)m(k)f(up)f(p)h(and)f(q)i(in)e(a)150 1062 y(sp)s(ecial)h(w)m(a)m (y)-8 b(.)150 1274 y Fo(De\014nition)35 b(6)46 b Fi(n)25 b Fn(=)g Fi(pq)35 b Fl(is)d(a)h(Blum)g(inte)-5 b(ger)33 b(i\013)f Fi(p)25 b Fn(=)g Fi(q)j Fn(=)d(3)33 b Fl(mo)-5 b(d)34 b Fn(4)f Fh(^)g Fi(p;)15 b(q)35 b Fl(ar)-5 b(e)34 b(primes.)291 1487 y Fn(No)m(w)d(w)m(e)g(kno)m(w)f(that)h(for)g(a)f (Blum)g(in)m(teger)g Fi(n)p Fn(,)g Fh(\000)p Fn(1)h(is)e(not)i(a)g (square)f(mo)s(dulo)f Fi(n)g Fn(but)3277 1413 y Fd(\000)3329 1451 y Fg(\000)p Fe(1)p 3329 1466 91 4 v 3352 1518 a Ff(n)3429 1413 y Fd(\001)3496 1487 y Fn(=)c(1.)150 1699 y Fo(Theorem)34 b(7)46 b Fl(Squaring)33 b(is)f(a)h(p)-5 b(ermutation)35 b(over)e Fi(QR)q Fn(\()p Fi(n)p Fn(\))p Fl(.)150 1912 y Fo(Pro)s(of)e Fn(of)g(Theorem)f(7:)122 b(It)30 b(is)g(su\016cien)m(t)f(to)j(sho)m(w)e(that)h(it)f(is)f(1)21 b Fh(\000)f Fn(1.)291 2025 y(Claim)29 b(1:)41 b(Let)31 b Fi(a)f Fn(b)s(e)g(in)f Fi(QR)q Fn(\()p Fi(n)p Fn(\).)40 b(Then)30 b(at)h(least)g(one)f(square)g(ro)s(ot)h(of)g Fi(a)f Fn(is)f(a)i(quadratic)f(residue.)291 2138 y(Wh)m(y:)36 b(By)21 b(fact)g(10)g(of)f(lecture)h(3,)i Fi(v)1476 2105 y Fe(2)1541 2138 y Fh(\021)h Fi(a)i Fn(mo)s(d)e Fi(n)c Fn(if)f(and)h(only)f(if)h Fi(v)2534 2105 y Fe(2)2573 2138 y Fi(eq)s(uiv)s(a)27 b Fn(mo)s(d)d Fi(p)c Fn(and)g Fi(v)3343 2105 y Fe(2)3382 2138 y Fi(eq)s(uiv)s(a)27 b Fn(mo)s(d)150 2251 y Fi(q)s Fn(.)53 b(Supp)s(ose)33 b Fi(v)38 b Fn(is)c(not)h(a)g(square)f(mo)s(dulo)f Fi(n)p Fn(.)53 b(Then)34 b(either)g(it)g(is)g(not)h(a)g(square)f(mo)s(dulo)f Fi(p)p Fn(,)j(or)f(it)f(is)150 2364 y(not)k(a)g(square)g(mo)s(dulo)e Fi(q)s Fn(,)k(or)e(b)s(oth.)63 b(Set)38 b Fi(v)1741 2331 y Fg(0)1738 2386 y Ff(p)1815 2364 y Fn(=)g Fi(v)28 b Fn(mo)s(d)d Fi(p)38 b Fn(if)e Fi(v)42 b Fn(is)37 b(a)h(square)f(mo)s (dulo)g Fi(p)p Fn(,)j(otherwise)150 2476 y Fi(v)197 2443 y Fg(0)194 2499 y Ff(p)269 2476 y Fn(=)34 b Fh(\000)p Fi(v)29 b Fn(mo)s(d)24 b Fi(p)36 b Fn(and)f(analogously)h(for)g Fi(v)1672 2443 y Fg(0)1669 2499 y Ff(q)1707 2476 y Fn(.)58 b(Note)38 b(that)e Fi(v)2265 2443 y Fg(0)2262 2499 y Ff(p)2338 2476 y Fn(is)f(suc)m(h)h(that)h(\()p Fi(v)2931 2443 y Fg(0)2928 2499 y Ff(p)2968 2476 y Fn(\))3003 2443 y Fe(2)3078 2476 y Fh(\021)d Fi(v)3230 2443 y Fe(2)3296 2476 y Fn(mo)s(d)24 b Fi(p)p Fn(,)38 b(and)150 2589 y(analogously)29 b(for)h Fi(v)828 2556 y Fg(0)825 2612 y Ff(q)863 2589 y Fn(.)41 b(Y)-8 b(et)31 b(note)g(that)f Fi(v)1540 2556 y Fg(0)1537 2612 y Ff(p)1607 2589 y Fn(is)f(a)h(square)g(mo)s(dulo)e Fi(p)p Fn(.)40 b(Wh)m(y?)h(Let's)31 b(compute)f(its)f(Legen-)150 2739 y(dre)j(sym)m(b)s(ol.)45 b(Supp)s(ose)30 b Fi(v)35 b Fn(is)d(a)g(square)g(mo)s(dulo)f Fi(p)p Fn(,)h(then)g(clearly)2490 2638 y Fd(\020)2554 2684 y Ff(v)2590 2661 y Fc(0)2588 2701 y Fb(p)p 2554 2718 71 4 v 2572 2770 a Ff(p)2635 2638 y Fd(\021)2717 2739 y Fn(=)2816 2638 y Fd(\020)2880 2703 y Ff(v)p 2880 2718 37 4 v 2881 2770 a(p)2927 2638 y Fd(\021)3010 2739 y Fn(=)c(1.)46 b(Supp)s(ose,)31 b Fi(v)36 b Fn(is)150 2876 y(not)c(a)g(square)f(mo)s(dulo)f Fi(p)p Fn(,)i(so)g Fi(v)1267 2843 y Fg(0)1264 2898 y Ff(p)1331 2876 y Fn(=)26 b Fh(\000)p Fi(v)j Fn(mo)s(d)24 b Fi(p)p Fn(.)44 b(Then)31 b(b)m(y)g(F)-8 b(act)34 b(8)e(of)f(lecture)h (3,)g(and)f(recalling)f(that)150 2989 y Fi(p)25 b Fn(=)g(4)p Fi(m)20 b Fn(+)g(3)478 3119 y Fd(\022)555 3176 y Fi(v)602 3143 y Fg(0)599 3199 y Ff(p)p 555 3226 84 4 v 574 3309 a Fi(p)649 3119 y Fd(\023)741 3247 y Fn(=)837 3119 y Fd(\022)914 3185 y Fh(\000)p Fn(1)p 914 3226 117 4 v 949 3309 a Fi(p)1040 3119 y Fd(\023)15 b(\022)1199 3185 y Fi(v)p 1199 3226 48 4 v 1200 3309 a(p)1256 3119 y Fd(\023)1348 3247 y Fn(=)25 b(\()p Fh(\000)p Fn(1\))1630 3209 y Fe(\()p Ff(p)p Fg(\000)p Fe(1\))p Ff(=)p Fe(2)1886 3247 y Fn(\()p Fh(\000)p Fn(1\))i(=)d(\()p Fh(\000)p Fn(1\))2380 3209 y Fe(2)p Ff(m)p Fe(+1)2573 3247 y Fn(\()p Fh(\000)p Fn(1\))j(=)d(\()p Fh(\000)p Fn(1\)\()p Fh(\000)p Fn(1\))j(=)e(1)150 3503 y(Analogously)-8 b(,)40 b Fi(v)740 3470 y Fg(0)737 3525 y Ff(q)812 3503 y Fn(is)d(also)h(a)g(square)g(mo)s(dulo)e Fi(q)s Fn(.)63 b(By)38 b(the)g(Chinese)e(Remainder)h(Theorem)g(\(CR)-8 b(T\),)150 3616 y(there)31 b(exists)f(a)g(unique)f Fi(v)1052 3583 y Fg(0)1101 3616 y Fh(2)c Fi(QR)q Fn(\()p Fi(n)p Fn(\))30 b(suc)m(h)g(that)h Fi(v)1933 3583 y Fg(0)1982 3616 y Fh(\021)25 b Fi(v)2125 3583 y Fg(0)2122 3638 y Ff(p)2187 3616 y Fn(mo)s(d)f Fi(p)30 b Fn(and)g Fi(v)2686 3583 y Fg(0)2735 3616 y Fh(\021)25 b Fi(v)2878 3583 y Fg(0)2875 3638 y Ff(q)2938 3616 y Fn(mo)s(d)f Fi(q)s Fn(.)41 b(And,)30 b(also)g(b)m(y)150 3729 y(CR)-8 b(T,)30 b(\()p Fi(v)478 3696 y Fg(0)502 3729 y Fn(\))537 3696 y Fe(2)602 3729 y Fn(mo)s(d)25 b Fi(n)g Fn(=)g Fi(v)1025 3696 y Fe(2)1090 3729 y Fn(mo)s(d)f Fi(n)h Fn(=)g Fi(a)p Fn(.)291 3842 y(Claim)30 b(2:)42 b(Let)32 b Fi(a)f Fn(b)s(e)g(in)e Fi(QR)q Fn(\()p Fi(n)p Fn(\).)43 b(A)m(t)32 b(most)g(one)f(square)g(ro) s(ot)h(of)f Fi(a)g Fn(can)h(b)s(e)e(a)i(quadratic)e(residue.)150 3954 y(Pro)s(of:)49 b(this)34 b(is)f(just)i(b)m(y)f(coun)m(ting:)49 b(if)34 b(some)h(quadratic)f(residue)g(had)g(more)h(than)f(one)h (square)g(ro)s(ot)150 4067 y(that)d(w)m(as)f(itself)f(a)i(square,)f (then)g(some)g(quadratic)g(residue)f(w)m(ould)g(ha)m(v)m(e)i(no)f(suc)m (h)g(square)g(ro)s(ot,)h(and)150 4180 y(this)d(w)m(ould)g(con)m (tradict)i(claim)f(1.)p 1378 4180 59 59 v 291 4393 a(Wh)m(y)h(is)e (this)g(p)s(erm)m(utation)g(hard)g(to)i(in)m(v)m(ert?)40 b(Let)31 b(me)f(giv)m(e)h(a)f(reduction)f(that)i(uses)e(assumption)150 4506 y(that)i(factoring)g(of)f(Blum)f(in)m(tegers)i(is)e(hard.)150 4619 y Fo(Pro)s(of)35 b Fn(of)e(Hard)h(to)g(In)m(v)m(ert:)126 b(Supp)s(ose)31 b(one)j(could)f(in)m(v)m(ert)h(it,)g(i.e.)50 b(there)34 b(is)f(an)h(adv)m(ersary)g(A)f(that)150 4732 y(on)d(input)e(\()p Fi(n;)15 b(x)p Fn(\))30 b(outputs)g Fi(u)f Fn(suc)m(h)h(that)g Fi(u)1633 4699 y Fe(2)1698 4732 y Fn(=)25 b Fi(x)30 b Fn(mo)s(d)f Fi(n)g Fn(with)2428 4696 y Fe(1)p 2381 4711 129 4 v 2381 4763 a Ff(pol)q(y)2549 4732 y Fn(probabilit)m(y)-8 b(.)39 b(Then)29 b(I)g(claim)g(that)150 4844 y(the)i(follo)m(wing)d(reduction)i(B)g(pro)s(duces)f(adv)m(ersary) i(that)g(factors)g(n.)291 4957 y(B:)g(On)f(input)e(\(n\):)150 5070 y(Run)h(A\(n,)i Fi(y)606 5037 y Fe(2)645 5070 y Fn(\))g(for)f(random)f Fi(y)1231 5037 y Fe(2)1271 5070 y Fn(.)1867 5416 y(12-2)p eop %%Page: 3 3 3 2 bop 150 159 a Fn(If)30 b Fi(u)25 b Fn(=)g Fi(y)33 b Fn(or)e Fi(u)25 b Fn(=)g Fh(\000)p Fi(y)s Fn(,)30 b(rep)s(eat)150 271 y(Otherwise)f(output)h Fi(p)25 b Fn(=)g Fi(g)s(cd)p Fn(\()p Fi(n;)15 b(u)22 b Fh(\000)e Fi(y)s Fn(\))30 b(as)h(a)g(factor)g (of)f(n.)291 384 y(Let)j(me)f(pro)m(v)m(e)h(that)g(this)e(reduction)g (giv)m(es)i(a)g(factor)g(of)f Fi(n)p Fn(.)46 b(So)32 b(in)f(the)i(last)f(case)h(w)m(e)g(get)g(that)g(A)150 497 y(pro)s(duces)c(some)i Fi(u)f Fn(suc)m(h)g(that:)516 701 y Fi(u)568 664 y Fe(2)628 701 y Fh(\000)19 b Fi(y)766 664 y Fe(2)831 701 y Fn(=)25 b(0)31 b(mo)s(d)e Fi(n)c Fh(\()-15 b(\))25 b Fn(\()p Fi(u)c Fh(\000)f Fi(y)s Fn(\)\()p Fi(u)g Fn(+)g Fi(y)s Fn(\))26 b(=)f(0)30 b(mo)s(d)g Fi(n)25 b Fh(\()-15 b(\))25 b Fi(n)p Fh(j)p Fn(\()p Fi(u)20 b Fh(\000)g Fi(y)s Fn(\)\()p Fi(u)h Fn(+)f Fi(y)s Fn(\).)150 906 y(But)33 b(no)m(w)h(b)s(ecause)f Fi(u)d Fh(6)p Fn(=)g Fi(y)25 b Fh(^)d Fi(u)30 b Fh(6)p Fn(=)g Fh(\000)p Fi(y)36 b Fn(it)d(follo)m(ws)f(that)i Fi(n)f Fn(do)s(es)g(not)g(divide)e(b)s (oth)i Fi(u)22 b Fh(\000)g Fi(y)36 b Fn(and)d Fi(u)22 b Fn(+)g Fi(y)s Fn(.)150 1019 y(So)43 b(w)m(e)h(m)m(ust)f(ha)m(v)m(e)h (that)g(one)g(of)f(t)m(w)m(o)i(prime)d(divisors)f(of)i(n,)j(sa)m(y)e Fi(p)p Fn(,)i(divides)41 b Fi(u)29 b Fh(\000)g Fi(y)s Fn(.)79 b(But)43 b(then)150 1132 y Fi(p)25 b Fn(=)g Fi(g)s(cd)p Fn(\()p Fi(n;)15 b(u)22 b Fh(\000)e Fi(y)s Fn(\).)p 893 1132 59 59 v 291 1344 a(Please)28 b(note)h(that)g(w)m(e)f(do)g(not)h (use)e(the)i(fact)g(that)f Fi(n)g Fn(is)f(Blum)g(in)m(teger)i(in)d(pro) s(of)i(of)g(correctness)h(of)150 1457 y(this)j(reduction.)49 b(This)31 b(fact)k(matters)f(for)f(making)f(p)s(erm)m(utation)h(o)m(v)m (er)h Fi(QR)q Fn(\()p Fi(n)p Fn(\).)50 b(The)33 b(theorem)g(that)150 1570 y(computing)c(square)i(ro)s(ots)f(is)g(as)g(hard)g(as)g(factoring) h(is)e(due)h(to)h(Rabin.)150 1782 y Fo(Theorem)j(8)46 b Fl(LSB)23 b(of)h(this)g(p)-5 b(ermutation)25 b(is)e(har)-5 b(dc)g(or)g(e.)42 b(\(Due)23 b(to)h(A)n(lexi,)g(Chor,)i(Goldr)-5 b(eich,)28 b(Schnorr.\))291 1995 y Fn(This)h(is)g(what)i(Blum,)e(Blum,) h(Sh)m(ub)f(PR)m(G)i(\(the)g(b)s(est)f(PR)m(G\))h(is)f(based)g(on.)41 b(The)30 b(pro)s(of)g(based)g(on)150 2108 y(hardness)d(of)i(factoring)g (is)e(di\016cult,)g(so)i(here's)f(a)h(simple)e(pro)s(of)h(based)g(on)g (QRA.)h(It)f(is)g(due)g(to)h(Blum,)150 2221 y(Blum)g(and)h(Sh)m(ub.)150 2334 y Fo(Pro)s(of)h Fn(of)g(Theorem)f(8:)122 b(Reduction)29 b(follo)m(ws.)291 2447 y(W)-8 b(e)32 b(kno)m(w)f(that)g(under)e(QRA)i (it)f(is)g(hard)g(to)h(sa)m(y)g(on)g(input)e(\()p Fi(n;)15 b(a)p Fn(\))31 b(if)f Fi(a)c Fh(2)f Fi(QR)q Fn(\()p Fi(n)p Fn(\).)42 b(Supp)s(ose)29 b(no)m(w)150 2560 y(that)i(w)m(e)g(ha)m(v)m (e)g(some)g(adv)m(ersary)g Fi(A)p Fn(\()p Fi(n;)15 b(x)1581 2527 y Fe(2)1621 2560 y Fn(\))30 b(that)h(can)g(tell)f(us)f(the)i(LSB)f (of)g(x)h(suc)m(h)f(that)1507 2764 y Fi(x)25 b Fh(2)g Fi(QR)q Fn(\()p Fi(n)p Fn(\))20 b Fh(^)g Fi(x)2090 2726 y Fe(2)2154 2764 y Fn(=)25 b Fi(a)2298 2726 y Fe(2)2338 2764 y Fn(,)150 2968 y(with)i(probabilit)m(y)828 2932 y Fe(1)p 828 2947 36 4 v 828 2999 a(2)889 2968 y Fn(+)16 b Fi(\017)p Fn(.)40 b(Then)27 b(the)i(follo)m(wing)d(adv)m(ersary)j Fi(B)k Fn(solv)m(es)28 b(QR)g(problem)f(with)g(probabilt)m(y)150 3081 y(non-negligibly)g(b)s(etter)k(than)1231 3045 y Fe(1)p 1231 3060 V 1231 3112 a(2)1276 3081 y Fn(:)150 3194 y(B:)g(On)e(input)g(\()p Fi(n;)15 b(a)p Fn(\),)933 3120 y Fd(\000)987 3158 y Ff(a)p 985 3173 43 4 v 985 3225 a(n)1038 3120 y Fd(\001)1105 3194 y Fn(=)24 b(1:)150 3307 y(Run)29 b Fi(A)p Fn(\()p Fi(n;)15 b(a)594 3274 y Fe(2)634 3307 y Fn(\))26 b Fh(!)f Fi(b)150 3420 y Fn(If)30 b Fi(b)25 b Fn(=)g Fi(LS)5 b(B)g Fn(\()p Fi(a)p Fn(\))31 b(output)f(\\SQUARE")g(otherwise)g(guess)g(it)g(is)g(not)g(a)h(square) 291 3645 y(Observ)m(e)g(no)m(w)f(that)i(if)d Fi(x)d Fh(2)f Fi(Z)1344 3612 y Fg(\003)1337 3668 y Ff(n)1384 3645 y Fn(,)31 b Fi(n)g Fn(is)e(Blum)h(in)m(teger)h(then)f Fi(LS)5 b(B)g Fn(\()p Fi(x)p Fn(\))26 b Fh(6)p Fn(=)g Fi(LS)5 b(B)g Fn(\()p Fh(\000)p Fi(x)p Fn(\).)41 b(If)30 b(n)g(is)g(Blum)150 3758 y(and)g(Jacobi)g(sym)m(b)s(ol)f(of)i(x)f(o)m(v)m(er)h(n)f(is)g(1)g (then)h(exactly)g(one)f(of)h(x,)f(-x)h(is)e(in)g(QR\(n\).)291 3871 y(No)m(w)21 b(see)g(that)g Fi(p)858 3838 y Fg(\000)p Fe(1)858 3894 y Ff(n)952 3871 y Fn(\()p Fi(a)1035 3838 y Fe(2)1075 3871 y Fn(\))k(=)g Fi(a)c Fn(if)e Fi(a)25 b Fh(2)g Fi(QR)q Fn(\()p Fi(n)p Fn(\))20 b(and)g(is)f Fh(\000)p Fi(a)h Fn(otherwise)g(and)f(also)i(that)f Fi(LS)5 b(B)g Fn(\()p Fi(p)3404 3838 y Fg(\000)p Fe(1)3404 3894 y Ff(n)3498 3871 y Fn(\()p Fi(a)3581 3838 y Fe(2)3621 3871 y Fn(\))26 b(=)150 3984 y Fi(LS)5 b(B)g Fn(\()p Fi(a)p Fn(\))30 b(if)g Fi(a)25 b Fh(2)g Fi(QR)q Fn(\()p Fi(n)p Fn(\))30 b(and)g Fi(LS)5 b(B)g Fn(\()p Fh(\000)p Fi(a)p Fn(\))30 b(otherwise)g(whic)m(h)f(is)g(equal)h(to)h(1)21 b Fh(\000)f Fi(LS)5 b(B)g Fn(\()p Fi(a)p Fn(\).)p 3273 3984 59 59 v 291 4197 a(Actually)-8 b(,)26 b(w)m(e)g(can)g(sho)m(w)f (that)h(squaring)e(is)g(a)i(O)m(WF)g(using)e(a)h(w)m(eak)m(er)i (assumption,)e(whic)m(h)f(is)g(that)150 4310 y(factoring)29 b(cannot)h(b)s(e)e(done)h(e\016cien)m(tly)g(\(but)f(not)h(necessarily)f (for)h(Blum)f(in)m(tegers\).)41 b(More)29 b(precisely)150 4423 y(this)g(assumption)g(lo)s(oks)h(lik)m(e:)483 4664 y Fh(8)p Fi(A)602 4679 y Ff(k)644 4664 y Fh(9)g Fn(p)s(olynomial)e Fi(p)p Fn(\()p Fi(k)s Fn(\))i(s.t.)42 b Fi(P)13 b(r)s Fn([)p Fi(n)24 b Fh( )h(f)p Fn(0)p Fi(;)15 b Fn(1)p Fh(g)2109 4626 y Ff(k)2154 4664 y Fn(;)g Fi(u)26 b Fh(6)p Fn(=)p Fh( )f Fi(A)2527 4679 y Ff(k)2570 4664 y Fn(\()p Fi(n)p Fn(\))g(:)h Fi(u)p Fh(j)p Fi(n)p Fn(])f Fh(\024)g Fn(1)c Fh(\000)3276 4602 y Fn(1)p 3216 4643 167 4 v 3216 4726 a Fi(p)p Fn(\()p Fi(k)s Fn(\))3392 4664 y(.)150 4874 y(This)i(assumption)f(is)i(equiv)-5 b(alen)m(t)24 b(that)h(there)f(are) h(alw)m(a)m(ys)g(some)g(in)m(tegers)f(that)h(w)m(e)g(cannot)g(factor)h (and)150 4987 y(it)34 b(is)f(strictly)g(w)m(eak)m(er)i(than)f(an)m(y)h (of)f(the)g(ab)s(o)m(v)m(e)h(men)m(tioned.)52 b(But)34 b(note)h(that)g(if)e(it)g(is)g Fl(p)-5 b(ermutations)150 5100 y Fn(that)31 b(w)m(e)g(w)m(an)m(t,)g(this)f(is)f(insu\016cien)m (t.)1867 5416 y(12-3)p eop %%Page: 4 4 4 3 bop 150 159 a Fk(3)135 b(Other)45 b(in)l(teresting)i(facts)e(with)h (no)e(pro)t(ofs)150 365 y Fj(3.1)112 b(W)-9 b(eak)38 b(one-w)m(a)m(y)g(functions)150 536 y Fn(Some)28 b(ma)m(y)g(still)e(b)s (e)h(sk)m(eptical,)h(and)f(sa)m(y:)40 b(I)28 b(am)f(willing)e(to)j(b)s (eliev)m(e)f(that)h(y)m(ou)g(cannot)g(alw)m(a)m(ys)h(factor,)150 649 y(I)24 b(am)g(ev)m(en)h(willing)c(to)k(b)s(eliev)m(e)e(that)i(for)f (an)m(y)h(pro)s(cedure,)f(there)h(is)e(a)i(p)s(olynomial)c(fraction)j (of)h(in)m(tegers)150 762 y(that)32 b(this)f(pro)s(cedure)f(cannot)j (factor.)45 b(But)32 b(I)g(don't)f(b)s(eliev)m(e)g(they)h(happ)s(en)e (to)j(ha)m(v)m(e)g(an)m(y)f(particular)150 875 y(structure)e(lik)m(e)g (Blum)f(in)m(tegers.)291 988 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Fr(can)g(distinguish)d Fm(p)p Fr(\()p Fm(k)s Fr(\))j(random)g (strings)f(from)g Fm(p)p Fr(\()p Fm(k)s Fr(\))i(pseudorandom)e (strings,)i(then)e(it)377 271 y(can)31 b(tell)f(if)f(a)i(giv)m(en)f (string)f(is)h(random)f(or)i(pseudorandom.)291 459 y(Claim)e(2)i(can)f (b)s(e)g(sho)m(wn)g(with)f(a)i(h)m(ybrid)d(argumen)m(t)j(and)e(will)f (not)j(b)s(e)e(pro)m(v)m(ed)i(here.)291 572 y(Claim)e(1)i(can)f(b)s(e)g (sho)m(w)g(b)m(y)g(creating)h(a)g(reduction.)291 685 y(The)e(goal)g(is)f(to)i(create)h(a)e(reduction)f Fm(R)q Fr(,)h(whic)m(h)f(giv)m(en)h(strings)f Fm(S)2579 699 y Fj(1)2618 685 y Fm(;)15 b(:)g(:)g(:)i(S)2836 703 y Fk(p)p Fj(\()p Fk(k)r Fj(\))2968 685 y Fr(,)30 b(pro)s(duces)e Fm(F)3475 652 y Fk(i)3462 712 y(S)3542 685 y Fr(if)g(the)150 798 y Fm(S)206 812 y Fk(j)242 798 y Fr('s)j(are)f(random)g(or)g(pro)s (duces)f Fm(F)1383 765 y Fk(i)p Fj(+1)1370 820 y Fk(s)1532 798 y Fr(if)h(they)g(are)h(pseudorandom.)291 911 y(On)f(input)e Fm(S)741 925 y Fj(1)780 911 y Fm(;)15 b(:)g(:)g(:)i(S)998 929 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Fr(and)e Fl(G)3136 4505 y Fi(1)3176 4491 y Fr(.)57 b Fl(G)3329 4505 y Fi(0)3369 4491 y Fr(\()p Fl(x)p Fr(\))36 b(is)f(the)150 4604 y(\014rst)30 b Fl(k)j Fr(bits)c(of)i Fl(G)p Fr(\()p Fl(x)p Fr(\))g(and)f Fl(G)1171 4618 y Fi(1)1211 4604 y Fr(\()p Fl(x)p Fr(\))h(is)e(the)i (last)f Fl(k)k Fr(bits)29 b(of)i Fl(G)p Fr(\()p Fl(x)p Fr(\).)291 4717 y(Let)h Fl(S)g Fm( )26 b(f)p Fr(0)p Fl(;)15 b Fr(1)p Fm(g)880 4684 y Fj(k)957 4717 y Fr(b)s(e)30 b(the)i(random)e(seed)i(and)e Fl(x)d Fr(=)g Fl(x)2178 4731 y Fi(1)2238 4717 y Fm(\016)21 b Fl(x)2356 4731 y Fi(2)2417 4717 y Fm(\016)g Fl(:)15 b(:)g(:)22 b Fm(\016)f Fl(x)2728 4732 y Fj(k)2802 4717 y Fr(b)s(e)31 b(a)g Fl(k)s Fr(-bit)g(long)g(n)m(um)m(b)s(er.)1867 5416 y(14-1)p eop %%Page: 2 2 2 1 bop 150 159 a Fr(Then)29 b Fl(F)445 173 y Fj(s)483 159 y Fr(\()p Fl(x)p Fr(\))d Fg(,)d Fl(l)753 173 y Fj(x)828 159 y Fr(where)1306 360 y Fl(l)1333 374 y Fj(x)1463 360 y Fr(=)85 b Fl(l)1646 374 y Fj(x)1686 383 y Ff(1)1721 374 y Fk(\016)p Fj(x)1796 383 y Ff(2)1830 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Fl(S)5 b Fr(\))1311 1150 y Fl(l)1338 1164 y Fi(1)1463 1150 y Fr(=)85 b Fl(G)1690 1164 y Fi(1)1730 1150 y Fr(\()p Fl(S)5 b Fr(\))291 1347 y(W)-8 b(e)37 b(pro)m(v)m(ed)g(that)g(this)e (giv)m(es)h(us)g(a)g(PRF)g(using)f(a)i(h)m(ybrid)d(argumen)m(t)i(whic)m (h)f(in)m(v)m(olv)m(ed)h(v)m(ertical)150 1460 y(h)m(ybrids.)62 b(In)37 b(fact,)k(the)e(same)f(pro)s(of)g(sho)m(ws)g(that)g(if)f Fl(x)i Fr(had)e(a)i(p)s(olynomial)c(length)j Fl(p)p Fr(\()p Fl(k)s Fr(\),)j(then)c(w)m(e)150 1573 y(w)m(ould)29 b(still)f(get)k(a)f (PRF)f Fl(F)38 b Fr(:)26 b Fm(f)p Fr(0)p Fl(;)15 b Fr(1)p Fm(g)1405 1540 y Fj(p)p Fi(\()p Fj(k)r Fi(\))1566 1573 y Fm(!)25 b(f)p Fr(0)p Fl(;)15 b Fr(1)p Fm(g)1902 1540 y Fj(k)1947 1573 y Fr(.)150 1859 y Fo(2)135 b(The)29 b(Decisional)j(Di\016e-Hellman)g(assumption)e(and)g(applications)150 2065 y Fn(2.1)112 b(The)38 b(Decisional)e(Di\016e-Hellman)f(assumption) 150 2237 y Fr(Let)k Fl(P)13 b Fr(\(1)472 2204 y Fj(k)515 2237 y Fr(\))38 b(b)s(e)g(an)g(algorithm)f(that)h(generates)i(a)e (prime)e Fl(p)i Fr(of)g(length)g Fl(k)j Fr(and)c(an)h(elemen)m(t)h Fl(g)i Fm(2)d Fd(Z)3715 2204 y Fk(\003)3715 2260 y Fj(p)150 2350 y Fr(suc)m(h)c(that)g Fl(g)k Fr(has)c(large)g(m)m(ultiplicativ)m (e)d(order)j Fl(q)s Fr(.)51 b(\(Let's)35 b(sa)m(y)g(that)f Fl(q)h Fr(=)c(\012\()2873 2293 y Fm(p)p 2948 2293 46 4 v 2948 2350 a Fl(p)p Fr(\).\))52 b(Since)33 b(the)h(order)150 2463 y(divides)g Fl(p)24 b Fm(\000)g Fr(1)37 b(b)m(y)g(F)-8 b(ermat's)37 b(Little)f(theorem,)j(w)m(e)e(can)g(write)f Fl(p)24 b Fm(\000)g Fr(1)36 b(=)f Fl(aq)k Fr(or)e Fl(p)e Fr(=)g Fl(aq)27 b Fr(+)d(1.)60 b(Then)150 2576 y(sa)m(ying)33 b Fl(q)j Fr(is)c(large)h(is)g(the)g(same)h(as)f(sa)m(ying)g Fl(a)g Fr(is)f(small)g(\(i.e.)49 b Fl(O)s Fr(\()2430 2519 y Fm(p)p 2506 2519 V 57 x Fl(p)p Fr(\).)g(Let)34 b Fl(G)c Fr(=)p Fl(<)g(g)j(>)p Fr(.)49 b(Let's)34 b(write)150 2689 y(\()p Fl(p;)15 b(q)s(;)g(g)s Fr(\))27 b Fm( )e Fl(P)13 b Fr(\(1)730 2656 y Fj(k)774 2689 y Fr(\).)291 2914 y(The)34 b Fs(decisional)40 b(Di\016e-Hellman)33 b Fr(\(DDH\))j(assumption)d(is)h(that,)i(for)e(all)f(adv)m(ersaries)h Fm(f)p Fl(A)3636 2929 y Fj(k)3679 2914 y Fm(g)p Fr(,)150 3027 y(there)d(exists)f(a)g(negligible)e(function)h Fl(\027)6 b Fr(\()p Fl(k)s Fr(\))31 b(suc)m(h)f(that)150 3240 y(Pr)o([\()p Fl(p;)15 b(q)s(;)g(g)s Fr(\))28 b Fm( )d Fl(P)13 b Fr(\(1)853 3207 y Fj(k)896 3240 y Fr(\);)i Fl(x)26 b Fm( )f Fd(Z)1230 3254 y Fj(q)1264 3240 y Fr(;)273 3353 y Fl(y)j Fm( )d Fl(Z)524 3367 y Fj(q)562 3353 y Fr(;)15 b Fl(z)644 3367 y Fi(0)709 3353 y Fr(=)25 b Fl(xy)j Fr(mo)s(d)d Fl(q)s Fr(;)15 b Fl(z)1256 3367 y Fi(1)1321 3353 y Fm( )25 b Fd(Z)1502 3367 y Fj(q)1536 3353 y Fr(;)15 b Fl(b)26 b Fm( )f(f)p Fr(0)p Fl(;)15 b Fr(1)p Fm(g)p Fr(;)273 3466 y Fl(b)312 3433 y Fk(0)360 3466 y Fm( )25 b Fl(A)544 3481 y Fj(k)587 3466 y Fr(\()p Fl(p;)15 b(q)s(;)g(g)s(;)g(g)924 3433 y Fj(x)971 3466 y Fl(;)g(g)1057 3433 y Fj(y)1099 3466 y Fl(;)g(g)1185 3433 y Fj(z)1218 3445 y Fh(b)1254 3466 y Fr(\))26 b(:)g Fl(b)f Fr(=)g Fl(b)1565 3433 y Fk(0)1588 3466 y Fr(])h Fm(\024)f Fr(1)p Fl(=)p Fr(2)c(+)f Fl(\027)6 b Fr(\()p Fl(k)s Fr(\))291 3800 y(In)33 b(con)m(trast,)k(the) d Fs(Di\016e-Hellman)k(assumption)c Fr(\(DH\))h(is)e(that,)j(if)d Fl(P)2925 3767 y Fk(0)2948 3800 y Fr(\(1)3028 3767 y Fj(k)3072 3800 y Fr(\))h(is)f(an)h(algorithm)150 3912 y(that)c(generates)i(a)e(prime)e Fl(p)i Fr(and)f(an)h(elemen)m(t)g Fl(g)f Fm(2)c Fd(Z)2014 3880 y Fk(\003)2014 3935 y Fj(p)2049 3912 y Fr(,)30 b(then)f(for)h(all)f(adv)m(ersaries)g Fm(f)p Fl(A)3153 3927 y Fj(k)3197 3912 y Fm(g)p Fr(,)h(there)g(exists) 150 4025 y(a)h(negligible)d(function)h Fl(\027)6 b Fr(\()p Fl(k)s Fr(\))31 b(suc)m(h)f(that,)150 4238 y(Pr)o([\()p Fl(p;)15 b(g)s Fr(\))27 b Fm( )f Fl(P)689 4205 y Fk(0)712 4238 y Fr(\(1)792 4205 y Fj(k)835 4238 y Fr(\);)15 b Fl(x)26 b Fm( )f Fd(Z)1169 4252 y Fj(p)1205 4238 y Fr(;)15 b Fl(y)28 b Fm( )d Fl(Z)1496 4252 y Fj(p)1536 4238 y Fr(;)273 4351 y Fl(Z)32 b Fm( )25 b Fl(A)551 4366 y Fj(k)594 4351 y Fr(\()p Fl(p;)15 b(g)s(;)g(g)847 4318 y Fj(x)893 4351 y Fl(;)g(g)979 4318 y Fj(y)1021 4351 y Fr(\))26 b(:)f Fl(Z)32 b Fr(=)25 b Fl(g)1368 4318 y Fj(xy)1450 4351 y Fr(])g Fm(\024)g Fl(\027)6 b Fr(\()p Fl(k)s Fr(\))291 4676 y(That)30 b(is,)g(it's)g(hard)f(to)j(compute)e Fl(g)1525 4643 y Fj(xy)1637 4676 y Fr(giv)m(en)h Fl(p;)15 b(g)s(;)g(g)2093 4643 y Fj(x)2138 4676 y Fl(;)g(g)2224 4643 y Fj(y)2267 4676 y Fr(.)291 4902 y(Finally)-8 b(,)32 b(the)h Fs(discrete)38 b(log)33 b Fr(\(DL\))h(assumption)d(is)h(that)i(giv)m(en)e Fl(p;)15 b(g)33 b Fm(2)c Fd(Z)2901 4869 y Fk(\003)2901 4925 y Fj(p)2969 4902 y Fr(a)k(generator,)i(and)e Fl(g)3706 4869 y Fj(x)150 5015 y Fr(it's)d(hard)f(to)j(compute)e Fl(x)25 b Fm(2)g Fd(Z)1222 5029 y Fj(p)1258 5015 y Fr(.)40 b(That)30 b(is,)1867 5416 y(14-2)p eop %%Page: 3 3 3 2 bop 150 159 a Fm(8)30 b Fr(PPTF)g Fm(f)p Fl(A)623 174 y Fj(k)666 159 y Fm(g)p Fr(,)h Fm(9)15 b Fl(\027)6 b Fr(\()p Fl(k)s Fr(\))31 b(negligible,)150 271 y(Pr)o([\()p Fl(p;)15 b(g)s Fr(\))27 b Fm( )f Fl(P)689 239 y Fk(0)712 271 y Fr(\(1)792 239 y Fj(k)835 271 y Fr(\);)15 b Fl(x)26 b Fm( )f Fd(Z)1169 285 y Fj(p)1205 271 y Fr(;)273 384 y Fl(z)k Fm( )c Fl(A)528 399 y Fj(k)571 384 y Fr(\()p Fl(p;)15 b(g)s(;)g(g)824 351 y Fj(x)870 384 y Fr(\))26 b(:)f Fl(z)30 b Fr(=)25 b Fl(x)p Fr(])g Fm(\024)g Fl(\027)6 b Fr(\()p Fl(k)s Fr(\))291 710 y(If)28 b(an)h(adv)m(ersary)g(can)g (break)g(the)g(Di\016e-Hellman)e(assumption,)h(then)g(it)h(can)g (compute)g Fl(g)3463 677 y Fj(xy)3573 710 y Fr(with)150 823 y(non-negligible)d(probabilit)m(y)-8 b(,)27 b(and)g(so)i(it)f(can)g (b)s(e)g(used)f(to)i(mak)m(e)g(an)g(adv)m(ersary)f(whic)m(h)f(can)i (break)f(the)150 936 y(decisional)f(Di\016e-Hellman)h(assumption.)39 b(In)28 b(fact,)j(using)c(similar)g(reasoning,)i(w)m(e)g(get)h(the)f (follo)m(wing)150 1049 y(string)g(of)i(implications)c(:)41 b(DDH)32 b Fm(\))e Fr(DH)h Fm(\))f Fr(DL)g Fm(\))h Fr(\(P)f Fm(6)p Fr(=)g(NP\).)291 1161 y(Ho)m(w)e(w)m(ould)e(w)m(e)h(use)g(P)g(=) f(NP)h(to)h(break)f(DL?)g(W)-8 b(e)28 b(w)m(an)m(t)g(to)g(answ)m(er)f (the)g(question)f(\\What)j(is)d Fl(x)p Fr(?")150 1274 y(if)f(w)m(e're)i(giv)m(en)g Fl(y)h Fr(=)d Fl(g)909 1241 y Fj(x)953 1274 y Fr(.)40 b(So)26 b(w)m(e)h(try)f(to)h(guess)g Fl(x)f Fr(bit)m(wise.)38 b(W)-8 b(e)27 b(ask)g(\\Is)f(there)h(an)f Fl(x)g Fr(=)e Fl(x)3199 1288 y Fi(1)3251 1274 y Fm(\016)12 b Fl(:::)g Fm(\016)g Fl(x)3504 1289 y Fj(k)3575 1274 y Fr(suc)m(h)150 1387 y(that)29 b(it's)g(\014rst)f(bit)g Fl(x)874 1401 y Fi(1)938 1387 y Fr(=)d(1)k(and)f Fl(g)1329 1354 y Fj(x)1399 1387 y Fr(=)d Fl(y)s Fr(.)40 b(This)27 b(problem)g(is)h(in)g(NP)-8 b(,)29 b(and)f(if)g(P)g(=)h(NP)-8 b(,)29 b(it's)g(p)s(olynomial)150 1500 y(time)f(decidable.)39 b(So)28 b(w)m(e)h(can)g(\014nd)e(out)i(whether)f(or)g(not)h(the)g (\014rst)f(bit)f(is)h(1.)40 b(And)28 b(so)h(on)f(to)h(guess)g(the)150 1613 y(other)j(bits)f(of)h Fl(x)p Fr(.)45 b(Since)30 b Fl(x)i Fr(has)g(p)s(olynomially)c(man)m(y)k(bits,)f(w)m(e)i(can)f (\014nd)e(the)i(discrete)f(logarithm)g(in)150 1726 y(p)s(olynomial)g (time.)51 b(\(In)33 b(fact,)j(since)d(w)m(e)i(only)e(need)g(the)h(bits) f(with)g(high)f(probabilit)m(y)-8 b(,)33 b(all)g(w)m(e)h(really)150 1839 y(need)c(in)f(fact)i(is)f(BPP)g(=)g(NP\).)150 2082 y Fn(2.2)112 b(The)38 b(PR)m(G)f(construction)150 2254 y Fr(W)-8 b(e)32 b(w)m(an)m(t)g(to)g(\014rst)f(construct)h(a)f(PR)m(G)h (using)e(the)h(DDH)h(assumption.)42 b(Supp)s(ose)29 b Fl(p;)15 b(q)s(;)g(g)36 b Fr(are)31 b(\014xed)g(as)150 2367 y(in)e(the)i(DDH)g(construction.)40 b(As)31 b(a)f(\014rst)g(try)-8 b(,)31 b(let's)f(tak)m(e)1474 2684 y Fl(G)p Fr(\()p Fl(x;)15 b(y)s Fr(\))26 b(=)f(\()p Fl(g)1958 2647 y Fj(x)2023 2684 y Fm(\016)c Fl(g)2135 2647 y Fj(y)2197 2684 y Fm(\016)g Fl(g)2309 2647 y Fj(xy)2391 2684 y Fr(\))150 2888 y(The)32 b(in)m(tuition)e(is)h(that)i(if)f(one)g(can't)h(tell)f Fl(g)1692 2855 y Fj(xy)1806 2888 y Fr(from)g(a)g(random)g Fl(g)2482 2855 y Fj(z)2555 2888 y Fr(\(under)f(DDH\),)j(then)e(this)f (could)150 3001 y(b)s(e)c(pseudorandom.)38 b(Ho)m(w)m(ev)m(er,)31 b(this)c(is)f(not)i(a)h(PR)m(G)f(in)e(the)i(usual)f(sense.)39 b(Recall)28 b(that)g(a)g(PR)m(G)g(tak)m(es)150 3114 y Fm(f)p Fr(0)p Fl(;)15 b Fr(1)p Fm(g)370 3081 y Fj(l)q Fi(\()p Fj(k)r Fi(\))528 3114 y Fr(in)m(to)37 b Fm(f)p Fr(0)p Fl(;)15 b Fr(1)p Fm(g)939 3081 y Fj(m)p Fi(\()p Fj(k)r Fi(\))1138 3114 y Fr(and)36 b(more)g(or)h(less)f(uniformly)e (distributes)g(it.)59 b(Ho)m(w)m(ev)m(er,)40 b(here)d Fl(g)3518 3081 y Fj(x)3598 3114 y Fl(<)e(p)150 3227 y Fr(and)28 b(so)g(the)h(\014rst)e(part)i(of)f(the)g(output)g(only)g(co)m (v)m(ers)i(a)e(fraction)h Fl(p=)p Fr(2)2532 3194 y Fj(k)2603 3227 y Fr(of)g(the)f(whole)g(space)g Fm(f)p Fr(0)p Fl(;)15 b Fr(1)p Fm(g)3575 3194 y Fj(k)3620 3227 y Fr(,)29 b(in)150 3340 y(fact)i(a)g(particular)d(p)s(ortion)h(of)h(the)h(space,)g(namely) e(those)i(n)m(um)m(b)s(ers)e Fl(y)k Fr(whic)m(h)28 b(are)j(less)e(than) h Fl(p)p Fr(.)41 b(This)150 3453 y(fact)27 b(migh)m(t)f(enable)g(an)g (adv)m(ersary)g(to)h(distinguish)c(b)s(et)m(w)m(een)j(the)h(output)f (of)g Fl(G)h Fr(from)e(a)i(truly)e(random)150 3566 y(distribution.)291 3679 y(W)-8 b(e)34 b(could)d(argue)h(that)h(if)e Fl(p)h Fr(is)g(v)m(ery)g(close)h(to)g(2)1978 3646 y Fj(k)2053 3679 y Fr(then)f(the)g(ab)s(o)m(v)m(e)i(con)m(tradiction)e(do)s(esn't)g (hold.)150 3792 y(Ho)m(w)m(ev)m(er,)j(there's)d(another)h(problem)d (with)h(this)g(generator.)48 b(Since)31 b Fl(g)k Fr(is)d(of)g(order)g Fl(q)s Fr(,)g Fl(g)3267 3759 y Fj(x)3344 3792 y Fr(is)f(of)h(order)150 3904 y(dividing)26 b Fl(q)s Fr(.)41 b(Ho)m(w)m(ev)m(er,)32 b(it's)d(easy)i(to)f(sho)m(w)g(that)g(a)g(random)f(elemen)m(t)i(of)f Fd(Z)2795 3872 y Fk(\003)2795 3927 y Fj(p)2860 3904 y Fr(has)f(order)g Fl(\036)p Fr(\()p Fl(p)p Fr(\))d(=)f Fl(p)19 b Fm(\000)g Fr(1)150 4017 y(\(the)37 b(Euler)e(phi)g (function\))g(with)h(go)s(o)s(d)g(probabilit)m(y)-8 b(.)57 b(In)36 b(fact,)j(the)e(n)m(um)m(b)s(er)e(of)i(elemen)m(ts)f(of)h Fd(Z)3605 4031 y Fj(p)3677 4017 y Fr(of)150 4130 y(order)26 b Fl(\036)p Fr(\()p Fl(p)p Fr(\))h(is)e Fl(\036)p Fr(\()p 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Fp(or)i(from)g Fj(X)551 2750 y Fg(2)591 2736 y Fp(,)h(no)f(matter)h(ho)m(w)f(man)m(y)h(samples)e(he)i(obtains.) 291 2849 y(The)36 b(formal)g(de\014nition)e(is)h(lik)m(e)h(this:)52 b(t)m(w)m(o)37 b(distributions)c Fj(X)2471 2863 y Fg(1)2547 2849 y Fp(and)j Fj(X)2805 2863 y Fg(2)2881 2849 y Fp(are)h(p)s (erfectly)e(indistin-)150 2962 y(guishable)28 b(if)i(they)g(are)h(iden) m(tical.)150 3175 y Fq(De\014nition)k(2)46 b Fj(X)40 b Fn(and)33 b Fj(Y)53 b Fn(ar)-5 b(e)33 b(p)-5 b(erfe)g(ctly)34 b(indistinguishable)g(if)e Fk(8)p Fj(a)25 b Fk(2)g Fj(D)18 b Fp(Pr)o([)p Fj(X)33 b Fp(=)25 b Fj(a)p Fp(])h(=)f(Pr)o([)p Fj(Y)46 b Fp(=)25 b Fj(a)p Fp(])p Fn(.)150 3387 y Fq(Theorem)34 b(3)46 b Fn(L)-5 b(et)33 b Fj(Z)911 3401 y Fg(0)983 3387 y Fn(and)g Fj(Z)1221 3401 y Fg(1)1293 3387 y Fn(b)-5 b(e)33 b(p)-5 b(erfe)g(ctly)34 b(indistinguishable.)42 b(Then,)33 b Fk(8)p Fj(A)150 3500 y Fp(Pr)o([)p Fj(b)26 b Fk( )f(f)p Fp(0)p Fj(;)15 b Fp(1)p Fk(g)p Fp(;)g Fj(a)28 b Fk( )d Fj(Z)967 3515 y Fh(b)1002 3500 y Fp(;)15 b Fj(b)1081 3467 y Fi(0)1130 3500 y Fp(=)25 b Fj(A)p Fp(\()p Fj(a)p Fp(\))h(:)f Fj(b)1527 3467 y Fi(0)1576 3500 y Fp(=)g Fj(b)p Fp(])g(=)g(1)p Fj(=)p Fp(2)150 3713 y Fq(Pro)s(of)119 b Fp(First,)28 b(let)g(us)f(sho)m(w)g(that)h(if)f Fj(P)13 b(r)s Fp([)p Fj(b)1686 3680 y Fi(0)1734 3713 y Fp(=)25 b Fj(b)p Fk(j)p Fj(b)h Fp(=)f(0])h(=)e Fj(\013)k Fp(then)g Fj(P)13 b(r)s Fp([)p Fj(b)2716 3680 y Fi(0)2764 3713 y Fp(=)25 b Fj(b)p Fk(j)p Fj(b)h Fp(=)f(1])h(=)f(1)15 b Fk(\000)g Fj(\013)p Fp(.)39 b(More)150 3825 y(formally)23 b(the)j(\014rst)e(probabilit)m(y)f(is)h(Pr)o([)p Fj(a)i Fk( )f Fj(Z)1757 3839 y Fg(0)1796 3825 y Fp(;)15 b Fj(b)1875 3793 y Fi(0)1924 3825 y Fp(=)25 b Fj(A)p Fp(\()p Fj(a)p Fp(\))h(:)g Fj(b)2322 3793 y Fi(0)2370 3825 y Fp(=)f(0].)40 b(Since)24 b Fj(Z)2895 3839 y Fg(0)2959 3825 y Fp(and)h Fj(Z)3193 3839 y Fg(1)3257 3825 y Fp(are)h(p)s(erfectly)150 3938 y(indistinguishable)i(this)k(probabilit)m(y)f(is)h(the)h(same)h (as)f(Pr[)p Fj(a)d Fk( )g Fj(Z)2481 3952 y Fg(1)2520 3938 y Fp(;)15 b Fj(b)2599 3905 y Fi(0)2653 3938 y Fp(=)29 b Fj(A)p Fp(\()p Fj(a)p Fp(\))i(:)f Fj(b)3064 3905 y Fi(0)3118 3938 y Fp(=)f(0].)50 b(Since)32 b(this)150 4051 y(probabilit)m(y)e(is)h(just)h Fj(\013)h Fp(w)m(e)f(no)m(w)h(ha)m (v)m(e)g Fj(P)13 b(r)s Fp([)p Fj(b)1700 4018 y Fi(0)1752 4051 y Fp(=)28 b Fj(b)p Fk(j)p Fj(b)g Fp(=)h(1])g(=)f(1)22 b Fk(\000)f Fj(\013)p Fp(.)47 b(No)m(w)33 b(w)m(e)g(decomp)s(ose)f(Pr[) p Fj(b)3535 4018 y Fi(0)3587 4051 y Fp(=)c Fj(b)p Fp(])150 4164 y(in)m(to)g(Pr)o([)p Fj(b)493 4131 y Fi(0)542 4164 y Fp(=)d Fj(b)p Fk(j)p Fj(b)h Fp(=)f(0])15 b(Pr[)p Fj(b)26 b Fp(=)e(0])17 b(+)e(Pr)o([)p Fj(b)1565 4131 y Fi(0)1614 4164 y Fp(=)25 b Fj(b)p Fk(j)p Fj(b)g Fp(=)g(1])15 b(Pr)q([)p Fj(b)25 b Fp(=)g(1])h(=)f Fj(\013)16 b Fk(\002)f Fp(1)p Fj(=)p Fp(2)i(+)e(\(1)h Fk(\000)f Fj(\013)p Fp(\))h Fk(\002)f Fp(1)p Fj(=)p Fp(2)27 b(=)e(1)p Fj(=)p Fp(2)p 3692 4164 59 59 v 150 4507 a Fl(2.2)112 b(Statistical)35 b(indistinguishabilit)m (y)150 4679 y Fp(The)40 b(in)m(tuition)f(here)h(is)g(that)h(ev)m(en)g (an)g(un)m(b)s(ounded)d(adv)m(ersary)-8 b(,)44 b(giv)m(en)d(only)f(p)s (olynomially)d(man)m(y)150 4792 y(samples)30 b(from)g(either)f Fj(X)1037 4806 y Fg(1)1108 4792 y Fp(or)h Fj(X)1294 4806 y Fg(2)1334 4792 y Fp(,)h(cannot)g(tell)e(whic)m(h)h(it)g(is)f(with)g (an)i(adv)-5 b(an)m(tage)32 b(b)s(etter)f(than)f(negli-)150 4905 y(gible.)150 5117 y Fq(De\014nition)35 b(4)46 b Fj(X)40 b Fn(and)33 b Fj(Y)53 b Fn(ar)-5 b(e)33 b Fj(\017)p Fn(-close)g(if)1668 5049 y Fa(P)1764 5144 y Fh(a)p Fi(2)p Fh(D)1928 5117 y Fk(j)15 b Fp(Pr[)p Fj(X)33 b Fp(=)25 b Fj(a)p Fp(])20 b Fk(\000)g Fp(Pr[)p Fj(Y)45 b Fp(=)25 b Fj(a)p Fp(])p Fk(j)h(\024)f Fj(\017)p Fn(.)1867 5416 y Fp(15-2)p eop %%Page: 3 3 3 2 bop 150 159 a Fq(De\014nition)35 b(5)46 b Fk(f)p Fj(X)856 174 y Fh(k)899 159 y Fk(g)36 b Fn(and)h Fk(f)p Fj(Y)1258 174 y Fh(k)1301 159 y Fk(g)f Fn(ar)-5 b(e)37 b(statistic)-5 b(al)5 b(ly)38 b(indistinguishable)e(families)h(of)f(pr) -5 b(ob)g(ability)38 b(dis-)150 271 y(tributions)31 b(over)g(sets)f Fk(f)p Fj(D)1055 286 y Fh(k)1098 271 y Fk(g)h Fn(if)f(for)h(al)5 b(l)31 b Fj(k)s Fn(,)f Fj(X)1714 286 y Fh(k)1787 271 y Fn(and)i Fj(Y)2015 286 y Fh(k)2087 271 y Fn(ar)-5 b(e)31 b Fj(\027)6 b Fp(\()p Fj(k)s Fp(\))p Fn(-close)31 b(for)g(some)g(ne)-5 b(gligible)30 b(function)150 384 y Fj(\027)6 b Fp(\()p Fj(k)s Fp(\))p Fn(.)150 597 y Fq(Theorem)34 b(6)46 b Fn(L)-5 b(et)26 b Fk(f)p Fj(X)962 612 y Fh(k)1005 597 y Fk(g)h Fn(and)g Fk(f)p Fj(Y)1345 612 y Fh(k)1388 597 y Fk(g)f Fn(b)-5 b(e)26 b(statistic)-5 b(al)5 b(ly)28 b(indistinguishable.)41 b(Then,)28 b(for)e(any)h(p)-5 b(olynomial)150 710 y Fj(p)p Fn(,)31 b(given)g Fj(p)p Fp(\()p Fj(k)s Fp(\))i Fn(indep)-5 b(endently)33 b(chosen)g(no)f (adversary)h(c)-5 b(an)33 b(tel)5 b(l)31 b Fj(X)2494 725 y Fh(k)2569 710 y Fn(fr)-5 b(om)33 b Fj(Y)2836 725 y Fh(k)2910 710 y Fn(with)f(advantage)h(b)-5 b(etter)150 823 y(than)34 b(ne)-5 b(gligible.)150 1035 y Fq(Pro)s(of)121 b Fp(First,)30 b(let)f(us)g(sho)m(w)h(that)g(if)e Fj(X)37 b Fp(and)29 b Fj(Y)50 b Fp(are)30 b Fj(\017)p Fp(-close,)g(then,)g(giv) m(en)g(one)g(sample)e(from)h(either)150 1148 y Fj(X)39 b Fp(or)31 b Fj(Y)20 b Fp(,)31 b(giv)m(en)h(just)e(one)i(sample,)f(the) g(adv)-5 b(an)m(tage)33 b(in)d(distinguishing)d(whic)m(h)j (distribution)d(it)k(came)150 1261 y(from)h(is)g Fj(\017)p Fp(.)47 b(Wh)m(y?)g(Because)34 b(the)e(b)s(est)g(strategy)i(that)f(an)m (y)g Fj(A)g Fp(can)g(use)f(to)h(determine)f(whic)m(h)f(distri-)150 1374 y(bution)g(a)h(sample)g(came)h(from)f(is)f(to)i(c)m(ho)s(ose)g (the)g(one)f(with)f(the)h(maxim)m(um)g(lik)m(eliho)s(o)s(d.)43 b(Using)31 b(this)150 1487 y(strategy)38 b(Pr[)p Fj(A)15 b(cor)s(r)s(ect)p Fk(j)p Fj(a)p Fp(])37 b(=)f Fj(max)p Fp(\(Pr)o([)p Fj(X)7 b Fk(j)p Fj(a)p Fp(])p Fj(;)15 b Fp(Pr)r([)p Fj(Y)20 b Fk(j)p Fj(a)p Fp(]\))37 b(=)f(1)p Fj(=)p Fp(2)26 b(+)e(1)p Fj(=)p Fp(2)p Fk(j)15 b Fp(Pr)q([)p Fj(X)7 b Fk(j)p Fj(a)p Fp(])26 b Fk(\000)e Fp(Pr[)p Fj(Y)c Fk(j)p Fj(a)p Fp(])p Fk(j)37 b Fp(and)g(so)150 1600 y Fk(8)p Fj(A)15 b Fp(Pr)o([)p Fj(A)g(cor)s(r)s(ect)p Fk(j)p Fj(a)p Fp(])27 b Fk(\024)e Fp(1)p Fj(=)p Fp(2)c(+)f(1)p Fj(=)p Fp(2)p Fk(j)15 b Fp(Pr)q([)p Fj(X)7 b Fk(j)p Fj(a)p Fp(])22 b Fk(\000)e Fp(Pr)o([)p Fj(Y)g Fk(j)p Fj(a)p Fp(])p Fk(j)p Fp(.)150 1713 y(Pr)o([)p Fj(X)7 b Fk(j)p Fj(a)p Fp(])27 b(=)e(Pr)o([)p Fj(X)r(;)15 b(a)p Fp(])p Fj(=)g Fp(Pr)r([)p Fj(a)p Fp(])26 b(=)f(Pr)o([)p Fj(a)p Fk(j)p Fj(X)7 b Fp(])15 b(Pr)q([)p Fj(X)7 b Fp(])p Fj(=)15 b Fp(Pr)q([)p Fj(a)p Fp(])26 b(=)f(Pr)o([)p Fj(a)p Fk(j)p Fj(X)7 b Fp(]1)p Fj(=)p Fp(2)p Fj(=)15 b Fp(Pr)s([)p Fj(a)p Fp(])31 b(and)f(so)150 1826 y Fj(P)13 b(r)s Fp([)p Fj(A)i(cor)s(r)s(ect)p Fp(])26 b(=)805 1757 y Fa(P)901 1852 y Fh(a)p Fi(2)p Fh(D)1064 1826 y Fp(Pr[)p Fj(A)15 b(cor)s(r)s(ect)p Fk(j)p Fj(a)p Fp(])g(Pr[)p Fj(a)p Fp(])150 1939 y Fk(\024)246 1870 y Fa(P)342 1965 y Fh(a)p Fi(2)p Fh(D)490 1939 y Fp(\(1)p Fj(=)p Fp(2)22 b(+)e(1)p Fj(=)p Fp(2)p Fk(j)15 b Fp(Pr)r([)p Fj(X)7 b Fk(j)p Fj(a)p Fp(])21 b Fk(\000)f Fp(Pr)o([)p Fj(Y)h Fk(j)p Fj(a)p Fp(])p Fk(j)p Fp(\))15 b(Pr[)p Fj(a)p Fp(])150 2051 y Fk(\024)25 b Fp(1)p Fj(=)p Fp(2)d(+)d(1)p Fj(=)p Fp(2)643 1983 y Fa(P)741 2078 y Fh(a)p Fi(2)p Fh(D)905 2051 y Fk(j)c Fp(Pr)o([)p Fj(a)p Fk(j)p Fj(X)7 b Fp(]1)p Fj(=)p Fp(2)p Fj(=)15 b Fp(Pr)s([)p Fj(a)p Fp(])21 b Fk(\000)f Fp(Pr)o([)p Fj(a)p Fk(j)p Fj(Y)h Fp(]1)p Fj(=)p Fp(2)p Fj(=)15 b Fp(Pr)r([)p Fj(a)p Fp(])p Fk(j)g Fp(Pr[)p Fj(a)p Fp(])150 2164 y Fk(\024)25 b Fp(1)p Fj(=)p Fp(2)d(+)d(1)p Fj(=)p Fp(4)643 2096 y Fa(P)741 2191 y Fh(a)p Fi(2)p Fh(D)905 2164 y Fk(j)c Fp(Pr)o([)p Fj(a)p Fk(j)p Fj(X)7 b Fp(])22 b Fk(\000)e Fp(Pr)o([)p Fj(a)p Fk(j)p Fj(Y)h Fp(])p Fk(j)150 2277 y(\024)k Fp(1)p Fj(=)p Fp(2)d(+)d(1)p Fj(=)p Fp(4)p Fj(\017)291 2503 y Fp(And)32 b(no)m(w)g(w)m(e)h(can)g(use)f(the)h(h)m (ybrid)d(argumen)m(t)j(to)g(\014nish)d(the)j(pro)s(of.)46 b(Let)33 b(our)f Fj(p)p Fp(\()p Fj(k)s Fp(\))h(h)m(ybrids)e(b)s(e)150 2616 y(as)g(follo)m(ws:)150 2729 y Fj(H)226 2743 y Fg(0)290 2729 y Fp(=)25 b Fj(X)468 2696 y Fh(p)p Fg(\()p Fh(k)r Fg(\))150 2842 y Fj(H)226 2856 y Fg(1)290 2842 y Fp(=)g Fj(Y)40 b Fk(\016)21 b Fj(X)627 2809 y Fh(p)p Fg(\()p Fh(k)r Fg(\))p Fi(\000)p Fg(1)150 2955 y Fj(H)226 2969 y Fh(i)279 2955 y Fp(=)k Fj(Y)40 b Fk(\016)21 b Fj(X)616 2922 y Fh(p)p Fg(\()p Fh(k)r Fg(\))p Fi(\000)p Fh(i)150 3068 y Fj(H)226 3086 y Fh(p)p Fg(\()p Fh(k)r Fg(\))384 3068 y Fp(=)k Fj(Y)553 3035 y Fh(p)p Fg(\()p Fh(k)r Fg(\))291 3293 y Fp(Supp)s(ose)e(that)j(the)f(sets)h(of)f(samples)f(from)h Fj(X)32 b Fp(and)25 b Fj(Y)45 b Fp(are)25 b Fj(\016)s Fp(-close)i(for)d Fj(\016)30 b(>)24 b(\027)6 b Fp(\()p Fj(k)s Fp(\))p Fj(p)p Fp(\()p Fj(k)s Fp(\).)41 b(The)24 b(h)m(ybrid)150 3406 y(argumen)m(t)38 b(sa)m(ys)f(that)h(at)g(least)f (on)g(pair)f(of)h(adjacen)m(t)h(h)m(ybrids)d(is)h(not)h Fj(\027)6 b Fp(\()p Fj(k)s Fp(\)-close)39 b(and)d(since)g(these)150 3519 y(h)m(ybrids)28 b(only)h(di\013er)h(b)m(y)g(one)h(sample)e Fj(X)38 b Fp(and)30 b Fj(Y)50 b Fp(cannot)31 b(b)s(e)f Fj(\027)6 b 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y(Since)35 b Fn(G)p Ft(\()p Fn(x)p Ft(\))i(is)d(length)i(2)p Fn(k)s Ft(,)i(w)m(e)e(can)g(divide)d(it)j(in)e(half)h(and)g(create)i Fn(G)2807 4216 y Fk(0)2883 4202 y Ft(and)e Fn(G)3136 4216 y Fk(1)3176 4202 y Ft(.)57 b Fn(G)3329 4216 y Fk(0)3369 4202 y Ft(\()p Fn(x)p Ft(\))36 b(is)f(the)150 4315 y(\014rst)30 b Fn(k)j Ft(bits)c(of)i Fn(G)p Ft(\()p Fn(x)p Ft(\))g(and)f Fn(G)1171 4329 y Fk(1)1211 4315 y Ft(\()p Fn(x)p Ft(\))h(is)e(the)i (last)f Fn(k)k Ft(bits)29 b(of)i Fn(G)p Ft(\()p Fn(x)p Ft(\))291 4428 y(Let)g Fn(S)f Fo( )25 b(f)p Ft(0)p Fn(;)15 b Ft(1)p Fo(g)876 4395 y Fl(k)951 4428 y Ft(b)s(e)29 b(the)h(random)f(seed)h(and)g Fn(x)25 b Ft(=)g Fn(x)2162 4442 y Fk(1)2220 4428 y Fo(\016)20 b Fn(x)2337 4442 y Fk(2)2396 4428 y Fo(\016)g Fn(:)15 b(:)g(:)20 b Fo(\016)g Fn(x)2703 4443 y Fl(k)2775 4428 y Ft(.)41 b(Then)28 b Fn(F)3135 4442 y Fl(s)3173 4428 y Ft(\()p Fn(x)p Ft(\))e Fi(,)e Fn(l)3444 4442 y Fl(x)3517 4428 y Ft(where)1318 4614 y Fn(l)1345 4628 y Fl(x)1472 4614 y Ft(=)82 b Fn(l)1652 4628 y Fl(x)1692 4637 y Fh(1)1727 4628 y Fm(\016)p Fl(x)1802 4637 y Fh(2)1836 4628 y Fm(\016)p Fl(:::)p Fm(\016)p Fl(x)2006 4640 y Fj(k)1472 4727 y Ft(=)g Fn(G)1696 4741 y Fl(x)1736 4753 y Fj(k)1779 4727 y Ft(\()p Fn(l)1841 4741 y Fl(x)1881 4750 y Fh(1)1916 4741 y Fm(\016)p Fl(x)1991 4750 y Fh(2)2025 4741 y Fm(\016)p Fl(:::)o Fm(\016)p Fl(x)2194 4753 y Fj(k)q Fg(\000)p Fh(1)2315 4727 y Ft(\))1472 4840 y(=)g Fn(G)1696 4854 y Fl(x)1736 4866 y Fj(k)1779 4840 y Fn(G)1850 4854 y Fl(x)1890 4866 y Fj(k)q Fg(\000)p Fh(1)2011 4840 y Ft(\()p Fn(l)2073 4854 y Fl(x)2113 4863 y Fh(1)2148 4854 y Fm(\016)p Fl(x)2223 4863 y Fh(2)2257 4854 y Fm(\016)p Fl(:::)o Fm(\016)p Fl(x)2426 4866 y Fj(k)q Fg(\000)p Fh(2)2547 4840 y Ft(\))1322 5066 y Fn(l)1349 5080 y Fk(0)1472 5066 y Ft(=)g Fn(G)1696 5080 y Fk(0)1736 5066 y Ft(\()p Fn(S)5 b Ft(\))1322 5179 y Fn(l)1349 5193 y Fk(1)1472 5179 y Ft(=)82 b Fn(G)1696 5193 y Fk(1)1736 5179 y Ft(\()p Fn(S)5 b Ft(\))1867 5416 y(16-1)p eop %%Page: 2 2 2 1 bop 291 159 a Ft(Wh)m(y)25 b(is)f Fn(F)649 173 y Fl(s)711 159 y Ft(a)h(PRF?)g(W)-8 b(e)26 b(can)f(sho)m(w)g(that)h Fn(F)1819 173 y Fl(s)1881 159 y Ft(is)d(a)j(PRF)f(using)e(a)i(h)m (ybrid)e(argumen)m(t.)39 b(The)24 b(h)m(ybrids)150 271 y(cannot)30 b(b)s(e)f(generated)i(horizon)m(tally)e(b)s(ecause)g(there) h(w)m(ould)f(b)s(e)g(an)h(exp)s(onen)m(tially)e(large)h(n)m(um)m(b)s (er)g(of)150 384 y(h)m(ybrids,)f(but)i(w)m(e)h(can)f(generate)i(them)f (v)m(ertically)-8 b(.)291 497 y(The)30 b(h)m(ybrids)e(are)j(de\014ned)e (in)g(this)g(w)m(a)m(y:)377 683 y(0)422 650 y Fl(th)523 683 y Ft(h)m(ybrid:)39 b Fn(F)905 697 y Fl(S)986 683 y Ft(for)31 b(some)f(random)g Fn(S)5 b Ft(.)377 870 y Fn(i)408 837 y Fl(th)515 870 y Ft(h)m(ybrid:)51 b(Ev)m(erything)36 b(on)g(lev)m(el)h Fn(i)f Ft(is)g(random.)58 b(On)36 b(all)f(lev)m(els)h Fn(j)42 b(>)35 b(i)p Ft(,)j(lab)s(els)d(are)i(deriv)m(ed)377 983 y(from)30 b(lab)s(els)f(in)g(lev)m(el)h Fn(i)p Ft(.)377 1170 y Fn(k)427 1137 y Fl(th)528 1170 y Ft(h)m(ybrid:)39 b(Some)30 b(random)g(function.)291 1356 y Fu(Claim:)62 b Ft(If)41 b Fn(A)825 1371 y Fl(k)910 1356 y Ft(distinguishes)d Fn(F)1516 1370 y Fl(S)1609 1356 y Ft(from)j(a)h(random)f(function)g Fn(R)q Ft(,)j(then,)h(for)c(some)h Fn(i)p Ft(,)j(it)d(can)150 1468 y(distinguish)27 b Fn(F)682 1435 y Fl(i)669 1495 y(S)750 1468 y Ft(from)j Fn(F)1036 1430 y Fl(i)p Fk(+1)1023 1497 y Fl(S)1154 1468 y Ft(.)291 1581 y(This)g(claim)h(is)f(true)i(b)s (ecause)g(there)f(are)i(only)d(a)i(p)s(olynomial)d(n)m(um)m(b)s(er)i (of)h(h)m(ybrids)d(and)i(b)s(ecause)150 1694 y(of)e(the)f(claim)g(w)m (e)h(pro)m(v)m(ed)g(the)g(\014rst)e(time)i(that)g(h)m(ybrids)d(w)m(ere) j(in)m(tro)s(duced.)39 b(F)-8 b(rom)29 b(the)f(problem)f(sets,)150 1807 y(w)m(e)k(also)f(kno)m(w)h(that)g(w)m(e)f(can)h(c)m(ho)s(ose)g Fn(i)g Ft(at)g(random.)150 2050 y Fp(1.3)112 b(Using)37 b Ff(A)795 2065 y Fl(k)876 2050 y Fp(to)g(distinguish)e Ff(G)p Fe(\()p Fd(\001)p Fe(\))i Fp(from)g(a)h(random)g(string)150 2222 y Ft(There)30 b(are)h(t)m(w)m(o)g(imp)s(ortan)m(t)f(claims:)261 2408 y(1.)46 b Fn(A)445 2423 y Fl(k)525 2408 y Ft(can)38 b(b)s(e)e(used)h(to)g(distinguish)d Fn(p)p Ft(\()p Fn(k)s Ft(\))k(random)e(2)p Fn(k)s Ft(-bit)h(strings)f(from)h Fn(p)p Ft(\()p Fn(k)s Ft(\))g(pseudorandom)377 2521 y(2)p Fn(k)s Ft(-bit)24 b(strings.)37 b(\(Here)25 b Fn(p)p Ft(\()p Fn(k)s Ft(\))f(is)f(an)h(upp)s(er)d(b)s(ound)h(on)i(the)g(n)m (um)m(b)s(er)e(of)i(queries)f(that)h Fn(A)3380 2536 y Fl(k)3446 2521 y Ft(mak)m(es\).)261 2708 y(2.)46 b(If)37 b Fn(A)543 2723 y Fl(k)623 2708 y Ft(can)g(distinguish)d Fn(p)p Ft(\()p Fn(k)s Ft(\))j(random)g(strings)f(from)g Fn(p)p Ft(\()p Fn(k)s Ft(\))i(pseudorandom)e(strings,)i(then)e(it)377 2820 y(can)31 b(tell)f(if)f(a)i(giv)m(en)f(string)f(is)h(random)f(or)i (pseudorandom.)150 3107 y Fq(2)135 b(The)44 b(Naor-Reingold)j (Construction)150 3313 y Fp(2.1)112 b(The)38 b(DDH)f(assumption)150 3484 y Ft(Let)27 b Fn(P)13 b Ft(\(1)460 3451 y Fl(k)504 3484 y Ft(\))27 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b(more)g(or)h(less)f(uniformly)e(distributes)g(it.)59 b(Ho)m(w)m(ev)m(er,)40 b(here)d Fn(g)3518 760 y Fl(x)3598 793 y Fn(<)e(p)150 906 y Ft(and)28 b(so)g(the)h(\014rst)e(part)i(of)f (the)g(output)g(only)g(co)m(v)m(ers)i(a)e(fraction)h Fn(p=)p Ft(2)2532 873 y Fl(k)2603 906 y Ft(of)g(the)f(whole)g(space)g Fo(f)p Ft(0)p Fn(;)15 b Ft(1)p Fo(g)3575 873 y Fl(k)3620 906 y Ft(,)29 b(in)150 1019 y(fact)i(a)g(particular)d(p)s(ortion)h(of)h (the)h(space,)g(namely)e(those)i(n)m(um)m(b)s(ers)e Fn(y)k Ft(whic)m(h)28 b(are)j(less)e(than)h Fn(p)p Ft(.)41 b(This)150 1132 y(fact)27 b(migh)m(t)f(enable)g(an)g(adv)m(ersary)g(to)h (distinguish)c(b)s(et)m(w)m(een)j(the)h(output)f(of)g Fn(G)h Ft(from)e(a)i(truly)e(random)150 1244 y(distribution.)291 1357 y(W)-8 b(e)34 b(could)d(argue)h(that)h(if)e Fn(p)h Ft(is)g(v)m(ery)g(close)h(to)g(2)1978 1324 y Fl(k)2053 1357 y Ft(then)f(the)g(ab)s(o)m(v)m(e)i(con)m(tradiction)e(do)s(esn't)g (hold.)150 1470 y(Ho)m(w)m(ev)m(er,)j(there's)d(another)h(problem)d (with)h(this)g(generator.)48 b(Since)31 b Fn(g)k Ft(is)d(of)g(order)g Fn(q)s Ft(,)g Fn(g)3267 1437 y Fl(x)3344 1470 y Ft(is)f(of)h(order)150 1583 y(dividing)26 b Fn(q)s Ft(.)41 b(Ho)m(w)m(ev)m(er,)32 b(it's)d(easy)i(to)f(sho)m(w)g(that)g(a)g(random)f(elemen)m(t)i(of)f Fc(Z)2795 1550 y Fm(\003)2795 1606 y Fl(p)2860 1583 y Ft(has)f(order)g Fn(\036)p Ft(\()p Fn(p)p Ft(\))d(=)f Fn(p)19 b Fo(\000)g Ft(1)150 1696 y(\(the)37 b(Euler)e(phi)g (function\))g(with)h(go)s(o)s(d)g(probabilit)m(y)-8 b(.)57 b(In)36 b(fact,)j(the)e(n)m(um)m(b)s(er)e(of)i(elemen)m(ts)f(of)h Fc(Z)3605 1710 y Fl(p)3677 1696 y Ft(of)150 1809 y(order)g Fn(\036)p Ft(\()p Fn(p)p Ft(\))i(is)e Fn(\036)p Ft(\()p Fn(\036)p Ft(\()p Fn(p)p Ft(\)\))i(=)e Fn(\036)p Ft(\()p Fn(p)26 b Fo(\000)f Ft(1\))38 b(=)g(\012\()1729 1743 y Fo(p)p 1804 1743 203 4 v 1804 1809 a Fn(p)20 b Fo(\000)g Ft(1)q(\).)63 b(So)38 b(the)g(adv)m(ersary)g(could)f(just)g(c)m(hec)m (k)i(if)e(it's)150 1922 y(input)31 b(had)h(order)g(dividing)d Fn(q)35 b 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Ft(=)28 b Fn(g)2379 3772 y Fl(z)2419 3805 y Ft(\).)46 b(W)-8 b(e)34 b(wish)c(to)i(determine)g(whether)150 3918 y Fn(z)e Ft(=)24 b Fn(xy)29 b Ft(mo)s(d)24 b Fn(q)s Ft(,)30 b(or)g Fn(z)35 b Ft(is)29 b(random)h(from)f Fc(Z)p Fn(q)o Ft(.)40 b(T)-8 b(o)31 b(that)g(end,)f(w)m(e)g(can)h(randomize)e (the)i(tuple)e(in)g(suc)m(h)h(a)150 4031 y(w)m(a)m(y)k(as)g(to)g (obtain)e(\()p Fn(Y)960 3998 y Fm(0)1013 4031 y Ft(=)d Fn(g)1159 3998 y Fl(y)1196 3974 y Fg(0)1224 4031 y Fn(;)15 b(Z)1333 3998 y Fm(0)1386 4031 y Ft(=)30 b Fn(g)1533 3998 y Fl(z)1569 3974 y Fg(0)1596 4031 y Ft(\))j(suc)m(h)g(that)h Fn(y)2120 3998 y Fm(0)2176 4031 y Ft(is)e(completely)h(random,)h(and)e (in)g(the)h(ev)m(en)m(t)150 4144 y(that)e Fn(z)e Ft(=)c Fn(xy)s Ft(,)31 b Fn(z)716 4111 y Fm(0)765 4144 y Ft(=)25 b Fn(xy)961 4111 y Fm(0)984 4144 y Ft(,)30 b(while)e(on)j(the)f(other)h (hand,)e(if)h Fn(z)k Ft(w)m(as)d(random,)f(then)g Fn(z)2996 4111 y Fm(0)3050 4144 y Ft(is)f(also)h(completely)150 4256 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Fl(bxy)r Fk(+)p Fl(axb)1609 271 y Ft(=)g(\()p Fn(g)1786 238 y Fl(by)r Fk(+)p Fl(ab)1982 271 y Ft(\))2017 238 y Fl(x)2086 271 y Ft(=)g(\()p Fn(Y)2290 238 y Fm(0)2314 271 y Ft(\))2349 238 y Fl(x)2418 271 y Ft(=)g Fn(g)2560 238 y Fl(xy)2637 215 y Fg(0)2664 271 y Ft(,)31 b(whic)m(h)e(is)g(what)i (w)m(e)f(w)m(an)m(t.)291 384 y(Case)42 b(2:)63 b(supp)s(ose)40 b(\()p Fn(X)r(;)15 b(Y)5 b(;)15 b(Z)7 b Ft(\))45 b(=)e(\()p Fn(g)1600 351 y Fl(x)1645 384 y Fn(;)15 b(g)1731 351 y Fl(y)1773 384 y Fn(;)g(g)1859 351 y Fl(z)1900 384 y Ft(\))42 b(where)f Fn(z)47 b Fo(6)p Ft(=)d Fn(xy)28 b Ft(mo)s(d)c Fn(q)s Ft(.)74 b(Then)40 b Fn(Y)3243 351 y Fm(0)3308 384 y Ft(and)g Fn(Z)3564 351 y Fm(0)3629 384 y Ft(are)150 497 y(completely)k(random.)80 b(As)44 b(sho)m(wn)f(ab)s(o)m(v)m(e,)49 b Fn(y)1815 464 y Fm(0)1886 497 y Ft(=)e(\()p Fn(y)33 b Ft(+)c Fn(a)p Ft(\))p Fn(b)44 b Ft(and)f Fn(z)2619 464 y Fm(0)2691 497 y Ft(=)k(\()p Fn(z)34 b Ft(+)29 b Fn(ax)p Ft(\))p Fn(b)p Ft(.)82 b(Simplifying)150 610 y Fn(xy)250 577 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Fo(\000)c Ft(1\))2789 936 y Fk(2)2866 969 y Ft(pairs)34 b(of)i(\()p Fn(y)3286 936 y Fm(0)3310 969 y Fn(;)15 b(z)3396 936 y Fm(0)3420 969 y Ft(\))36 b(in)f(this)150 1082 y(case,)d(and)d (therefore)i Fn(Y)999 1049 y Fm(0)1053 1082 y Ft(and)f Fn(Z)1299 1049 y Fm(0)1352 1082 y Ft(are)h(completely)f(random.)150 1325 y Fp(2.4)112 b(The)38 b(NR)f(construction)150 1497 y Ft(The)d(NR)g(is)f(a)h(construction)g(of)g(a)g(\\relaxed")h(PRF)f(in) m(to)g(the)g(domain)f Fn(G)f Ft(=)f Fn(Z)2977 1464 y Fm(\003)2970 1519 y Fl(p)3016 1497 y Ft(.)52 b(The)33 b(structure)h(of)150 1610 y(the)h(construction)f(is)f(similar)f(to)j (the)f(tree-based)i(GGM)f(construction)f(of)g(a)h(PRF)g(from)f(a)g(PR)m (G)h(in)150 1723 y(lecture)30 b(13.)291 1835 y(Cho)s(ose)g(some)h Fn(p;)15 b(q)s(;)g(g)34 b Ft(as)c(in)f(the)h(Decisional)f (Di\016e-Hellman)g(assumption)g(ab)s(o)m(v)m(e.)42 b(Let)31 b(the)f(seed)150 1948 y(of)36 b(this)e(PRF)i Fn(F)723 1962 y Fl(s)795 1948 y Ft(b)s(e)f(\()p Fn(x)1011 1962 y Fk(0)1051 1948 y Fn(;)15 b(x)1143 1962 y Fk(1)1183 1948 y Fn(;)g(:)g(:)g(:)h(;)f(x)1436 1963 y Fl(k)1479 1948 y Ft(\),)38 b(where)d Fn(x)1897 1962 y Fl(i)1960 1948 y Ft(are)h(random)f(elemen)m(ts)h(of)f Fn(Z)3003 1962 y Fl(q)3077 1948 y Fo(8)g Fn(i)p Ft(.)56 b(Then)35 b(de\014ne)150 2061 y(the)c(PRF)f Fn(F)583 2075 y Fl(s)651 2061 y Ft(as)1058 2266 y Fn(F)1116 2280 y Fl(s)1153 2266 y Ft(\()p Fn(m)p Ft(\))c(=)f Fn(F)1483 2280 y Fl(s)1520 2266 y Ft(\()p Fn(m)1635 2280 y Fk(0)1695 2266 y Fo(\016)20 b Fn(m)1840 2280 y Fk(1)1900 2266 y Fo(\016)g(\001)15 b(\001)g(\001)22 b(\016)e Fn(m)2237 2281 y Fl(k)2280 2266 y Ft(\))25 b(=)g Fn(g)2482 2228 y Fl(x)2522 2237 y Fh(0)2557 2228 y Fk(\005)2610 2205 y Fj(k)2610 2249 y(i)p Fh(=1)2714 2228 y Fl(x)2754 2189 y Fj(m)2808 2205 y(i)2754 2250 y(i)291 2470 y Ft(The)30 b(\014rst)g(few)g(lev)m(els)g (of)g(this)g(constructions)f(are)i(demonstrated)g(in)e(this)g(diagram:) 2075 2678 y Fn(g)2121 2645 y Fl(x)2161 2654 y Fh(0)1992 2803 y Fb(\000)1950 2844 y(\000)-83 b(\011)2158 2803 y(@)2199 2844 y(@)g(R)1825 2927 y Fn(g)1871 2894 y Fl(x)1911 2903 y Fh(0)1742 3052 y Fb(\000)1701 3093 y(\000)g(\011)1576 3177 y Fn(g)1622 3144 y Fl(x)1662 3153 y Fh(0)1867 3052 y Fb(@)1908 3093 y(@)g(R)1908 3177 y Fn(g)1954 3144 y Fl(x)1994 3153 y Fh(0)2029 3144 y Fl(x)2069 3153 y Fh(2)2324 2927 y Fn(g)2370 2894 y Fl(x)2410 2903 y Fh(0)2445 2894 y Fl(x)2485 2903 y Fh(1)2282 3052 y Fb(\000)2241 3093 y(\000)g(\011)2241 3177 y Fn(g)2287 3144 y Fl(x)2327 3153 y Fh(0)2362 3144 y Fl(x)2402 3153 y Fh(1)2407 3052 y Fb(@)2448 3093 y(@)g(R)2573 3177 y Fn(g)2619 3144 y Fl(x)2659 3153 y Fh(0)2694 3144 y Fl(x)2734 3153 y Fh(1)2768 3144 y Fl(x)2808 3153 y Fh(2)1576 3343 y Fn(:)15 b(:)g(:)394 b(:)15 b(:)g(:)393 b(:)15 b(:)g(:)1120 2678 y Ft(Lev)m(el)30 b(0)1120 2927 y(Lev)m(el)g(1)1120 3177 y(Lev)m(el)g(2)291 3638 y(The)43 b(b)s(ottom)h(lev)m(el)f(of)g(the)g(tree,)48 b(whic)m(h)42 b(has)h(depth)f Fn(k)s Ft(,)47 b(w)m(ould)42 b(ha)m(v)m(e)j(elemen)m(ts)e(of)h(the)f(form)150 3751 y Fn(g)196 3718 y Fl(x)236 3727 y Fh(0)275 3751 y Fn(;)15 b(g)361 3718 y Fl(x)401 3727 y Fh(0)436 3718 y Fl(x)476 3730 y Fj(k)518 3751 y Fn(;)g(:)g(:)g(:)i(;)e(g)766 3718 y Fl(x)806 3727 y Fh(0)841 3718 y Fl(x)881 3727 y Fh(1)916 3718 y Fl(:::)n(x)1014 3730 y Fj(k)1056 3751 y Ft(.)150 3994 y Fp(2.5)112 b(Pro)s(of)37 b(of)h(NR)f(construction)f(PRF)150 4166 y Ft(As)31 b(with)g(the)g(pro)s(of)g(of)h(the)g(GGM)g (construction,)g(a)g(v)m(ertical)f(h)m(ybrid)e(argumen)m(t)j(can)g(b)s (e)f(used)g(here.)150 4279 y(Supp)s(ose)i(that)j(there)f(exists)g(an)g (adv)m(ersary)h(whic)m(h)e(can)h(distinguish)d(b)s(et)m(w)m(een)j Fn(F)3070 4293 y Fl(s)3108 4279 y Ft(\()p Fn(m)p Ft(\))g(and)g(a)g (truly)150 4392 y(random)d(function,)h(with)e(adv)-5 b(an)m(tage)35 b Fn(\017)p Ft(\()p Fn(k)s Ft(\).)49 b(Then)32 b(it)h(is)f(p)s(ossible)e(to)k(use)e(this)g(adv)m(ersary)h(to)h (distin-)150 4505 y(guish)29 b(b)s(et)m(w)m(een)i(one)g(of)f(these)h(h) m(ybrids,)d(b)m(y)i(the)h(h)m(ybrid)d(argumen)m(t:)291 4618 y Fn(H)367 4632 y Fk(0)431 4618 y Ft(=)d Fn(F)585 4632 y Fl(s)623 4618 y Ft(\()p Fn(m)p Ft(\))291 4731 y Fn(H)367 4745 y Fl(i)422 4731 y Ft(is)h(a)i(h)m(ybrid)d(tree)j(where) f(the)g Fn(i)p Ft(th)g(lev)m(el)g(of)h(the)f(tree)h(is)e(truly)g (random.)39 b(That)27 b(is,)g(it)g(consists)g(of)150 4844 y(elemen)m(ts)g(of)g(the)f(form)g Fn(g)1027 4811 y Fl(z)1068 4844 y Ft(,)h(where)f Fn(z)31 b Ft(is)26 b(c)m(hosen)h(at)g(random)f(from)g Fn(Z)2537 4858 y Fl(q)2575 4844 y Ft(.)39 b(The)26 b Fn(i)13 b Ft(+)g(1st)27 b(lev)m(el)f(is)g (pro)s(duced)150 4956 y(according)k(to)h(the)g(ab)s(o)m(v)m(e)h (construction.)291 5069 y Fn(H)367 5084 y Fl(k)r Fk(+1)525 5069 y Ft(=)25 b Fn(R)q Ft(,)30 b(a)h(truly)e(random)g(function.)1867 5416 y(16-4)p eop %%Page: 5 5 5 4 bop 291 159 a Ft(Let)35 b Fn(A)526 126 y Fm(0)526 186 y Fl(k)604 159 y Ft(b)s(e)e(the)i(adv)m(ersary)g(that)g(can)g (distinguish)c(b)s(et)m(w)m(een)k(some)g(t)m(w)m(o)h(h)m(ybrids)c Fn(H)3313 173 y Fl(i)3375 159 y Ft(and)i Fn(H)3632 173 y Fl(i)p Fk(+1)150 271 y Ft(with)27 b(adv)-5 b(an)m(tage)29 b Fn(\017)p Ft(\()p Fn(k)s Ft(\))p Fn(=k)s Ft(.)41 b(It)28 b(can)g(b)s(e)g(reduced)f(to)h(a)g(distinguisher)d(b)s(et)m(w)m(een)j (the)g(t)m(w)m(o)h(kinds)d(of)i(DDH)150 384 y(tuples)h(with)g(adv)-5 b(an)m(tage)33 b Fn(\017)p Ft(\()p Fn(k)s Ft(\))p Fn(=k)s Ft(,)f(as)e(demonstrated)h(b)s(elo)m(w.)291 497 y(The)f(reduction)g(is) f(giv)m(en)i(\()p Fn(p;)15 b(q)s(;)g(g)s(;)g(X)r(;)g(Y)5 b(;)15 b(Z)7 b Ft(\).)44 b(By)30 b(the)h(DDH)h(random)d (self-reducibilit)m(y)e(theorem,)150 610 y(w)m(e)i(can)f(sample)g(as)g (man)m(y)h(indep)s(enden)m(t)d(tuples)h(\()p Fn(Y)1990 577 y Fm(0)2014 610 y Fn(;)15 b(Z)2123 577 y Fm(0)2146 610 y Ft(\))29 b(as)f(w)m(e)h(w)m(an)m(t.)41 b(Construct)28 b(a)g(tree,)i(as)f(in)e(the)150 723 y(NR)33 b(construction,)g(with)e(v) -5 b(alues)32 b Fn(Y)1422 690 y Fm(0)1477 723 y Ft(on)h(lev)m(el)f Fn(i)h Ft(and)f Fn(Z)2127 690 y Fm(0)2182 723 y Ft(on)h(lev)m(el)f Fn(i)22 b Ft(+)f(1)33 b(\(the)g(left)g(c)m(hild)e(of)h Fn(Y)3499 690 y Fm(0)3555 723 y Ft(m)m(ust)150 836 y(b)s(e)k Fn(Y)353 803 y Fm(0)377 836 y Ft(,)i(and)f(the)g(righ)m(t)f(c)m(hild)f (w)m(ould)h(b)s(e)g Fn(Z)1710 803 y Fm(0)1733 836 y Ft(\).)61 b(Ev)m(en)37 b(though)f(w)m(e)i(need)e(2)2807 803 y Fl(i)2873 836 y Ft(indep)s(enden)m(t)e(tuples)i(at)150 949 y(the)g Fn(i)p Ft(th)g(lev)m(el,)h(the)e(random)g(self-reducibilit)m(y)d(allo)m (ws)j(us)g(to)i(accomplish)d(this)h(without)f(losing)h(the)150 1062 y(randomness)29 b(at)i(the)g Fn(i)p Ft(th)f(lev)m(el.)291 1175 y(Giv)m(en)42 b(this)f(tree,)46 b Fn(A)1045 1142 y Fm(0)1045 1202 y Fl(k)1130 1175 y Ft(can)d(tell)e(whether)h(the)g (randomness)f(starts)i(on)f(lev)m(el)f Fn(i)i Ft(or)f Fn(i)28 b Ft(+)g(1.)77 b(By)150 1288 y(construction,)30 b(these)h(t)m(w)m(o)h(cases)f(map)f(exactly)h(on)m(to)h(the)e(t)m(w)m (o)i(t)m(yp)s(es)e(of)h(DDH)g(input)e(tuples.)291 1401 y(Supp)s(ose)41 b(the)j(input)d(w)m(as)j(a)f(tuple)f(where)h Fn(z)51 b Ft(=)46 b Fn(xy)s Ft(.)79 b(Then)42 b Fn(Y)2619 1368 y Fm(0)2689 1401 y Ft(=)k Fn(g)2852 1368 y Fl(y)2889 1344 y Fg(0)2916 1401 y Fn(;)15 b(Z)3025 1368 y Fm(0)3095 1401 y Ft(=)47 b Fn(g)3259 1368 y Fl(xy)3336 1344 y Fg(0)3363 1401 y Ft(,)f(and)d(the)150 1513 y(constructed)35 b(tree)g(is)f (exactly)h(the)g Fn(i)p Ft(th)g(h)m(ybrid,)e Fn(H)1951 1527 y Fl(i)1979 1513 y Ft(.)53 b(No)m(w)35 b(supp)s(ose)f(the)g(input) f(w)m(as)i(a)g(tuple)e(where)150 1626 y Fn(z)d Fo(6)p Ft(=)24 b Fn(xy)s Ft(.)41 b(Then)28 b(the)h(random)g(self-reducibilit)m (y)c(guaran)m(tees)31 b(that)f Fn(Y)2566 1593 y Fm(0)2618 1626 y Ft(and)f Fn(Z)2863 1593 y Fm(0)2915 1626 y Ft(will)d(b)s(oth)j (b)s(e)f(random,)150 1739 y(and)i(therefore)h(w)m(e)f(will)e(ha)m(v)m (e)k(constructed)e(a)h(tree)g(equiv)-5 b(alen)m(t)30 b(to)h(the)g Fn(i)20 b Ft(+)g(1st)31 b(h)m(ybrid)d Fn(H)3325 1753 y Fl(i)p Fk(+1)3443 1739 y Ft(.)291 1852 y(W)-8 b(e)31 b(ha)m(v)m(e)g(demonstrated)f(that)g(b)m(y)f(using)g Fn(A)1854 1819 y Fm(0)1854 1880 y Fl(k)1926 1852 y Ft(it)g(is)g(p)s (ossible)e(to)k(distinguish)26 b(b)s(et)m(w)m(een)k(t)m(w)m(o)h(t)m(yp) s(es)150 1965 y(of)23 b(DDH)h(tuples,)g(con)m(tradicting)e(the)i (Decisional)e(Di\016e-Hellman)g(assumption.)36 b(Clearly)22 b(this)g(implies)150 2078 y(that)31 b(the)g(NR)f(construction)g(m)m (ust)g(b)s(e)g(a)h(PRF)f(under)f(the)h(DDH)i(assumption.)291 2191 y(It's)c(w)m(orth)f(noting)g(that,)i(although)e(the)h(GGM)g (construction)f(also)h(demonstrates)g(the)f(existence)150 2304 y(of)42 b(a)h(PRF)f(giv)m(en)g(a)h(PR)m(G)f(algorithm,)j(the)d(NR) g(construction)g(only)f(uses)h(one)g(\(v)m(ertical\))h(h)m(ybrid)150 2417 y(argumen)m(t.)65 b(The)37 b(GGM)i(construction)f(pro)s(of)g (relied)e(on)i(t)m(w)m(o)i(\(one)f(horizon)m(tal)e(and)h(one)g(v)m (ertical\))150 2530 y(h)m(ybrid)h(argumen)m(ts,)45 b(whic)m(h)40 b(mak)m(es)j(the)e(required)f(adv)m(ersary)i(1)p Fn(=t)p Ft(\()p Fn(k)s Ft(\))h(w)m(eak)m(er.)75 b(While)41 b(in)f(theory)150 2643 y(b)s(oth)30 b(are)g(correct,)i(in)d(practice)i(the)g(NR)f (construction)g(o\013ers)h(a)f(higher)f(margin)h(of)g(randomness.)150 2929 y Fq(3)135 b(Pseudorandom)45 b(p)t(erm)l(utations)150 3132 y Ft(References:)c(Lub)m(y-Rac)m(k)m(o\013.)i(This)28 b(presen)m(tation)j(is)e(from)h(Goldreic)m(h's)g(b)s(o)s(ok.)150 3375 y Fp(3.1)112 b(De\014nition)150 3547 y Fu(De\014nition)35 b(2)46 b Fr(A)34 b(family)i(of)f(p)-5 b(ermutations)37 b Fn(P)1833 3561 y Fm(\001)1857 3547 y Ft(\()p Fo(\001)p Ft(\))f Fr(is)f(pseudo-r)-5 b(andom)38 b(if)c(b)-5 b(oth)36 b Fn(P)48 b Fr(and)36 b Fn(P)3362 3514 y Fm(\000)p Fk(1)3491 3547 y Fr(ar)-5 b(e)36 b(ef-)150 3660 y(\014ciently)c(c)-5 b(omputable,)34 b(and)f(the)g(output)g(is)f(indistinguishable)h(fr)-5 b(om)34 b(a)e(r)-5 b(andom)35 b(p)-5 b(ermutation,)35 b(given)150 3773 y(only)k(or)-5 b(acle)39 b(ac)-5 b(c)g(ess)38 b(to)h Fn(P)1071 3787 y Fl(s)1108 3773 y Fr(.)57 b(Mor)-5 b(e)38 b(pr)-5 b(e)g(cisely,)40 b(let)e Ft(\005\(1)2115 3740 y Fl(k)2158 3773 y Ft(\))g Fr(b)-5 b(e)38 b(the)g(uniform)h (distribution)f(of)g(al)5 b(l)39 b(p)-5 b(er-)150 3886 y(mutations)31 b(over)g Fo(f)p Ft(0)p Fn(;)15 b Ft(1)p Fo(g)991 3853 y Fl(k)1036 3886 y Fr(.)40 b(Then)31 b Fn(P)1397 3900 y Fm(\001)1421 3886 y Ft(\()p Fo(\001)p Ft(\))f Fr(is)g(a)g(pseudo-r)-5 b(andom)33 b(p)-5 b(ermutation)33 b(if)c Fo(8)h Fr(PPTF)f Fo(f)p Fn(A)3409 3901 y Fl(k)3452 3886 y Fo(g)p Fr(,)i(ther)-5 b(e)150 3999 y(exists)33 b(a)g(ne)-5 b(gligible)32 b(function)h Fn(\027)6 b Ft(\()p Fn(k)s Ft(\))p Fr(,)33 b(such)g(that)733 4316 y Ft(Pr)o([)p Fn(s)26 b Fo( )f(f)p Ft(0)p Fn(;)15 b Ft(1)p Fo(g)1260 4278 y Fl(k)1305 4316 y Ft(;)g Fn(O)1414 4330 y Fk(0)1479 4316 y Ft(=)25 b Fn(P)1633 4330 y Fl(s)1670 4316 y Ft(;)15 b Fn(O)1779 4330 y Fk(1)1845 4316 y Fo( )25 b Ft(\005\(1)2109 4278 y Fl(k)2152 4316 y Ft(\))84 b(;)1225 4454 y Fn(b)25 b Fo( )g(f)p Ft(0)p Fn(;)15 b Ft(1)p Fo(g)p Ft(;)g Fn(b)1704 4416 y Fm(0)1755 4454 y Fo( )26 b Fn(A)1940 4411 y Fl(O)1994 4423 y Fj(b)1940 4483 y Fl(k)2029 4454 y Ft(\(1)2109 4416 y Fl(k)2152 4454 y Ft(\))84 b(:)f Fn(b)25 b Ft(=)g Fn(b)2578 4416 y Fm(0)2602 4454 y Ft(])g Fo(\024)g Ft(1)p Fn(=)p Ft(2)d(+)d Fn(\027)6 b Ft(\()p Fn(k)s Ft(\))150 4666 y Fu(De\014nition)35 b(3)46 b Fr(A)e(family)h(of)f(p)-5 b(ermutations)47 b Fn(P)1871 4680 y Fm(\001)1895 4666 y Ft(\()p Fo(\001)p Ft(\))e Fr(is)g(str)-5 b(ongly)46 b(pseudo-r)-5 b(andom)47 b(if)d(it)g(is)g(pseudo-)150 4779 y(r)-5 b(andom,)34 b(and)d(its)g(output)h(is)f(indistinguishable)h (fr)-5 b(om)32 b(r)-5 b(andom)33 b(given)e(or)-5 b(acle)32 b(ac)-5 b(c)g(ess)32 b(to)f(b)-5 b(oth)32 b Fn(P)3538 4793 y Fl(s)3606 4779 y Fr(and)150 4892 y Fn(P)221 4859 y Fm(\000)p Fk(1)208 4914 y Fl(s)315 4892 y Fr(.)42 b(Mor)-5 b(e)33 b(pr)-5 b(e)g(cisely,)1867 5416 y Ft(16-5)p eop %%Page: 6 6 6 5 bop 733 345 a Ft(Pr)o([)p Fn(s)26 b Fo( )f(f)p Ft(0)p Fn(;)15 b Ft(1)p Fo(g)1260 308 y Fl(k)1305 345 y Ft(;)g Fn(O)1414 359 y Fk(0)1479 345 y Ft(=)25 b Fn(P)1633 359 y Fl(s)1670 345 y Ft(;)15 b Fn(O)1779 359 y Fk(1)1845 345 y Fo( )25 b Ft(\005\(1)2109 308 y Fl(k)2152 345 y Ft(\))84 b(;)1067 518 y Fn(b)25 b Fo( )h(f)p Ft(0)p Fn(;)15 b Ft(1)p Fo(g)p Ft(;)g Fn(b)1547 481 y Fm(0)1598 518 y Fo( )25 b Fn(A)1782 463 y Fl(O)1836 475 y Fj(b)1867 463 y Fl(;O)1943 433 y Fg(\000)p Fh(1)1941 487 y Fj(b)1782 548 y Fl(k)2029 518 y Ft(\(1)2109 481 y Fl(k)2152 518 y Ft(\))84 b(:)f Fn(b)25 b Ft(=)g Fn(b)2578 481 y Fm(0)2602 518 y Ft(])g Fo(\024)g Ft(1)p Fn(=)p Ft(2)d(+)d Fn(\027)6 b Ft(\()p Fn(k)s Ft(\))150 759 y Fp(3.2)112 b(Construction)150 931 y Ft(The)38 b(follo)m(wing)f(construction)i(of)g(a)g(PRP)f(on)h (the)g(basis)e(of)i(a)g(one-w)m(a)m(y)i(pseudo-random)c(function)150 1044 y Fn(F)208 1058 y Fl(s)245 1044 y Ft(\()p Fn(x)p Ft(\))26 b(:)g Fo(f)p Ft(0)p Fn(;)15 b Ft(1)p Fo(g)664 1011 y Fk(2)p Fl(k)769 1044 y Fo(!)25 b(f)p Ft(0)p Fn(;)15 b Ft(1)p Fo(g)1105 1011 y Fk(2)p Fl(k)1214 1044 y Ft(is)27 b(due)h(to)h(Lub)m(y)f(and)g(Rac)m(k)m(o\013.)42 b(First,)28 b(separate)i(the)e(input)f Fn(x)h Ft(in)m(to)h(t)m(w)m(o)150 1156 y(parts,)f(denoted)e(b)m(y)h Fn(L)930 1170 y 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Fi(\001)p Fp(\))f Fn(is)g(a)g(pseudo-r)-5 b(andom)33 b(p)-5 b(ermutation)33 b(if)c Fi(8)h Fn(PPTF)f Fi(f)p Fk(A)3409 1959 y Fg(k)3452 1944 y Fi(g)p Fn(,)i(ther)-5 b(e)150 2056 y(exists)33 b(a)g(ne)-5 b(gligible)32 b(function)h Fk(\027)6 b Fp(\()p Fk(k)s Fp(\))p Fn(,)33 b(such)g(that)733 2374 y Fp(Pr)o([)p Fk(s)26 b Fi( )f(f)p Fp(0)p Fk(;)15 b Fp(1)p Fi(g)1260 2336 y Fg(k)1305 2374 y Fp(;)g Fk(O)1414 2388 y Fh(0)1479 2374 y Fp(=)25 b Fk(P)1633 2388 y Fg(s)1670 2374 y Fp(;)15 b Fk(O)1779 2388 y Fh(1)1845 2374 y Fi( )25 b Fp(\005\(1)2109 2336 y Fg(k)2152 2374 y Fp(\))84 b(;)1225 2511 y Fk(b)25 b Fi( )g(f)p Fp(0)p Fk(;)15 b Fp(1)p Fi(g)p Fp(;)g Fk(b)1704 2474 y Fj(0)1755 2511 y Fi( )26 b Fk(A)1940 2468 y Fg(O)1994 2480 y Ff(b)1940 2541 y Fg(k)2029 2511 y Fp(\(1)2109 2474 y Fg(k)2152 2511 y Fp(\))84 b(:)f Fk(b)25 b Fp(=)g Fk(b)2578 2474 y Fj(0)2602 2511 y Fp(])g Fi(\024)g Fp(1)p Fk(=)p Fp(2)d(+)d Fk(\027)6 b Fp(\()p Fk(k)s Fp(\))150 2724 y Fq(De\014nition)35 b(2)46 b Fn(A)e(family)h(of) f(p)-5 b(ermutations)47 b Fk(P)1871 2738 y Fj(\001)1895 2724 y Fp(\()p Fi(\001)p Fp(\))e Fn(is)g(str)-5 b(ongly)46 b(pseudo-r)-5 b(andom)47 b(if)d(it)g(is)g(pseudo-)150 2837 y(r)-5 b(andom,)34 b(and)d(its)g(output)h(is)f(indistinguishable)h (fr)-5 b(om)32 b(r)-5 b(andom)33 b(given)e(or)-5 b(acle)32 b(ac)-5 b(c)g(ess)32 b(to)f(b)-5 b(oth)32 b Fk(P)3538 2851 y Fg(s)3606 2837 y Fn(and)150 2950 y Fk(P)221 2917 y Fj(\000)p Fh(1)208 2972 y Fg(s)315 2950 y Fn(.)42 b(Mor)-5 b(e)33 b(pr)-5 b(e)g(cisely,)733 3267 y Fp(Pr)o([)p Fk(s)26 b Fi( )f(f)p Fp(0)p Fk(;)15 b Fp(1)p Fi(g)1260 3229 y Fg(k)1305 3267 y Fp(;)g Fk(O)1414 3281 y Fh(0)1479 3267 y Fp(=)25 b Fk(P)1633 3281 y Fg(s)1670 3267 y Fp(;)15 b Fk(O)1779 3281 y Fh(1)1845 3267 y Fi( )25 b Fp(\005\(1)2109 3229 y Fg(k)2152 3267 y Fp(\))84 b(;)1067 3440 y Fk(b)25 b Fi( )h(f)p Fp(0)p Fk(;)15 b Fp(1)p Fi(g)p Fp(;)g Fk(b)1547 3402 y Fj(0)1598 3440 y Fi( )25 b Fk(A)1782 3385 y Fg(O)1836 3397 y Ff(b)1867 3385 y Fg(;O)1943 3355 y Fe(\000)p Fd(1)1941 3408 y Ff(b)1782 3469 y Fg(k)2029 3440 y Fp(\(1)2109 3402 y Fg(k)2152 3440 y Fp(\))84 b(:)f Fk(b)25 b Fp(=)g Fk(b)2578 3402 y Fj(0)2602 3440 y Fp(])g Fi(\024)g Fp(1)p Fk(=)p Fp(2)d(+)d Fk(\027)6 b Fp(\()p Fk(k)s Fp(\))150 3683 y Fl(1.2)112 b(Construction)150 3855 y Fp(The)38 b(follo)m(wing)f(construction)i(of)g(a)g(PRP)f(on)h(the)g(basis)e(of)i (a)g(one-w)m(a)m(y)i(pseudo-random)c(function)150 3968 y Fk(F)208 3982 y Fg(s)245 3968 y Fp(\()p Fk(x)p Fp(\))26 b(:)g Fi(f)p Fp(0)p Fk(;)15 b Fp(1)p Fi(g)664 3935 y Fh(2)p Fg(k)769 3968 y Fi(!)25 b(f)p Fp(0)p Fk(;)15 b Fp(1)p Fi(g)1105 3935 y Fh(2)p Fg(k)1214 3968 y Fp(is)27 b(due)h(to)h(Lub)m(y)f(and)g(Rac)m(k)m(o\013.)42 b(First,)28 b(separate)i(the)e(input)f Fk(x)h Fp(in)m(to)h(t)m(w)m(o)150 4081 y(parts,)f(denoted)e(b)m(y)h Fk(L)930 4095 y Fh(0)996 4081 y Fp(and)g Fk(R)1239 4095 y Fh(0)1278 4081 y Fp(,)h(of)f(equal)f (length:)39 b Fk(x)25 b Fp(=)g Fk(L)2214 4095 y Fh(0)2266 4081 y Fi(\016)13 b Fk(R)2393 4095 y Fh(0)2433 4081 y Fk(;)i Fi(j)p Fk(L)2560 4095 y Fh(0)2600 4081 y Fi(j)26 b Fp(=)f Fi(j)p Fk(R)2841 4095 y Fh(0)2880 4081 y Fi(j)p Fp(.)40 b(Then)26 b(w)m(e)h(can)g(obtain)150 4193 y(\()p Fk(L)247 4207 y Fg(i)p Fh(+1)366 4193 y Fk(;)15 b(R)475 4207 y Fg(i)p Fh(+1)593 4193 y Fp(\))31 b(from)f(\()p Fk(L)971 4207 y Fg(i)999 4193 y Fk(;)15 b(R)1108 4207 y Fg(i)1137 4193 y Fp(\))31 b(as)f(follo)m(ws:)1867 5416 y(17-1)p eop %%Page: 2 2 2 1 bop 1452 150 a Fk(L)1514 164 y Fg(i)2407 150 y Fk(R)2476 164 y Fg(i)p 2446 516 4 324 v 2448 516 a Fc(?)2314 783 y Fb(\036\035)2314 517 y(\037)o(\034)2390 673 y Fk(F)2448 687 y Fg(s)2481 697 y Ff(i)p 2446 897 4 117 v 2448 897 a Fc(?)2448 939 y Fb(i)p 2446 980 4 84 v 2407 941 84 4 v 2446 1063 4 84 v 2448 1063 a Fc(?)2365 1146 y Fk(R)2434 1160 y Fg(i)p Fh(+1)1535 295 y Fc(Z)1618 358 y(Z)1701 420 y(Z)1784 482 y(Z)1867 545 y(Z)1950 607 y(Z)2033 669 y(Z)2116 731 y(Z)2199 794 y(Z)2282 856 y(Z)2340 899 y(Z)-83 b(~)2365 275 y(\000)2282 358 y(\000)2199 441 y(\000)2116 524 y(\000)2033 607 y(\000)1950 690 y(\000)1867 773 y(\000)1784 856 y(\000)1701 939 y(\000)1618 1022 y(\000)1593 1047 y(\000)g(\011)1452 1146 y Fk(L)1514 1160 y Fg(i)p Fh(+1)291 1525 y Fp(More)36 b(precisely)-8 b(,)36 b(\()p Fk(L)1024 1539 y Fg(i)p Fh(+1)1143 1525 y Fk(;)15 b(R)1252 1539 y Fg(i)p Fh(+1)1371 1525 y Fp(\))34 b(=)f(\()p Fk(R)1648 1539 y Fg(i)1676 1525 y Fk(;)15 b(L)1778 1539 y Fg(i)1830 1525 y Fi(\010)24 b Fk(F)1983 1539 y Fg(s)2016 1549 y Ff(i)2046 1525 y Fp(\()p Fk(R)2150 1539 y Fg(i)2179 1525 y Fp(\)\),)37 b(where)e Fk(s)2622 1539 y Fh(0)2685 1525 y Fi(\016)24 b Fk(s)2797 1539 y Fh(1)2860 1525 y Fi(\016)g(\001)15 b(\001)g(\001)34 b Fp(=)g Fk(s)p Fp(,)i(the)g(seed)f(for)150 1638 y(the)k(PRP)-8 b(.)39 b(The)f(Lub)m(y-Rac)m(k)m(o\013)i (construction)e(uses)g(rep)s(eated)h(applications)e(of)i(this)e(round)g (\(often)150 1751 y(referred)30 b(to)h(as)f(a)h(F)-8 b(eistel)31 b(round\))e(to)i(construct)g(a)g(PRP)f(and)f(SPRP)-8 b(.)291 1864 y(W)g(e)32 b(sa)m(w)f(last)f(time)h(that)g(this)f(results) f(in)g(a)i(p)s(erm)m(utation.)41 b(No)m(w)31 b(let)g(us)f(see)h(wh)m(y) f(rep)s(eating)g(this)150 1977 y(will)e(result)h(in)g(a)i(PRP)-8 b(.)291 2090 y(Lub)m(y)30 b(and)f(Rac)m(k)m(o\013)k(pro)m(v)m(ed)e(the) f(follo)m(wing)f(t)m(w)m(o)j(theorems:)150 2302 y Fq(Theorem)i(3)46 b Fk(LR)20 b Fi(\000)g Fp(3)26 b(=)f Fk(P)1159 2316 y Fh(3)1199 2302 y Fp(\()p Fk(x;)15 b(y)s Fp(\))33 b Fn(is)g(a)g(PRP,)f (for)h(r)-5 b(andom)35 b Fk(s)2388 2316 y Fh(1)2427 2302 y Fk(;)15 b(s)2510 2316 y Fh(2)2550 2302 y Fk(;)g(s)2633 2316 y Fh(3)2672 2302 y Fn(.)150 2515 y Fq(Theorem)34 b(4)150 2665 y Fk(LR)21 b Fi(\000)f Fp(4)25 b(=)g Fk(P)617 2679 y Fh(4)657 2665 y Fp(\()p Fk(x;)15 b(y)s Fp(\))34 b Fn(is)e(a)h(SPRP,)f(for)i(r)-5 b(andom)35 b Fk(s)1898 2679 y Fh(1)1937 2665 y Fk(;)15 b(:)g(:)g(:)h(;)f(s)2181 2679 y Fh(4)2221 2665 y Fn(.)291 2877 y Fp(W)-8 b(e)32 b(will)27 b(no)m(w)k(pro)m(v)m(e)g(Theorem)f(3;)h(Theorem)f(4)h(will)d (b)s(e)h(a)i(homew)m(ork)g(problem.)150 3164 y Fm(2)135 b(Pro)t(of)45 b(for)g(three)g(rounds)150 3367 y Fp(The)30 b(pro)s(of)f(is)h(brok)m(en)g(up)f(in)m(to)i(the)f(follo)m(wing)f (three)i(claims:)150 3579 y Fq(Claim)i(5)46 b Fn(If)33 b(the)h(PRFs)f Fk(F)1117 3593 y Fh(1)1157 3579 y Fn(,)g Fk(F)1276 3593 y Fh(2)1349 3579 y Fn(and)i Fk(F)1585 3593 y Fh(3)1658 3579 y Fn(ar)-5 b(e)34 b(r)-5 b(eplac)g(e)g(d)35 b(with)f(r)-5 b(andom)36 b(functions)e(in)f(the)g(LR-3)h(c)-5 b(on-)150 3692 y(struction,)27 b(then)f(the)g(r)-5 b(esulting)26 b(function)f(wil)5 b(l)26 b(b)-5 b(e)25 b(indistinguishable)h(fr)-5 b(om)26 b(a)g(r)-5 b(andom)28 b(function.)39 b(\(Cal)5 b(l)150 3805 y(this)47 b(r)-5 b(esulting)46 b(c)-5 b(onstruction)48 b(LR-3R\).)f(Mor)-5 b(e)46 b(pr)-5 b(e)g(cisely,)51 b(let)46 b Fi(F)9 b Fp(\()p Fk(n)p Fp(\))47 b Fn(b)-5 b(e)46 b(the)g(set)h(of)f (al)5 b(l)47 b(functions)150 3918 y Fk(F)89 b Fp(:)76 b Fi(f)p Fp(0)p Fk(;)15 b Fp(1)p Fi(g)621 3876 y Fg(n)704 3918 y Fi(7!)37 b(f)p Fp(0)p Fk(;)15 b Fp(1)p Fi(g)1055 3876 y Fg(n)1102 3918 y Fn(.)61 b(L)-5 b(et)39 b Fi(LR)p Fn(-3R)p Fp(\(2)p Fk(n)p Fp(\))h Fn(b)-5 b(e)39 b(the)g(set)h(of)f(al)5 b(l)39 b(functions)h(obtaine)-5 b(d)40 b(by)f(cho)-5 b(osing)150 4031 y Fk(F)208 4045 y Fg(i)263 4031 y Fi( )27 b(F)9 b Fp(\()p Fk(n)p Fp(\))p Fn(,)34 b Fp(1)27 b Fi(\024)g Fk(i)g Fi(\024)f Fp(3)34 b Fn(and)g(using)f(these)h Fk(F)1752 4045 y Fh(1)1792 4031 y Fn(,)f Fk(F)1911 4045 y Fh(2)1951 4031 y Fn(,)h Fk(F)2071 4045 y Fh(3)2144 4031 y Fn(in)f(the)h(LR-3R)g (c)-5 b(onstruction.)46 b(Then)33 b(for)h(al)5 b(l)150 4144 y Fk(m)p Fn(,)32 b(for)h(al)5 b(l)34 b(or)-5 b(acle)34 b(machines)f Fi(D)1304 4111 y Fh(?)1374 4144 y Fn(that)h(make)f Fk(m)f Fn(queries)411 4348 y Fp(Pr)o([)p Fk(O)602 4362 y Fh(0)668 4348 y Fi( )25 b(F)9 b Fp(\(2)p Fk(n)p Fp(\);)15 b Fk(O)1137 4362 y Fh(1)1203 4348 y Fi( )25 b(LR)p Fn(-3R)p Fp(\(2)p Fk(n)p Fp(\);)15 b Fk(b)27 b Fi( )e(f)p Fp(0)p Fk(;)15 b Fp(1)p Fi(g)r Fp(;)g Fk(b)2297 4311 y Fj(0)2346 4348 y Fi( )25 b(D)2535 4311 y Fg(O)2589 4323 y Ff(b)2624 4348 y Fp(\(1)2704 4311 y Fg(n)2752 4348 y Fp(\))h(:)f Fk(b)g Fp(=)g Fk(b)3062 4311 y Fj(0)3086 4348 y Fp(])g Fi(\024)g Fk(m)3312 4311 y Fh(2)3351 4348 y Fk(=)p Fp(2)3441 4311 y Fg(n)291 4561 y Fp(Pro)s(of)30 b(of)h(this)e(claim)g(will)f(tak) m(e)k(the)f(bulk)d(of)j(the)f(lecture.)150 4773 y Fq(Claim)j(6)46 b Fn(If)33 b(LR-3R)g(is)f(indistinguishable)i(fr)-5 b(om)34 b(a)f(r)-5 b(andom)35 b(function,)e(then)g(so)g(is)f(LR-3.)291 4986 y Fp(This)d(is)g(easy)i(to)g(see.)1867 5416 y(17-2)p eop %%Page: 3 3 3 2 bop 150 159 a Fq(Claim)33 b(7)46 b Fn(L)-5 b(et)36 b Fi(F)9 b Fp(\()p Fk(k)s Fp(\))37 b Fn(b)-5 b(e)35 b(the)h(set)f(of)h (al)5 b(l)36 b(functions)g Fk(F)79 b Fp(:)66 b Fi(f)p Fp(0)p Fk(;)15 b Fp(1)p Fi(g)2448 117 y Fg(k)2521 159 y Fi(7!)31 b(f)p Fp(0)p Fk(;)15 b Fp(1)p Fi(g)2865 117 y Fg(k)2907 159 y Fn(.)51 b(L)-5 b(et)36 b Fi(P)7 b Fp(\()p Fk(k)s Fp(\))37 b Fn(b)-5 b(e)35 b(the)h(set)150 271 y(of)d(al)5 b(l)34 b(p)-5 b(ermutations)35 b(over)e Fi(f)p Fp(0)p Fk(;)15 b Fp(1)p Fi(g)1358 230 y Fg(k)1401 271 y Fn(.)43 b(Then,)33 b(for)h(al)5 b(l)33 b(or)-5 b(acle)34 b(machines)g Fi(D)2752 238 y Fh(?)2822 271 y Fn(that)g(make)g(a)f(p)-5 b(olynomial)150 384 y(numb)g(er)33 b(of)g(queries,)f(ther)-5 b(e)33 b(exists)g(a)g(ne)-5 b(gligible)33 b(function)f Fk(\027)6 b Fp(\()p Fk(k)s Fp(\))34 b Fn(such)e(that)613 589 y Fp(Pr)o([)p Fk(O)804 603 y Fh(0)869 589 y Fi( )26 b(F)9 b Fp(\()p Fk(k)s Fp(\);)15 b Fk(O)1289 603 y Fh(1)1355 589 y Fi( )26 b(P)7 b Fp(\()p Fk(k)s Fp(\);)15 b Fk(b)27 b Fi( )e(f)p Fp(0)p Fk(;)15 b Fp(1)p Fi(g)r Fp(;)g Fk(b)2185 551 y Fj(0)2234 589 y Fi( )25 b(D)2423 551 y Fg(O)2477 563 y Ff(b)2512 589 y Fp(\(1)2592 551 y Fg(k)2636 589 y Fp(\))g(:)h Fk(b)f Fp(=)g Fk(b)2946 551 y Fj(0)2969 589 y Fp(])h Fi(\024)f Fk(\027)6 b Fp(\()p Fk(k)s Fp(\))291 801 y(This)29 b(last)h(claim)f(will)f(b)s(e)i(left)g(as)h(a)f(homew)m (ork)h(problem.)291 914 y(If)c(w)m(e)h(com)m(bine)g(the)f(claims,)h(w)m (e)g(get)g(our)g(theorem,)g(b)s(ecause)g(then)f(it)g(will)e(follo)m(w,) j(b)m(y)f(the)h(h)m(ybrid)150 1027 y(argumen)m(t,)39 b(that)e(an)f(adv)m(ersary)h(that)g(distinguishes)c(LR-3)k(from)f(a)h (random)f(p)s(erm)m(utation,)h(either)150 1140 y(distinguishes)g(LR-3) 42 b(from)e(LR-3R)h(\(hence)h(breaks)e(the)i(underlying)37 b(PRF,)42 b(b)m(y)e(Claim)g(6\))h(or)g(dis-)150 1253 y(tinguishes)33 b(LR-3R)j(form)f(a)h(random)f(function)f(\(whic)m(h)h (con)m(tradicts)h(Claim)e(5\))i(or)g(distinguishes)c(a)150 1366 y(random)e(function)f(from)h(a)g(random)g(p)s(erm)m(utation)f (\(whic)m(h)h(con)m(tradicts)h(Claim)d(7\).)150 1528 y Fq(Pro)s(of)36 b(of)f(Claim)e(5:)291 1641 y Fp(Supp)s(ose)j(a)i (distinguisher)c Fi(D)40 b Fp(mak)m(es)f Fk(m)e Fp(queries.)62 b(Assume)37 b(WLOG)h(that)g(all)f(his)f(queries)h(are)150 1754 y(distinct.)291 1867 y(Let)25 b Fk(A)f Fp(b)s(e)f(an)h(oracle)g (that)h(acts)g(lik)m(e)e(a)h(random)f(function)g(for)g(this)g (distinguisher)d(with)j Fk(m)h Fp(queries:)150 1980 y Fi(f)p Fp(0)p Fk(;)15 b Fp(1)p Fi(g)370 1947 y Fh(2)p Fg(n)480 1980 y Fi(7!)25 b(f)p Fp(0)p Fk(;)15 b Fp(1)p Fi(g)816 1947 y Fh(2)p Fg(n)900 1980 y Fp(.)40 b(W)-8 b(e)28 b(can)f(imagine)e(that)j Fk(A)f Fp(w)m(orks)f(lik)m(e)g(this:)38 b(Its)27 b(random)f(tap)s(e)h(consists)f(of)h(2)p Fk(m)150 2093 y Fp(strings)j(of)h(length)f Fk(n)p Fp(,)h(denote)h(them)f Fi(f)p Fk(Y)1563 2107 y Fg(i)1591 2093 y Fk(;)15 b(Z)1693 2107 y Fg(i)1779 2093 y Fp(:)58 b(1)27 b Fi(\024)f Fk(i)g Fi(\024)g Fk(m)p Fi(g)p Fp(.)43 b(On)31 b(input)e Fk(L)2836 2107 y Fg(i)2864 2093 y Fp(,)i Fk(R)2989 2107 y Fg(i)3017 2093 y Fp(,)h Fk(A)f Fp(will)d(output)j Fk(Y)3697 2107 y Fg(i)3725 2093 y Fp(,)150 2206 y Fk(Z)212 2220 y Fg(i)240 2206 y Fp(.)41 b(It's)30 b(clear)h(that)g Fk(A)p Fp('s)f(b)s(eha)m (vior)g(is)f(the)i(same)g(as)f(that)h(of)g(a)g(random)e(function.)291 2319 y(Let)24 b Fk(B)k Fp(b)s(e)23 b(an)g(oracle)h(that)g(acts)g(lik)m (e)f(the)g(three-round)g(Lub)m(y-Rac)m(k)m(o\013)i(cipher)d(with)g (three)i(random)150 2432 y(functions)29 b(sitting)g(in)g(it.)41 b(Let)30 b(us)g(lo)s(ok)g(at)h Fk(B)5 b Fp('s)30 b(co)s(de:)150 2644 y Fq(Fix)35 b(random)f(tap)s(e)45 b Fp(as)31 b(consisting)e(of)h (3)p Fk(m)h Fp(strings)e Fi(f)p Fk(X)2170 2658 y Fg(i)2199 2644 y Fk(;)15 b(Y)2292 2658 y Fg(i)2320 2644 y Fk(;)g(Z)2422 2658 y Fg(i)2507 2644 y Fp(:)55 b(1)26 b Fi(\024)f Fk(i)h Fi(\024)f Fk(m)p Fi(g)p Fp(.)150 2832 y Fq(F)-9 b(or)35 b Fp(1)26 b Fi(\024)f Fk(i)g Fi(\024)g Fk(m)p Fq(,)35 b(on)g(input)g(\()p Fk(L)1327 2846 y Fg(i)1355 2832 y Fq(,)g Fk(R)1488 2846 y Fg(i)1516 2832 y Fq(\))45 b Fp(do:)486 3019 y Fi(\017)h Fp(Let)31 b Fk(l)767 3033 y Fg(R)855 3019 y Fp(b)s(e)f(the)h(least)f Fk(l)i Fp(index)d(suc)m(h)h(that)h Fk(R)2123 3033 y Fg(i)2177 3019 y Fp(=)25 b Fk(R)2342 3034 y Fg(l)2368 3019 y Fp(.)41 b(Let)30 b Fk(\013)2654 3033 y Fg(i)2708 3019 y Fp(=)25 b Fk(L)2866 3033 y Fg(i)2914 3019 y Fi(\010)20 b Fk(X)3080 3034 y Fg(l)3101 3045 y Ff(R)3156 3019 y Fp(.)486 3166 y Fi(\017)46 b Fp(Let)31 b Fk(L)802 3180 y Fg(\013)882 3166 y Fp(b)s(e)f(the)g(least)h(index)e Fk(l)j Fp(suc)m(h)e(that)h Fk(\013)2139 3180 y Fg(i)2193 3166 y Fp(=)24 b Fk(\013)2346 3181 y Fg(l)2373 3166 y Fp(.)40 b(Let)31 b Fk(\014)2652 3180 y Fg(i)2706 3166 y Fp(=)25 b Fk(R)2871 3180 y Fg(i)2919 3166 y Fi(\010)20 b Fk(Y)3063 3181 y Fg(l)3084 3189 y Ff(\013)3132 3166 y Fp(.)486 3312 y Fi(\017)46 b Fp(Let)31 b Fk(L)802 3327 y Fg(\014)879 3312 y Fp(b)s(e)f(the)h(least)f(index)f Fk(l)k Fp(suc)m(h)d(that)h Fk(\014)2130 3326 y Fg(i)2183 3312 y Fp(=)25 b Fk(\014)2330 3327 y Fg(l)2357 3312 y Fp(.)40 b(Let)31 b Fk(\015)2632 3326 y Fg(i)2686 3312 y Fp(=)25 b Fk(\013)2840 3326 y Fg(i)2888 3312 y Fi(\010)20 b Fk(\014)3030 3327 y Fg(l)3051 3339 y Ff(\014)3097 3312 y Fp(.)486 3458 y Fi(\017)46 b Fp(Output)30 b(\()p Fk(\014)987 3472 y Fg(i)1015 3458 y Fk(;)15 b(\015)1102 3472 y Fg(i)1131 3458 y Fp(\).)291 3670 y(Consider)29 b(a)h(di\013eren)m(t)g (description)f(for)h Fk(B)5 b Fp(,)30 b(call)g(it)f Fk(B)2170 3637 y Fj(0)2193 3670 y Fp(:)150 3883 y Fq(Fix)35 b(random)f(tap)s(e)45 b Fp(as)31 b(consisting)e(of)h(3)p Fk(m)h Fp(strings)e Fi(f)p Fk(X)2170 3897 y Fg(i)2199 3883 y Fk(;)15 b(Y)2292 3897 y Fg(i)2320 3883 y Fk(;)g(Z)2422 3897 y Fg(i)2507 3883 y Fp(:)55 b(1)26 b Fi(\024)f Fk(i)h Fi(\024)f Fk(m)p Fi(g)p Fp(.)150 4071 y Fq(F)-9 b(or)35 b Fp(1)26 b Fi(\024)f Fk(i)g Fi(\024)g Fk(m)p Fq(,)35 b(on)g(input)g(\()p Fk(L)1327 4085 y Fg(i)1355 4071 y Fq(,)g Fk(R)1488 4085 y Fg(i)1516 4071 y Fq(\))45 b Fp(do:)486 4258 y Fi(\017)h Fp(Let)36 b Fk(l)772 4272 y Fg(R)864 4258 y Fp(b)s(e)e(the)h(least)h Fk(l)g Fp(index)e(suc)m(h)g(that)i Fk(R)2164 4272 y Fg(i)2224 4258 y Fp(=)d Fk(R)2397 4273 y Fg(l)2423 4258 y Fp(.)54 b(Let)35 b(If)g Fk(l)2792 4272 y Fg(R)2882 4258 y Fp(=)d Fk(i)k Fp(then)e(let)h Fk(\013)3457 4272 y Fg(i)3518 4258 y Fp(=)d Fk(X)3696 4272 y Fg(i)3725 4258 y Fp(.)577 4371 y(Otherwise,)e(let)g Fk(\013)1222 4385 y Fg(i)1275 4371 y Fp(=)25 b Fk(X)1446 4386 y Fg(l)1467 4397 y Ff(R)1542 4371 y Fi(\010)20 b Fk(L)1695 4386 y Fg(l)1716 4397 y Ff(R)1791 4371 y Fi(\010)g Fk(L)1944 4385 y Fg(i)1972 4371 y Fp(.)486 4517 y Fi(\017)46 b Fp(Let)e Fk(l)780 4531 y Fg(\013)872 4517 y Fp(b)s(e)e(the)h(least)g(index)e Fk(l)k Fp(suc)m(h)d(that)i Fk(\013)2216 4531 y Fg(i)2290 4517 y Fp(=)i Fk(\013)2465 4532 y Fg(l)2491 4517 y Fp(.)78 b(If)42 b Fk(l)2724 4531 y Fg(\013)2819 4517 y Fp(=)k Fk(i)p Fp(,)g(then)d(let)g Fk(\014)3453 4531 y Fg(i)3527 4517 y Fp(=)j Fk(Y)3697 4531 y Fg(i)3725 4517 y Fp(.)577 4630 y(Otherwise,)30 b(let)g Fk(\014)1215 4644 y Fg(i)1269 4630 y Fp(=)25 b Fk(Y)1418 4645 y Fg(l)1439 4653 y Ff(\013)1507 4630 y Fi(\010)20 b Fk(R)1667 4645 y Fg(l)1688 4653 y Ff(\013)1757 4630 y Fi(\010)g Fk(R)1917 4644 y Fg(i)1945 4630 y Fp(.)486 4776 y Fi(\017)46 b Fp(Let)e Fk(l)780 4791 y Fg(\014)871 4776 y Fp(b)s(e)e(the)i(least)f(index)f Fk(l)j Fp(suc)m(h)e(that)h Fk(\014)2211 4790 y Fg(i)2286 4776 y Fp(=)j Fk(\014)2455 4791 y Fg(l)2481 4776 y Fp(.)80 b(If)42 b Fk(l)2716 4791 y Fg(\014)2810 4776 y Fp(=)47 b Fk(i)p Fp(,)g(then)c(let)g Fk(\015)3442 4790 y Fg(i)3517 4776 y Fp(=)k Fk(Z)3697 4790 y Fg(i)3725 4776 y Fp(.)577 4889 y(Otherwise,)30 b(let)g Fk(\015)1211 4903 y Fg(i)1264 4889 y Fp(=)25 b Fk(\013)1418 4903 y Fg(i)1467 4889 y Fi(\010)20 b Fk(Z)1620 4904 y Fg(l)1641 4916 y Ff(\014)1707 4889 y Fi(\010)g Fk(\013)1856 4904 y Fg(l)1877 4916 y Ff(\014)1923 4889 y Fp(.)486 5035 y Fi(\017)46 b Fp(Output)30 b(\()p Fk(\014)987 5049 y Fg(i)1015 5035 y Fk(;)15 b(\015)1102 5049 y Fg(i)1131 5035 y Fp(\).)1867 5416 y(17-3)p eop %%Page: 4 4 4 3 bop 291 159 a Fk(B)365 126 y Fj(0)428 159 y Fp(is)39 b(equiv)-5 b(alen)m(t)40 b(to)g Fk(B)5 b Fp(:)60 b(for)40 b(ev)m(ery)h(setting)f(of)g(the)g(random)f(tap)s(e)i(for)f Fk(B)k Fp(there)c(is)f(a)i(unique)150 271 y(setting)30 b(of)h(a)g(random)e(tap)s(e)i(for)f Fk(B)1379 238 y Fj(0)1402 271 y Fp(,)g(so)h(their)e(b)s(eha)m(vior)h(is)f(iden)m(tical.)291 384 y(In)d(turn,)h Fk(B)700 351 y Fj(0)723 384 y Fp('s)g(b)s(eha)m (vior)f(is)g(iden)m(tical)g(to)h(that)h(of)f Fk(A)g Fp(whenev)m(er)g (for)g(all)e Fk(i)p Fp(,)k Fk(l)2892 398 y Fg(\013)2941 384 y Fp(\()p Fk(i)p Fp(\))d(=)f Fk(i)j Fp(and)e Fk(l)3423 399 y Fg(\014)3470 384 y Fp(\()p Fk(i)p Fp(\))g(=)f Fk(i)p Fp(.)150 497 y(This)34 b(is)g(simply)f(b)s(ecause,)j(whenev)m(er)g (that)g(is)e(the)h(case,)j Fk(B)2284 464 y Fj(0)2342 497 y Fp(outputs)d(the)h(con)m(ten)m(ts)h(of)e(its)g(random)150 610 y(tap)s(e,)c(just)f(as)g Fk(A)h Fp(w)m(ould)e(do.)291 723 y(So,)39 b(w)m(e)e(wish)e(to)j(sho)m(w)f(that)g(the)g(probabilit)m (y)e(that,)k(for)e(some)g(1)g Fi(\024)f Fk(l)i(<)e(i)g Fi(\024)g Fk(m)p Fp(,)j Fk(l)3309 737 y Fg(\013)3358 723 y Fp(\()p Fk(i)p Fp(\))e(=)f Fk(l)j Fp(or)150 836 y Fk(l)177 851 y Fg(\014)224 836 y Fp(\()p Fk(i)p Fp(\))27 b(=)e Fk(l)r Fp(,)30 b(is)f(at)j(most)e Fk(m)1037 803 y Fh(2)1077 836 y Fk(=)p Fp(2)1167 803 y Fg(n)1214 836 y Fp(.)291 949 y(Supp)s(ose)f(for)h(all)f Fk(l)e(<)e(i)p Fp(,)31 b Fk(l)1177 963 y Fg(\013)1227 949 y Fp(\()p Fk(l)r Fp(\))26 b(=)f Fk(l)32 b Fp(and)e Fk(l)1711 964 y Fg(\014)1758 949 y Fp(\()p Fk(l)r Fp(\))c(=)f Fk(l)r Fp(.)40 b(\(As)31 b(is)e(the)i(case)g(for)f Fk(i)c Fp(=)f(2.\))291 1062 y(What)45 b(is)e(the)i(probabilit)m(y)d(that,)48 b(for)d(a)f(sp)s(eci\014c)f Fk(l)51 b(<)d(i)p Fp(,)g Fk(\013)2486 1077 y Fg(l)2561 1062 y Fp(=)g Fk(\013)2738 1076 y Fg(i)2766 1062 y Fp(?)83 b(F)-8 b(rom)44 b(the)h(description)150 1175 y(of)e Fk(B)340 1142 y Fj(0)363 1175 y Fp(,)j Fk(\013)492 1190 y Fg(l)565 1175 y Fp(=)g Fk(L)744 1190 y Fg(l)798 1175 y Fi(\010)28 b Fk(X)972 1193 y Fg(l)993 1204 y Ff(R)1044 1193 y Fh(\()p Fg(l)q Fh(\))1153 1175 y Fi(\010)h Fk(L)1315 1193 y Fg(l)1336 1204 y Ff(R)1386 1193 y Fh(\()p Fg(l)q Fh(\))1467 1175 y Fp(.)78 b Fk(\013)1628 1189 y Fg(i)1703 1175 y Fp(=)46 b Fk(L)1882 1189 y Fg(i)1938 1175 y Fi(\010)29 b Fk(X)2113 1193 y Fg(l)2134 1204 y Ff(R)2184 1193 y Fh(\()p Fg(i)p Fh(\))2296 1175 y Fi(\010)f Fk(L)2457 1193 y Fg(l)2478 1204 y Ff(R)2528 1193 y Fh(\()p Fg(i)p Fh(\))2612 1175 y Fp(.)78 b(Supp)s(ose)41 b Fk(R)3153 1190 y Fg(l)3225 1175 y Fp(=)46 b Fk(R)3411 1189 y Fg(i)3440 1175 y Fp(.)78 b(Then)150 1288 y Fk(X)225 1306 y Fg(l)246 1317 y Ff(R)297 1306 y Fh(\()p Fg(l)q Fh(\))399 1288 y Fi(\010)20 b Fk(L)552 1306 y Fg(l)573 1317 y Ff(R)623 1306 y Fh(\()p Fg(l)q Fh(\))732 1288 y Fp(=)27 b Fk(X)905 1306 y Fg(l)926 1317 y Ff(R)976 1306 y Fh(\()p Fg(i)p Fh(\))1080 1288 y Fi(\010)21 b Fk(L)1234 1306 y Fg(l)1255 1317 y Ff(R)1305 1306 y Fh(\()p Fg(i)p Fh(\))1388 1288 y Fp(,)32 b(and)f(therefore,)i Fk(\013)2090 1302 y Fg(i)2145 1288 y Fi(6)p Fp(=)27 b Fk(\013)2301 1303 y Fg(l)2359 1288 y Fp(since)k Fk(L)2645 1303 y Fg(l)2698 1288 y Fi(6)p Fp(=)c Fk(L)2858 1302 y Fg(i)2886 1288 y Fp(,)32 b(b)s(ecause)g(w)m(e)g (assumed)150 1401 y(that)43 b(all)d(queries)h(w)m(ere)i(distinct.)73 b(So,)45 b(supp)s(ose)c Fk(R)2023 1416 y Fg(l)2094 1401 y Fi(6)p Fp(=)j Fk(R)2278 1415 y Fg(i)2306 1401 y Fp(.)75 b(Then)41 b Fk(R)q Fp(\()p Fk(l)r Fp(\))k Fi(6)p Fp(=)f Fk(R)q Fp(\()p Fk(i)p Fp(\),)i(so)c Fk(\013)3407 1415 y Fg(i)3478 1401 y Fp(and)f Fk(\013)3724 1416 y Fg(l)150 1513 y Fp(are)32 b(indep)s(enden)m(tly)c(c)m(hosen)k(random)e(v)-5 b(ariables:)41 b Fk(l)1959 1527 y Fg(R)2017 1513 y Fp(\()p Fk(l)r Fp(\))27 b Fi(6)p Fp(=)g Fk(l)2268 1527 y Fg(R)2326 1513 y Fp(\()p Fk(i)p Fp(\),)32 b(so)g Fk(X)2672 1532 y Fg(l)2693 1543 y Ff(R)2743 1532 y Fh(\()p Fg(l)q Fh(\))2856 1513 y Fp(and)e Fk(X)3108 1532 y Fg(l)3129 1543 y Ff(R)3180 1532 y Fh(\()p Fg(i)p Fh(\))3294 1513 y Fp(are)i(random)150 1626 y(v)-5 b(ariables)19 b(that)j(are)f(indep)s(enden)m(t)d(of)j(eac)m (h)h(other)f(and)f(also)h(w)m(ere)g(c)m(hosen)g(indep)s(enden)m(tly)d (of)j(the)g(v)-5 b(alues)150 1739 y Fk(L)212 1754 y Fg(l)238 1739 y Fk(;)15 b(R)347 1754 y Fg(l)373 1739 y Fk(;)g(L)475 1758 y Fg(l)496 1769 y Ff(R)547 1758 y Fh(\()p Fg(l)q Fh(\))628 1739 y Fk(;)g(L)730 1753 y Fg(i)758 1739 y Fk(;)g(R)867 1753 y Fg(i)896 1739 y Fk(;)g(L)998 1758 y Fg(l)1019 1769 y Ff(R)1069 1758 y Fh(\()p Fg(i)p 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Fj(P)j(K)q(;)i(m;)g(\033)s Fo(\))26 b(=)f(1])h Fk(\024)f Fj(\027)6 b Fo(\()p Fj(k)s Fo(\))291 5167 y(This)29 b(de\014nition)f(is)h(due)h(to)h(Goldw)m (asser,)f(Micali,)g(Riv)m(est)g([2)q(].)1867 5416 y(18-1)p eop %%Page: 2 2 2 1 bop 150 159 a Fl(2)135 b(Digital)47 b(Signature)e(Sc)l(hemes)150 361 y Fo(Di\016e)26 b(and)g(Hellman)f(prop)s(osed)h(a)g(heuristic)f (signature)h(based)g(on)g(T)-8 b(rap)s(do)s(or)25 b(P)m(erm)m(utations) i(\(TDP\).)150 474 y(As)h(an)g(example)g(consider)f(a)h(public)e(k)m (ey)j(cryptosystem)f(\()p Fj(G;)15 b(E)5 b(;)15 b(D)s Fo(\))31 b(based)c(on)h(a)h(trap)s(do)s(or)e(function.)150 587 y(A)36 b(message)g Fj(m)g Fo(can)f(b)s(e)g(signed)g(with)f Fj(D)s Fo(\()p Fj(m)p Fo(\))g(=)g Fj(f)1942 554 y Ff(\000)p Fg(1)2035 587 y Fo(\()p Fj(m)p Fo(\),)k(where)d Fj(f)45 b Fo(is)34 b(the)i(user's)f(trap)s(do)s(or)g(public)150 700 y(function.)43 b(The)31 b(user)f(is)h(the)h(only)e(one)i(who)f(kno) m(ws)g(the)h(trap)s(do)s(or)e(information.)43 b(Th)m(us)30 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y(The)h(public)d(k)m(ey)j(pair)f(is)g(the)h(pair) f(\()p Fj(n;)15 b(e)p Fo(\))32 b(where)g Fj(n)f Fo(is)g(the)h(pro)s (duct)f(of)h(t)m(w)m(o)h(large)f(primes)e(and)i Fj(e)150 1716 y Fo(is)c(relativ)m(ely)h(prime)e(to)j Fj(\036)p Fo(\()p Fj(n)p Fo(\).)41 b(The)28 b(secret)i(k)m(ey)g(is)f Fj(d)g Fo(suc)m(h)g(that)g Fj(ed)2533 1691 y Fk(\030)2533 1721 y Fo(=)2629 1716 y(1)p Fj(mod\036)p Fo(\()p Fj(n)p Fo(\).)42 b(This)27 b(is)h(the)h(usual)150 1829 y(RSA)d(k)m(ey)g(pair.) 38 b(T)-8 b(o)27 b(sign)e(a)h(message,)i(one)f(computes)f Fj(\033)s Fo(\()p Fj(m)p Fo(\))2291 1804 y Fk(\030)2291 1833 y Fo(=)2387 1829 y Fj(m)2467 1796 y Fh(d)2533 1829 y Fe(mod)f Fj(n)p Fo(.)39 b(T)-8 b(o)26 b(v)m(erify)-8 b(,)27 b(one)f(raises)g(the)150 1942 y(signature)32 b(to)h(the)f(p)s(o) m(w)m(er)h Fj(e)g Fo(and)e(compares)i(it)f(to)h(the)g(original)d (message.)48 b(This)31 b(system)i(is)e(forgable)150 2055 y(under)e(a)i(c)m(hosen)f(message)i(attac)m(k.)43 b(See)31 b([1])g(for)f(a)h(pro)s(of.)150 2243 y 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3994 y(T)-8 b(able)30 b(1:)41 b(Secret)31 b(Key)f(SK)p 1477 4024 917 4 v 1475 4137 4 113 v 1526 4103 a Fj(x)1578 4070 y Fg(0)1578 4127 y(1)p 1666 4137 V 1717 4103 a Fj(x)1769 4070 y Fg(0)1769 4127 y(2)p 1857 4137 V 1908 4103 a Fj(:)15 b(:)g(:)p 2193 4137 V 231 w(x)2296 4070 y Fg(0)2296 4125 y Fh(n)p 2391 4137 V 1477 4140 917 4 v 1475 4253 4 113 v 1526 4219 a Fj(x)1578 4186 y Fg(1)1578 4243 y(1)p 1666 4253 V 1717 4219 a Fj(x)1769 4186 y Fg(1)1769 4243 y(2)p 1857 4253 V 1908 4219 a Fj(:)g(:)g(:)p 2193 4253 V 231 w(x)2296 4186 y Fg(1)2296 4241 y Fh(n)p 2391 4253 V 1477 4256 917 4 v 1466 4510 a Fo(T)-8 b(able)30 b(2:)42 b(Public)28 b(Key)i(PK)p 1304 4540 1292 4 v 1302 4652 4 113 v 1354 4619 a Fj(f)10 b Fo(\()p Fj(x)1496 4586 y Fg(0)1496 4643 y(1)1535 4619 y Fo(\))p 1619 4652 V 100 w Fj(f)g Fo(\()p Fj(x)1812 4586 y Fg(0)1812 4643 y(2)1851 4619 y Fo(\))p 1935 4652 V 100 w Fj(:)15 b(:)g(:)p 2271 4652 V 231 w(f)10 b Fo(\()p Fj(x)2464 4586 y Fg(0)2464 4641 y Fh(n)2511 4619 y Fo(\))p 2594 4652 V 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b(return)f(\\PRP")i(else)f(return)g(\\random.")150 534 y(Our)f(probabilit)m(y)f(of)j(sucess)f(is)f(then:)291 647 y Fi(P)13 b(r)s Fn([)p Fi(S)487 661 y Fe(P)d(RP)653 647 y Fn(\()p Fi(A)756 614 y Fg(0)780 647 y Fn(\()p Fi(k)s Fn(\)\)])27 b(=)e(1)p Fi(=)p Fn(2)d Fh(\003)e Fi(P)13 b(r)s Fn([)p Fi(S)1501 661 y Fe(P)d(RP)1668 647 y Fn(\()p Fi(A)1771 614 y Fg(0)1794 647 y Fn(\()p Fi(k)s Fn(\)\))p Fh(jO)30 b Fn(=)25 b Fi(P)2230 661 y Fe(s)2267 647 y Fn(])c(+)f(1)p Fi(=)p Fn(2)h Fh(\003)g Fi(P)13 b(r)s Fn([)p Fi(S)2822 661 y Fe(P)d(RP)2988 647 y Fn(\()p Fi(A)3091 614 y Fg(0)3115 647 y Fn(\()p Fi(k)s Fn(\)\))p Fh(jO)30 b Fn(=)24 b Fi(\031)s Fn(])150 809 y Fo(Lemma)33 b(1:)46 b(W)-9 b(e)36 b(ha)m(v)m(e)f(an)f(adv)-6 b(an)m(tage)35 b(when)g Fh(O)28 b Fn(=)d Fi(P)2235 823 y Fe(s)150 922 y Fn(W)-8 b(e)35 b(kno)m(w)e(this)g(b)s(ecause)g(when)g(the)h(oracle)g (is)e Fi(P)1889 936 y Fe(s)1927 922 y Fn(,)i(w)m(e)g(succeed)g(with)e (the)i(same)g(probabilit)m(y)d(as)j Fi(A)3707 937 y Fe(k)150 1035 y Fn(whic)m(h)29 b(is)h(1)p Fi(=)p Fn(2)21 b(+)f(1)p Fi(=pol)r(y)s Fn(.)150 1198 y Fo(Lemma)39 b(2:)59 b(When)42 b(the)e(oracle)i(is)f(random,)h(w)m(e)f(ha)m(v)m(e)g(a)g(probabilit)m (y)h(of)f(1/2)h(-)f Fi(\027)6 b Fn(\()p Fi(k)s Fn(\))42 b Fo(of)150 1311 y(sucess.)291 1537 y Fn(The)36 b(idea)h(here)f(is)g (that)h(if)f(w)m(e)h(nev)m(er)g(giv)m(e)g Fh(O)j Fn(the)d(same)g(input) e(t)m(wice)i(then)f(w)m(e)i(migh)m(t)e(as)h(w)m(ell)150 1650 y(decide)25 b(whic)m(h)f(message)j(the)f(output)f(is)f(an)i (encryption)e(of)i(after)g(the)f(adv)m(ersary)h(\014nishes)e (computing.)150 1762 y(If)30 b(this)f(is)h(the)g(case,)i(then)e(the)g (probabilit)m(y)e(of)j(success)g(m)m(ust)f(b)s(e)f(1/2.)291 1875 y(Ho)m(w)m(ev)m(er,)i(if)c Fi(A)830 1890 y Fe(k)900 1875 y Fn(mak)m(es)i(some)f(n)m(um)m(b)s(er)f Fi(p)p Fn(\()p Fi(k)s Fn(\))h(of)g(queries)f(to)i Fi(C)7 b(B)e(C)20 b Fh(\000)15 b(\000O)s Fn(,)28 b(then)g(there)g(can)g(b)s(e)f(at)150 1988 y(most)33 b Fi(l)24 b Fh(\001)e Fi(p)p Fn(\()p Fi(k)s Fn(\))33 b(queries)e(to)j Fh(O)h Fn(in)m(v)m(ok)m(ed.)48 b(F)-8 b(urthermore,)33 b(ev)m(ery)h(query)e(to)h Fh(O)i Fn(is)d(random)g(unless)f(there)150 2101 y(has)39 b(b)s(een)f(a)i (collision.)64 b(Ev)m(ery)39 b(time)g(the)g(adv)m(ersary)h(asks)f(for)g (the)g(encryption)f(of)h(a)h(message)g Fi(m)p Fn(,)150 2214 y(w)m(e)35 b(generate)h(an)e(indep)s(enden)m(t,)f(random)g Fi(I)7 b(V)21 b Fn(,)35 b(and)f(query)f Fh(O)38 b Fn(on)c(input)e Fi(m)23 b Fh(\010)f Fi(I)7 b(V)21 b Fn(,)35 b(whic)m(h)e(if)g Fh(O)k Fn(is)d(a)150 2327 y(random)26 b(p)s(erm)m(utation,)g(will)e (giv)m(e)i(us)g(a)h(random)e(v)-5 b(alue)26 b(\(since)g Fi(m)12 b Fh(\010)g Fi(I)7 b(V)46 b Fn(has)27 b(a)f(uniform)e (distribution,)150 2440 y(regardless)29 b(of)g Fi(m)p Fn(.\))40 b(Th)m(us,)29 b(there)g(are)h(at)g(most)f Fi(l)1849 2407 y Ff(2)1906 2440 y Fh(\001)19 b Fi(p)p Fn(\()p Fi(k)s Fn(\))2116 2407 y Ff(2)2185 2440 y Fn(random)28 b(queries)g(to)i Fh(O)s Fn(,)g(and)e(so)i(there)f(is)f(at)150 2553 y(most)j(a)g (probabilit)m(y)d 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Fn(2)23 b(+)e(1)p Fi(=pol)r(y)36 b Fn(and)c(so)h Fi(P)2969 3019 y Fe(s)3038 3005 y Fn(cannot)g(b)s(e)f (a)h(PRP)-8 b(.)150 3117 y(Therefore)30 b(since)g Fi(P)843 3131 y Fe(s)910 3117 y Fn(is)g(a)g(PRP)-8 b(,)31 b(this)e(encyrption)h (sc)m(heme)h(is)e(secure.)291 3230 y(Th)m(us,)46 b(w)m(e)d(ha)m(v)m(e)h (ac)m(hiev)m(ed)g(a)g(sc)m(heme)f(whic)m(h)f(is)g(secure)h(against)h (adaptiv)m(e)f(c)m(hosen)g(plain)m(text)150 3343 y(attac)m(k.)150 3636 y Fj(2.5)112 b(Securit)m(y)37 b(against)g(CCA)150 3808 y Fn(Unfortunately)-8 b(,)28 b(if)f(w)m(e)h(allo)m(w)m(ed)g(a)g(c) m(hosen)h(ciphertext)e(attac)m(k)k(on)c(this)g(encryption)g(system,)i (it)e(w)m(ould)150 3921 y(no)i(longer)h(b)s(e)f(secure.)40 b(W)-8 b(e)31 b(can)f(easily)f(see)h(this)e(that)j(giv)m(en)e(an)h (encryption)e(\()p Fi(c;)15 b(I)7 b(V)22 b Fn(\))29 b(w)m(e)h(can)g (ask)g(for)150 4034 y Fi(D)s Fn(\()p Fi(c;)15 b(I)7 b(V)41 b Fh(\010)20 b Fn(1\))31 b(that)g(giv)m(es)g(us)f Fi(m)1302 4001 y Fg(0)1325 4034 y Fn(.)41 b(Then)29 b(w)m(e)i(can)f(get)i(bac)m (k)f(the)g(original)d(message)k Fi(m)e Fn(from)g Fi(m)3532 4001 y Fg(0)3575 4034 y Fh(\010)20 b Fn(1.)150 4320 y Fk(3)135 b(MA)l(Cs)150 4527 y Fj(3.1)112 b(De\014nition)150 4698 y Fn(A)25 b(function)f(family)f Fh(f)p Fi(M)999 4712 y Fe(s)1037 4698 y Fn(\()p Fh(\016)p Fn(\))p Fh(g)j Fn(is)e(a)h(MA)m(C)h(if)e(for)h(all)f(PPTF)g Fh(f)p Fi(A)2338 4712 y Fe(n)2386 4698 y Fh(g)p Fn(,)i(there)g(exists)e(a)i(negligible)c (function)150 4811 y Fi(\027)6 b Fn(\()p Fi(k)s Fn(\))31 b(suc)m(h)f(that)571 5015 y(Pr[)p Fi(s)25 b Fh( )g(f)p Fn(0)p Fi(;)15 b Fn(1)p Fh(g)1098 4978 y Fe(k)1143 5015 y Fn(;)g(\()p Fi(m;)g(x)p Fn(\))p Fi(;)g(Q)27 b Fh( )e Fi(A)1748 4976 y Fe(M)1816 4984 y Fd(s)1748 5045 y Fe(k)1879 5015 y Fn(:)g Fi(x)g Fn(=)g Fi(M)2190 5029 y Fe(s)2227 5015 y Fn(\()p Fi(m)p Fn(\))31 b(and)f(\()p Fi(m;)15 b(x)p Fn(\))36 b Fi(=)-55 b Fh(2)25 b Fi(Q)p Fn(])g Fh(\024)g Fi(\027)6 b Fn(\()p Fi(k)s Fn(\))1867 5416 y(21-5)p eop %%Page: 6 6 6 5 bop 150 159 a Fj(3.2)112 b(Blo)s(c)m(k-cipher)36 b(based)j(MA)m(C)150 330 y Fn(W)-8 b(e)35 b(can)f(use)f(a)h(PRF)g(as)g (a)g(MA)m(C)h(but)e(there)h(are)g(sev)m(eral)g(di\016culties)d(to)k (mak)m(e)f(it)g(w)m(ork.)50 b(The)34 b(\014rst)150 443 y(issue)k(is)h(that)h(a)g(PRF)g(ma)m(y)g(ha)m(v)m(e)h(an)e(output)g(of) h(an)m(y)g(length)f(and)g(it)g(ma)m(y)h(ev)m(en)g(b)s(e)f(longer)g (than)150 556 y(the)32 b(message)g(itself.)43 b(An)31 b(ideal)f(MA)m(C)i(w)m(ould)e(only)h(add)f(a)i(constan)m(t)h(n)m(um)m (b)s(er)d(\(prop)s(ortional)g(to)i(the)150 669 y(securit)m(y)g (parameter\))i(of)f(bits)e(to)j(the)e(message)i(so)f(that)g(the)g (message)h(do)s(esn't)e(gro)m(w)h(m)m(uc)m(h)g(longer.)150 782 y(Another)g(di\016cult)m(y)e(is)h(that)i(a)f(particular)e(PRF)j (only)e(accepts)i(messages)g(of)f(one)g(length,)g(therefore)150 895 y(w)m(e)e(could)e(not)i(create)h(a)f(MA)m(C)g(for)f(an)m(y)g (message)i(w)m(e)f(wish.)291 1008 y(So)25 b(one)h(idea)f(here)h(is)e (to)i(use)f(a)h(PRP)f(of)h(a)g(\014xed)f(length)f(\(sa)m(y)j 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b(force)i(this)e(input)f(to)j(b)s(e)e(an)m (ything)h(he)g(w)m(an)m(ts,)k(since)40 b(he)h(do)s(esn't)g(ha)m(v)m(e)i (access)f(to)g Fh(O)150 3878 y Fn(directly)-8 b(.)291 3991 y(Here,)37 b(w)m(e)f(hand-w)m(a)m(v)m(e)g(a)g(bit.)54 b(Basically)-8 b(,)36 b(if)e(the)h(adv)m(ersary)h(successfully)d (forges)j(a)f(pair)f(\()p Fi(m;)15 b(x)p Fn(\))150 4104 y(where)25 b(the)i(adv)m(ersary)f(nev)m(er)g(queried)f Fi(C)7 b(B)e(C)i(M)j(AC)16 b Fh(\000)11 b(\000O)29 b Fn(on)c Fi(m)p Fn(,)i(there)f(m)m(ust)g(b)s(e)f(a)i(pre\014x)e Fi(m)3467 4118 y Ff(1)3517 4104 y Fh(\016)11 b Fi(m)3653 4118 y Ff(2)3705 4104 y Fh(\016)150 4217 y Fi(:)k(:)g(:)f Fh(\016)f Fi(m)407 4231 y Fe(j)471 4217 y Fn(whic)m(h)26 b(w)m(as)h(the)g(shortest)g(pre\014x)f(suc)m(h)g(that)i(it)e(is)g(not)h (a)g(pre\014x)f(of)h(an)m(y)g(query)g(the)g(adv)m(ersary)150 4330 y(did)i(mak)m(e.)41 b(\(One)31 b(suc)m(h)f(pre\014x)f(m)m(ust)h (exist)g(or)h(the)f(adv)m(ersary)h(did)d(not)j(forge)g(for)f(a)h(new)f (message.\))291 4443 y(No)m(w,)38 b(starting)e(with)f Fi(c)1127 4457 y Fe(j)1164 4443 y Fn(,)i(eac)m(h)h(output)d(is)g(a)i (truly)d(random)h(v)-5 b(alue)36 b(since)f(the)h(adv)m(ersary)h(nev)m (er)150 4556 y(made)30 b Fh(O)i Fn(b)s(e)d(queried)f(on)i(these)g(v)-5 b(alues)29 b(b)s(efore)1810 4523 y Ff(1)1849 4556 y Fn(.)40 b(Th)m(us,)29 b(the)h(probabilit)m(y)d(that)k(the)e(blo)s(c)m(k)h Fi(m)3461 4571 y Fe(l)3505 4556 y Fh(\010)18 b Fi(c)3633 4571 y Fe(l)q Fg(\000)p Ff(1)150 4669 y Fn(w)m(as)38 b(one)g(whic)m(h)f(had)g(b)s(een)g(giv)m(en)g(to)i Fh(O)h Fn(b)s(efore)e(w)m(ould)e(b)s(e)h(negligible.)61 b(Since)36 b(it)i(is)e(bad)h(for)h(us)f(if)150 4782 y(the)e(adv)m(ersary)g(ev)m (er)h(manages)g(to)g(forge)f(a)h(message,)h(let)e(us)f(assume)h(that)h (if)e Fi(m)3051 4797 y Fe(l)3100 4782 y Fh(\010)23 b Fi(c)3233 4797 y Fe(l)q Fg(\000)p Ff(1)3384 4782 y Fn(had)35 b(b)s(een)150 4895 y(queried)42 b(b)s(efore,)47 b(the)c(adv)m(ersary)h (succeeds)g(with)e(probabilit)m(y)f(1,)48 b(and)43 b(otherwise)f (succeeds)i(with)150 5008 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926 y Fe(s)1204 912 y Fn(\()p Fi(m)1319 926 y Ff(1)1359 912 y Fn(\),)f(then)f(w)m(e)i(query)e(the)g(oracle)i(for)e Fi(P)2655 926 y Fe(s)2692 912 y Fn(\()p Fi(m)2807 926 y Ff(1)2847 912 y Fn(\))h(and)f(get)h Fi(P)3296 926 y Fe(s)3334 912 y Fn(\()p Fi(P)3427 926 y Fe(s)3464 912 y Fn(\()p Fi(m)3579 926 y Ff(1)3619 912 y Fn(\)\)\).)150 1025 y(W)-8 b(e)37 b(then)f(answ)m(er)f(that)i(\()p Fi(m)1149 1039 y Ff(1)1212 1025 y Fh(\016)24 b Fn(0)1326 992 y Fe(k)1370 1025 y Fi(;)15 b(P)1468 1039 y Fe(s)1505 1025 y Fn(\()p Fi(P)1598 1039 y Fe(s)1636 1025 y Fn(\()p Fi(m)1751 1039 y Ff(1)1790 1025 y Fn(\)\)\))37 b(is)e(a)h(v)-5 b(alid)34 b(message/MA)m(C)k(pair)d(b)s(ecause)h(0)3530 992 y Fe(k)3609 1025 y Fn(will)150 1138 y(not)31 b(c)m(hange)g(the)g(v) -5 b(alue)30 b(of)g Fi(P)1165 1152 y Fe(s)1202 1138 y Fn(\()p Fi(m)1317 1152 y Ff(1)1357 1138 y Fn(\))g(b)s(efore)g(it)g(en)m (ters)h Fi(P)2110 1152 y Fe(s)2178 1138 y Fn(for)f(the)g(second)h (time.)291 1251 y(So)j(ho)m(w)g(can)g(w)m(e)g(manage)h(to)g(mak)m(e)g (our)e(MA)m(C)h(w)m(ork)g(for)g(messages)h(of)f(arbitrary)e(length?)51 b(W)-8 b(e)150 1364 y(giv)m(e)31 b(three)f(secure)h(w)m(a)m(ys:)261 1576 y(1.)46 b(Let)31 b(the)g(k)m(ey)g(b)s(e)f(a)g(pair)f(\()p Fi(s;)15 b(s)1408 1543 y Fg(0)1432 1576 y Fn(\),)31 b(and)e(output)h Fi(P)2055 1592 y Fe(s)2088 1573 y Fa(0)2115 1576 y Fn(\()p Fi(C)7 b(B)e(C)i(M)j(AC)26 b Fh(\000)20 b(\000)p Fi(P)2845 1590 y Fe(s)2881 1576 y Fn(\)\()p Fi(m)p Fn(\).)261 1764 y(2.)46 b(Output)30 b Fi(C)7 b(B)e(C)i(M)j(AC)25 b Fh(\000)20 b(\000)p Fi(P)1395 1778 y Fe(s)1432 1764 y Fn(\()p Fi(m)g Fh(\016)h(j)p Fi(m)p Fh(j)p Fn(\).)261 1952 y(3.)46 b(Let)31 b Fi(s)583 1919 y Fg(0)631 1952 y Fh( )25 b Fi(P)805 1966 y Fe(s)843 1952 y Fn(\()p Fh(j)p Fi(m)p Fh(j)p Fn(\).)41 b(Then,)30 b(output)g Fi(C)7 b(B)e(C)i(M)j(AC)25 b Fh(\000)20 b(\000)p Fi(P)2364 1967 y Fe(s)2397 1948 y Fa(0)2423 1952 y Fn(\()p Fi(m)p Fn(\).)291 2164 y(T)-8 b(o)30 b(giv)m(e)g(a)f 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Grant, and David Carlisle. %% All rights reserved. %% %% This file is part of the PSfrag package. %% userdict begin /PSfragLib 90 dict def /PSfragDict 6 dict def /PSfrag { PSfragLib begin load exec end } bind def end PSfragLib begin /RO /readonly load def /CP /currentpoint load def /CM /currentmatrix load def /B { bind RO def } bind def /X { exch def } B /MD { { X } forall } B /OE { end exec PSfragLib begin } B /S false def /tstr 8 string def /islev2 { languagelevel } stopped { false } { 2 ge } ifelse def [ /sM /tM /srcM /dstM /dM /idM /srcFM /dstFM ] { matrix def } forall sM currentmatrix RO pop dM defaultmatrix RO idM invertmatrix RO pop srcFM identmatrix pop /Hide { gsave { CP } stopped not newpath clip { moveto } if } B /Unhide { { CP } stopped not grestore { moveto } if } B /setrepl islev2 {{ /glob currentglobal def true setglobal array astore globaldict exch /PSfrags exch put glob setglobal }} {{ array astore /PSfrags X }} ifelse B /getrepl islev2 {{ globaldict /PSfrags get aload length }} {{ PSfrags aload length }} ifelse B /convert { /src X src length string /c 0 def src length { dup c src c get dup 32 lt { pop 32 } if put /c c 1 add def } repeat } B /Begin { /saver save def srcFM exch 3 exch put 0 ne /debugMode X 0 setrepl dup /S exch dict def { S 3 1 roll exch convert exch put } repeat srcM CM dup invertmatrix pop mark { currentdict { end } stopped { pop exit } if } loop PSfragDict counttomark { begin } repeat pop } B /End { mark { currentdict end dup PSfragDict eq { pop exit } if } loop counttomark { begin } repeat pop getrepl saver restore 7 idiv dup /S exch dict def { 6 array astore /mtrx X tstr cvs /K X S K [ S K known { S K get aload pop } if mtrx ] put } repeat } B /Place { tstr cvs /K X S K known { bind /proc X tM CM pop CP /cY X /cX X 0 0 transform idtransform neg /aY X neg /aX X S K get dup length /maxiter X /iter 1 def { iter maxiter ne { /saver save def } if tM setmatrix aX aY translate [ exch aload pop idtransform ] concat cX neg cY neg translate cX cY moveto /proc load OE iter maxiter ne { saver restore /iter iter 1 add def } if } forall /noXY { CP /cY X /cX X } stopped def tM setmatrix noXY { newpath } { cX cY moveto } ifelse } { Hide OE Unhide } ifelse } B /normalize { 2 index dup mul 2 index dup mul add sqrt div dup 4 -1 roll exch mul 3 1 roll mul } B /replace { aload pop MD CP /bY X /lX X gsave sM setmatrix str stringwidth abs exch abs add dup 0 eq { pop } { 360 exch div dup scale } ifelse lX neg bY neg translate newpath lX bY moveto str { /ch X ( ) dup 0 ch put false charpath ch Kproc } forall flattenpath pathbbox [ /uY /uX /lY /lX ] MD CP grestore moveto currentfont /FontMatrix get dstFM copy dup 0 get 0 lt { uX lX /uX X /lX X } if 3 get 0 lt { uY lY /uY X /lY X } if /cX uX lX add 0.5 mul def /cY uY lY add 0.5 mul def debugMode { gsave 0 setgray 1 setlinewidth lX lY moveto lX uY lineto uX uY lineto uX lY lineto closepath lX bY moveto uX bY lineto lX cY moveto uX cY lineto cX lY moveto cX uY lineto stroke grestore } if dstFM dup invertmatrix dstM CM srcM 2 { dstM concatmatrix } repeat pop getrepl /temp X S str convert get { aload pop [ /rot /scl /loc /K ] MD /aX cX def /aY cY def loc { dup 66 eq { /aY bY def } { % B dup 98 eq { /aY lY def } { % b dup 108 eq { /aX lX def } { % l dup 114 eq { /aX uX def } { % r dup 116 eq { /aY uY def } % t if } ifelse } ifelse } ifelse } ifelse pop } forall K srcFM rot tM rotate dstM 2 { tM concatmatrix } repeat aload pop pop pop 2 { scl normalize 4 2 roll } repeat aX aY transform /temp temp 7 add def } forall temp setrepl } B /Rif { S 3 index convert known { pop replace } { exch pop OE } ifelse } B /XA { bind [ /Kproc /str } B /XC { ] 2 array astore def } B /xs { pop } XA XC /xks { /kern load OE } XA /kern XC /xas { pop ax ay rmoveto } XA /ay /ax XC /xws { c eq { cx cy rmoveto } if } XA /c /cy /cx XC /xaws { ax ay rmoveto c eq { cx cy rmoveto } if } XA /ay /ax /c /cy /cx XC /raws { xaws { awidthshow } Rif } B /rws { xws { widthshow } Rif } B /rks { xks { kshow } Rif } B /ras { xas { ashow } Rif } B /rs { xs { show } Rif } B /rrs { getrepl dup 2 add -1 roll //restore exec setrepl } B PSfragDict begin islev2 not { /restore { /rrs PSfrag } B } if /show { /rs PSfrag } B /kshow { /rks PSfrag } B /ashow { /ras PSfrag } B /widthshow { /rws PSfrag } B /awidthshow { /raws PSfrag } B end PSfragDict RO pop end %%EndProcSet %%BeginProcSet: special.pro %! 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tgifellipsedict /mtrx matrix put /tgifellipse { tgifellipsedict begin /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 0 360 arc savematrix setmatrix end } def /tgifarrowtipdict 8 dict def tgifarrowtipdict /mtrx matrix put /tgifarrowtip { tgifarrowtipdict begin /dy exch def /dx exch def /h exch def /w exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate dy dx atan rotate 0 0 moveto w neg h lineto w neg h neg lineto savematrix setmatrix end } def end %%PageBoundingBox: 0 735 89 791 tgifdict begin /tgifsavedpage save def 1 setmiterlimit 1 setlinewidth 0 setgray 72 0 mul 72 11.00 mul translate 72 128 div 100.000 mul 100 div dup neg scale gsave /tgiforigctm matrix currentmatrix def % TEXT 0 setgray /Courier findfont [17 0 0 -17 0 0] makefont setfont gsave 72 17 moveto (y) show grestore % OVAL 0 setgray newpath 72 28 2 2 tgifellipse fill gsave gsave newpath 72 28 2 2 tgifellipse [4 4] 0 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setlinewidth stroke [] 0 setdash grestore % TEXT 0 setgray /Courier findfont [14 0 0 -14 0 0] makefont setfont gsave 3 68 moveto (f) show grestore % TEXT 0 setgray /Courier findfont [14 0 0 -14 0 0] makefont setfont gsave 17 54 moveto (f) show grestore % TEXT 0 setgray /Courier findfont [14 0 0 -14 0 0] makefont setfont gsave 47 29 moveto (f) show grestore % TEXT 0 setgray /Courier findfont [8 0 0 -8 0 0] makefont setfont gsave 152 76 moveto (1) show grestore % TEXT 0 setgray /Courier findfont [8 0 0 -8 0 0] makefont setfont gsave 144 73 moveto (R') show grestore % TEXT 0 setgray /Courier findfont [14 0 0 -14 0 0] makefont setfont gsave 136 73 moveto (f) show grestore grestore tgifsavedpage restore end %MatchingCreationDate: Mon Nov 26 18:48:58 2001 %%EndDocument @endspecial 675 x @beginspecial 0 @llx 711 @lly 106 @urx 791 @ury 1060 @rwi @setspecial %%BeginDocument: branches.eps %! %%BoundingBox: 0 711 106 791 %%Title: branches %%CreationDate: Mon Nov 26 16:30:25 2001 %%Creator: Tgif-3.0-p13 by William Chia-Wei Cheng (william@cs.UCLA.edu) /tgifdict 6 dict def tgifdict begin /tgifellipsedict 6 dict def tgifellipsedict /mtrx matrix put /tgifellipse { tgifellipsedict begin /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate xrad yrad scale 0 0 1 0 360 arc savematrix setmatrix end } def /tgifarrowtipdict 8 dict def tgifarrowtipdict /mtrx matrix put /tgifarrowtip { tgifarrowtipdict begin /dy exch def /dx exch def /h exch def /w exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y translate dy dx atan rotate 0 0 moveto w neg h lineto w neg h neg lineto savematrix setmatrix end } def end %%PageBoundingBox: 0 711 106 791 tgifdict begin /tgifsavedpage save def 1 setmiterlimit 1 setlinewidth 0 setgray 72 0 mul 72 11.00 mul translate 72 128 div 100.000 mul 100 div dup neg scale gsave /tgiforigctm matrix currentmatrix def % OVAL 0 setgray gsave gsave newpath 81 61 16 16 tgifellipse stroke grestore 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e(w)m(a)m(y)i(that)150 2896 y(Ev)m(e)k(can)f(decrypt)g(is)f(b)m(y)h (actually)g(asking)g(Alice)f(to)i(decrypt.)52 b(This)32 b(is)h(called)h(\\Adaptiv)m(e)g(Chosen-)150 3008 y(Ciphertext)29 b(Securit)m(y")i(and)e(it)h(go)s(es)h(as)g(follo)m(ws:)40 b(the)30 b(adv)m(ersary)-8 b(,)32 b(Ev)m(e)286 3221 y Fg(\017)46 b Fl(Receiv)m(es)32 b(Alice's)e(Public)e(Key)286 3409 y Fg(\017)46 b Fl(Queries)29 b(Alice)h(for)g(decryptions)f(of)i (ciphertexts)f(of)g(her)g(c)m(hoice)286 3596 y Fg(\017)46 b Fl(Declares)35 b(that)g(she)g(can)f(succeed)h(in)e(\014guring)g(out)h (Bob's)h(message)h(if)d(it's)h(c)m(hosen)h(according)377 3709 y(to)28 b(the)f(Message)i(Space)e Fg(M)h Fl(\(Before,)h Fg(M)e Fl(w)m(as)g(\014xed)g(after)g(lo)s(oking)f(at)i(only)e(PK,)h (but)g(no)m(w)g(can)377 3822 y(use)j(Alice\))286 4010 y Fg(\017)46 b Fl(Challenge)29 b(encryption)h(is)f(issued)g(\()h Ff(m)c Fg( )f(M)p Ff(;)15 b(c)26 b Fg( )f Ff(E)5 b Fl(\()p Ff(m;)15 b(P)e(K)7 b Fl(\))31 b(\))286 4197 y Fg(\017)46 b Fl(Query)30 b(Alice)g(for)g(more)g(decryptions)f(\(Except)j(for)e Ff(c)p Fl(\))286 4385 y Fg(\017)46 b Fl(Output)30 b(a)g(guess)h(for)f Ff(m)291 4597 y Fl(What)k(Ev)m(e)f(is)f(going)h(to)h(try)e(and)h(do)f (is)g(b)m(ypass)h Ff(E)5 b Fl(\(\))33 b(so)g(that)h(she)e(giv)m(es)i (Alice)e(something)g(suc)m(h)150 4710 y(that)f(she)f(\014nds)f(out)h (something)g(to)h(help)e(her)h(crac)m(k)i(it.)1912 5416 y(-1)p eop %%Page: 2 2 2 1 bop 150 159 a Fh(Example)150 330 y Fl(Public)28 b(Key:)41 b Ff(P)13 b(K)37 b Fl(of)30 b(a)h(TDP)-8 b(,)31 b(to)g(encrypt)f Ff(m)p Fl(,)h(with)e Fg(j)p Ff(m)p Fg(j)c Fl(=)g Ff(n)p Fl(.)291 443 y(Cho)s(ose)h Ff(n)h Fl(random)e Ff(r)1055 457 y Fe(1)1095 443 y Ff(;)15 b(:::;)g(r)1291 457 y Fd(n)1366 443 y Fl(and)26 b Ff(R)1608 457 y Fe(1)1647 443 y Ff(;)15 b(:::;)g(R)1871 457 y Fd(n)1920 443 y Fl(,)27 b(all)f(from)g(the)h (domain)e(of)i(the)f(TDP;)h(then)f(output)150 556 y Ff(p)196 570 y Fd(P)10 b(K)319 556 y Fl(\()p Ff(r)395 570 y Fe(1)434 556 y Fl(\))p Ff(;)15 b(:::;)g(p)670 570 y Fd(P)10 b(K)795 556 y Fl(\()p Ff(r)871 570 y Fd(n)918 556 y Fl(\))27 b(and)g Ff(R)1223 570 y Fe(1)1275 556 y Fg(\001)13 b Ff(:)i(:)g(:)f Fg(\001)f Ff(R)1539 570 y Fd(n)1613 556 y Fl(and)27 b(let)f Ff(c)g Fl(=)f(\(\()p Ff(r)2186 570 y Fe(1)2239 556 y Fg(\001)13 b Ff(R)2346 570 y Fe(1)2386 556 y Fl(\))g Fg(\016)g Ff(:)i(:)g(:)g Fg(\016)e Fl(\()p Ff(r)2746 570 y Fd(n)2807 556 y Fg(\001)g Ff(R)2914 570 y Fd(n)2961 556 y Fl(\)\))g Fg(\010)g Ff(m)p Fl(.)40 b(This)25 b(creates)150 669 y(a)31 b(one-time)f(pad)g(using)f(the)i (Goldreic)m(h-Levin)d(hard)i(core)h(bit.)150 912 y Fh(Closer)37 b(Lo)s(ok)150 1084 y Fl(Create)g(a)f(random)g(pad)f Ff(U)10 b Fl(,)38 b(with)d Fg(j)p Ff(U)10 b Fg(j)35 b Fl(=)f Ff(n)p Fl(,)k(b)m(y)e(c)m(ho)s(osing)f(random)h Ff(r)2667 1098 y Fd(i)2695 1084 y Ff(;)15 b(R)2804 1098 y Fd(i)2833 1084 y Fl(;)39 b(and)c(letting)h Ff(U)44 b Fl(=)35 b(\()p Ff(r)3661 1098 y Fe(1)3725 1084 y Fg(\001)150 1197 y Ff(R)219 1211 y Fe(1)258 1197 y Fl(\))24 b Fg(\016)f Ff(:)15 b(:)g(:)23 b Fg(\016)h Fl(\()p Ff(r)658 1211 y Fd(n)728 1197 y Fg(\001)f Ff(R)845 1211 y Fd(n)892 1197 y Fl(\).)52 b(T)-8 b(o)34 b(encrypt)g Ff(m)g Fl(compute)h Ff(y)2008 1211 y Fd(i)2067 1197 y Fl(=)c Ff(p)2215 1211 y Fd(P)10 b(K)2338 1197 y Fl(\()p Ff(r)2414 1211 y Fd(i)2442 1197 y Fl(\))35 b(then)f(output)f Fg(f)p Ff(R)3138 1211 y Fd(i)3167 1197 y Fg(g)p Ff(;)15 b Fg(f)p Ff(y)3342 1211 y Fd(i)3371 1197 y Fg(g)p Ff(;)g(U)34 b Fg(\010)22 b Ff(m)p Fl(.)150 1310 y(The)30 b(one)h(time)f(pad)f(is)h(hidden)e (from)i(the)g(adv)m(ersary)-8 b(,)32 b(that)f(is)e(wh)m(y)h(this)f(is)h (seman)m(tically)f(secure.)291 1423 y(This)23 b(example)i(is)f(not)h (adaptiv)m(e)g(ho)m(w)m(ev)m(er,)j(since)c(\()p Ff(U)19 b Fg(\010)9 b Ff(m)p Fl(\))g Fg(\010)g Ff(J)g(U)h(N)g(K)32 b Fl(could)24 b(b)s(e)h(Ev)m(e's)g(ciphertext)150 1536 y(to)i(query)e(to)i(Alice,)g(and)f(she)f(w)m(ould)g(decrypt)h(giving)f (bac)m(k)i Ff(m)12 b Fg(\010)g Ff(J)d(U)h(N)g(K)d Fl(,)26 b(and)f(then)h(Ev)m(e)h(can)g Fg(\010)e Fl(that)150 1648 y(with)k Ff(J)9 b(U)h(N)g(K)37 b Fl(to)31 b(get)h Ff(m)p Fl(,)e(allo)m(wing)f(Ev)m(e)i(to)g(break)g(the)f(system.)291 1761 y(In)g(an)g(up)s(coming)f(homew)m(ork)h(problems,)f(w)m(e)i(will) 286 1974 y Fg(\017)46 b Fl(F)-8 b(ormalize)31 b(this)e(notion)h(of)h (securit)m(y)286 2162 y Fg(\017)46 b Fl(Sho)m(w)30 b(ho)m(w)h(to)g (break)f(more)g(:)41 b(El-Gamal)30 b(and)g(GM)291 2374 y(but)25 b(for)g(no)m(w,)i(w)m(e)e(will)e(presen)m(t)i(something)g (that)h(satis\014es)f(this)f(securit)m(y)h(using)f(this)g(the)i (informal)150 2487 y(de\014nition)i(from)i(ab)s(o)m(v)m(e.)150 2773 y Fi(2)135 b(Cramer-Shoup)44 b(Cryptosystem)150 2980 y Fh(2.1)112 b(Decisional)36 b(Di\016e-Hellman)150 3151 y Fl(Recall)30 b(the)h(DDH)g(assumption)e(and)h(setting.)40 b(Giv)m(en)31 b(prime)d Ff(p)i Fl(with:)941 3377 y Ff(p)25 b Fl(=)g Ff(aq)e Fl(+)d(1;)15 b Ff(g)29 b Fg(2)c Fc(Z)1619 3391 y Fd(p)1654 3340 y Fb(\003)1694 3377 y Fl(;)15 b Fg(j)p Ff(g)s Fg(j)27 b Fl(=)e Ff(q)s Fl(;)15 b Ff(G)26 b Fl(=)p Ff(<)e(g)29 b(>)p Fl(;)15 b Ff(x;)g(y)s(;)g(z)30 b Fg(2)25 b Fc(Z)2886 3391 y Fd(q)2920 3340 y Fb(\003)291 3544 y Fl(it)20 b(is)f(hard)h(to)h(distinguish,)d(or)j(tell)e(whether)h (the)g(tuple)f(\()p Ff(g)2267 3558 y Fe(1)2308 3544 y Ff(;)c(g)2391 3558 y Fe(2)2431 3544 y Ff(;)g(u)2523 3558 y Fe(1)2563 3544 y Ff(;)g(u)2655 3558 y Fe(2)2694 3544 y Fl(\))21 b(is)e(of)i(the)f(form)g(\()p Ff(g)3354 3558 y Fe(1)3394 3544 y Ff(;)15 b(g)3477 3558 y Fe(1)3517 3511 y Fd(x)3561 3544 y Ff(;)g(g)3644 3558 y Fe(1)3684 3511 y Fd(y)3726 3544 y Ff(;)g(g)3809 3558 y Fe(1)3849 3511 y Fd(z)3889 3544 y Fl(\))150 3657 y(or)30 b(\()p Ff(g)339 3671 y Fe(1)380 3657 y Ff(;)15 b(g)463 3671 y Fe(1)503 3624 y Fd(x)547 3657 y Ff(;)g(g)630 3671 y Fe(1)670 3624 y Fd(y)711 3657 y Ff(;)g(g)794 3671 y Fe(1)834 3624 y Fd(xy)916 3657 y Fl(\).)150 3900 y Fh(2.2)112 b(The)38 b(Cramer-Shoup)g(Cryptosystem)150 4072 y Fl(The)30 b(Cramer-Shoup)e(Cryptosystem)j(is)e(then)h(as)h(follo)m(ws:)291 4185 y(The)f(Public)e(Key)i(consists)g(of)471 4502 y Ff(p;)15 b(q)s(;)g(g)684 4516 y Fe(1)724 4502 y Ff(;)g(g)807 4516 y Fe(2)847 4502 y Ff(;)g(G)26 b Fl(=)p Ff(<)f(g)1194 4516 y Fe(1)1259 4502 y Ff(>)p Fl(=)p Ff(<)f(g)1539 4516 y Fe(2)1604 4502 y Ff(>;)15 b Fg(j)p Ff(G)p Fg(j)27 b Fl(=)e Ff(q)s(;)15 b(c)25 b Fl(=)g Ff(g)2249 4461 y Fd(x)2289 4470 y Fa(1)2246 4528 y Fe(1)2329 4502 y Ff(g)2375 4461 y Fd(x)2415 4470 y Fa(2)2372 4528 y Fe(2)2454 4502 y Ff(;)15 b(d)26 b Fl(=)f Ff(g)2709 4458 y Fd(y)2744 4467 y Fa(1)2706 4528 y Fe(1)2783 4502 y Ff(g)2829 4458 y Fd(y)2864 4467 y Fa(2)2826 4528 y Fe(2)2903 4502 y Ff(;)15 b(h)26 b Fl(=)f Ff(g)3163 4461 y Fd(z)3196 4470 y Fa(1)3160 4528 y Fe(1)3235 4502 y Ff(;)15 b(H)7 b Fl(\(\))291 4706 y(where)31 b Ff(H)7 b Fl(\(\))32 b(is)f(a)h(collision)d(resistan)m(t)j (hash)f(function)f(with)g(output)i(in)e Fc(Z)2880 4720 y Fd(q)2914 4670 y Fb(\003)2953 4706 y Fl(.)44 b(\(Actually)-8 b(,)33 b(an)e(ev)m(en)150 4819 y(w)m(eak)m(er)h(primitiv)m(e,)c(called) i(\\univ)m(ersal)f(one-w)m(a)m(y)j(hash)d(function,")h(is)g(su\016cien) m(t.\))291 4932 y(The)g(Secret)h(Key)f(consists)g(of)1528 5158 y Ff(x)1580 5172 y Fe(1)1620 5158 y Ff(;)15 b(x)1712 5172 y Fe(2)1752 5158 y Ff(;)g(y)1837 5172 y Fe(1)1876 5158 y Ff(;)g(y)1961 5172 y Fe(2)2000 5158 y Ff(;)g(z)2082 5172 y Fe(1)2148 5158 y Fg(2)25 b Fc(Z)2299 5172 y Fd(q)2332 5120 y Fb(\003)1912 5416 y Fl(-2)p eop %%Page: 3 3 3 2 bop 291 159 a Fl(T)-8 b(o)31 b(encrypt)f(a)h(message)g Ff(m)25 b Fg(2)g Ff(G)p Fl(,)31 b(pic)m(k)f(a)h(random)f Ff(r)d Fg( )e Fc(Z)2350 173 y Fd(q)2384 122 y Fb(\003)2424 159 y Fl(.)40 b(Then)30 b(let)861 384 y Ff(u)913 398 y Fe(1)978 384 y Fl(=)25 b Ff(g)1117 398 y Fe(1)1156 347 y Fd(r)1194 384 y Ff(;)15 b(u)1286 398 y Fe(2)1352 384 y Fl(=)24 b Ff(g)1490 398 y Fe(2)1530 347 y Fd(r)1568 384 y Ff(;)15 b(e)26 b Fl(=)f Ff(h)1824 347 y Fd(r)1863 384 y Ff(m;)15 b(\013)25 b Fl(=)g Ff(H)7 b Fl(\()p Ff(u)2332 398 y Fe(1)2372 384 y Ff(;)15 b(u)2464 398 y Fe(2)2504 384 y Ff(;)g(e)p Fl(\))p Ff(;)g(v)30 b Fl(=)25 b Ff(c)2870 347 y Fd(r)2908 384 y Ff(d)2955 347 y Fd(\013r)150 551 y Fl(and)30 b(then)g(the)g(ciphertext)g(w)m(ould)g(b)s(e)f(the)i(tuple) e(\()p Ff(u)1987 565 y Fe(1)2027 551 y Ff(;)15 b(u)2119 565 y Fe(2)2159 551 y Ff(;)g(e;)g(v)s Fl(\).)291 664 y(T)-8 b(o)36 b(decrypt)e(a)i(ciph)m(text)f(\()p Ff(u)1291 678 y Fe(1)1331 664 y Ff(;)15 b(u)1423 678 y Fe(2)1463 664 y Ff(;)g(e;)g(v)s Fl(\),)38 b(there)d(are)g(t)m(w)m(o)i(steps,)f (\014rst)e(c)m(hec)m(king)i(that)g(it's)e(a)i(v)-5 b(alid)150 777 y(encryption,)30 b(and)f(then)h(the)h(actual)g(decryption)e(and)h (retriv)-5 b(al)29 b(of)h(the)h(message.)291 890 y(T)-8 b(o)31 b(v)-5 b(alidate,)860 1116 y Ff(v)29 b Fl(=)c Ff(c)1068 1078 y Fd(r)1106 1116 y Ff(d)1153 1078 y Fd(\013r)1262 1116 y Fl(=)g(\()p Ff(g)1439 1075 y Fd(x)1479 1084 y Fa(1)1436 1142 y Fe(1)1519 1116 y Ff(g)1565 1075 y Fd(x)1605 1084 y Fa(2)1562 1142 y Fe(2)1644 1116 y Fl(\))1679 1078 y Fd(r)1717 1116 y Fl(\()p Ff(g)1798 1072 y Fd(y)1833 1081 y Fa(1)1795 1142 y Fe(1)1873 1116 y Ff(g)1919 1072 y Fd(y)1954 1081 y Fa(2)1916 1142 y Fe(2)1993 1116 y Fl(\))2028 1078 y Fd(\013r)2137 1116 y Fl(=)g(\()p Ff(g)2314 1078 y Fd(r)2311 1138 y Fe(1)2353 1116 y Fl(\))2388 1078 y Fd(x)2428 1087 y Fa(1)2462 1078 y Fe(+)p Fd(\013y)2597 1087 y Fa(1)2636 1116 y Fl(\()p Ff(g)2717 1078 y Fd(r)2714 1138 y Fe(2)2756 1116 y Fl(\))2791 1078 y Fd(x)2831 1087 y Fa(2)2866 1078 y Fe(+)p Fd(\013y)3001 1087 y Fa(2)291 1283 y Fl(so)31 b(c)m(hec)m(k)h(if)d Ff(v)g Fl(=)c Ff(u)952 1238 y Fd(x)992 1247 y Fa(1)1026 1238 y Fe(+)p Fd(\013y)1161 1247 y Fa(1)952 1309 y Fe(1)1200 1283 y Ff(u)1252 1238 y Fd(x)1292 1247 y Fa(2)1326 1238 y Fe(+)p Fd(\013y)1461 1247 y Fa(2)1252 1309 y Fe(2)1500 1283 y Fl(.)41 b(If)30 b(not)g(output)g(INV)-10 b(ALID)31 b(and)f(do)g(not)g(decrypt)h Ff(m)p Fl(.)291 1396 y(Otherwise,)e(con)m(tin)m(ue)i(and)f(output)1052 1600 y Ff(e=)p Fl(\()p Ff(u)1226 1559 y Fd(z)1259 1568 y Fa(1)1226 1626 y Fe(1)1299 1600 y Fl(\))c(=)f Ff(e=)p Fl(\(\()p 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(h/Details.asp?RID=894&NDPH=blue+ro)r(om&VID=1&PID=1&LID=3)150 4817 y(for)26 b(a)g(restauran)n(t)g(review)254 4877 y Fh(2)288 4908 y Fg(See)g(Blum,)f(M.,)i(\\Coin)g(Flipping)f(b)n(y)e(T)-6 b(elephone,")27 b(Pro)r(c.)35 b(IEEE)26 b(Spring)g(COMPCON)g(\(1982\),) h(133-137.)1867 5416 y Fm(22-1)p eop %%Page: 2 2 2 1 bop 150 159 a Ff(3.1)112 b(T)-9 b(ry)37 b(1)150 330 y Fn(Anna:)46 b Fm(Encrypts)29 b(a)i(0)f(and)g(a)h(1)g(and)e(sends)h (to)h(Adrian)e(in)g(random)g(order)150 518 y Fn(Adrian:)46 b Fm(Cho)s(oses)30 b(one)g(of)h(the)f(ciphertexts)150 705 y Fn(Anna:)46 b Fm(Decrypts)30 b(b)s(oth)g(ciphertexts)291 918 y(There)23 b(is)f(a)h(problem)f(in)g(this)g(sc)m(heme.)39 b(Encryption)21 b(is)h(not)i(designed)e(to)i(handle)d(this)h (application.)150 1031 y(W)-8 b(e)32 b(need)e(another)g(building)d(blo) s(c)m(k.)150 1274 y Ff(3.2)112 b(T)-9 b(ry)37 b(2{Commitmen)m(t)d(Sc)m (heme)150 1446 y Fn(Anna:)46 b Fm(Puts)29 b(a)i(0)g(and)f(a)g(1)h(in)m 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b(answ)m(er)f(question)g(1)h(ab)s(o)m(v)m(e,)h(w)m(e)f(b)s(egin)e(with) g(a)h(standard)g(T)-8 b(uring)30 b(mac)m(hine.)43 b(T)-8 b(o)32 b(b)s(e)f(in)m(teractiv)m(e,)150 3487 y(w)m(e)26 b(add)e(t)m(w)m(o)j(tap)s(es)e(in)f(addition)f(to)j(the)f(default)g (tap)s(e:)38 b Fe(T)2154 3502 y Fd(send)2328 3487 y Fm(and)24 b Fe(T)2552 3501 y Fd(r)r(eceiv)r(e)2780 3487 y Fm(.)39 b Fe(T)2897 3502 y Fd(send)3071 3487 y Fm(con)m(tains)25 b(the)g(data)150 3600 y(that)35 b(T)-8 b(uring)32 b(mac)m(hine)i Fe(T)46 b Fm(sends)33 b(to)i(others,)g(and)e Fe(T)2007 3614 y Fd(r)r(eceiv)r(e)2269 3600 y Fm(logs)h(the)g(data)h Fe(T)46 b Fm(receiv)m(es)35 b(from)f(others.)150 3713 y Fe(T)40 b Fm(op)s(erates)28 b(in)e(rounds.)38 b(That)27 b(is,)g(in)f(the)h(\\send")g(round,)g(it)g(writes)f(output)h(to)h(its)e Fe(T)3145 3728 y Fd(send)3294 3713 y Fm(,)i(and)f(in)f(the)150 3825 y(\\receiv)m(e")32 b(round,)d(it)h(reads)g(data)h(from)f(its)g Fe(T)1758 3839 y Fd(r)r(eceiv)r(e)1986 3825 y Fm(.)291 3938 y(Tw)m(o)h(ITMs,)f Fe(A)g Fm(and)g Fe(B)5 b Fm(,)30 b(are)h(comm)m(unicating)f(if)f Fe(A)p Fm('s)i Fe(T)2255 3953 y Fd(send)2434 3938 y Fm(is)e Fe(B)5 b Fm('s)30 b Fe(T)2743 3952 y Fd(r)r(eceiv)r(e)3001 3938 y Fm(and)g(vice)g(v)m (ersa.)150 4182 y Ff(4.2)112 b(Notation)150 4353 y Fm(If)30 b Fe(A)g Fm(and)g Fe(B)35 b Fm(are)c(ITMs,)f(then)1385 4671 y Fe(a)25 b Fc( )g Fm(\()p Fe(A)p Fm(\()p Fe(x)p Fm(\))h Fc($)g Fe(B)5 b Fm(\()p Fe(y)s Fm(\)\))25 b Fc(!)g Fe(b)1285 b Fm(\(1\))291 4875 y(The)33 b(notation)h(\\\()p Fe(A)p Fm(\()p Fe(x)p Fm(\))e Fc($)f Fe(B)5 b Fm(\()p Fe(y)s Fm(\)\)")34 b(means)f(that)i Fe(A)e Fm(and)g Fe(B)38 b Fm(are)c(comm)m(unicating.)50 b Fe(x)33 b Fm(and)g Fe(y)k Fm(are)150 4988 y(resp)s(ectiv)m(ely)h Fe(A)p Fm('s)h(and)f Fe(B)5 b Fm('s)38 b(input)f(on)i(their)e(normal,)j (default)e(tap)s(es.)66 b Fe(a)39 b Fm(is)e Fe(A)p Fm('s)i(output,)i Fe(b)e Fm(is)e Fe(B)5 b Fm('s)150 5101 y(output.)1867 5416 y(22-2)p eop %%Page: 3 3 3 2 bop 150 159 a Fj(5)135 b(De\014nition)45 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Fm(\))i(and)f(the)g(Receiv)m(er)h(\()p Fe(B)5 b Fm(\))30 b(suc)m(h)g(that)h(they)150 1039 y(ha)m(v)m(e)h(the)e (follo)m(wing)f(three)i(prop)s(erties:)150 1244 y Fn(Correctness)46 b Fm(If)e(w)m(e)g(put)g(a)h(0)g(in,)h(a)f(0)g(comes)g(out.)83 b(A)44 b(bidder)e(w)m(ould)h(w)m(an)m(t)i(to)g(ha)m(v)m(e)h(his)d(bid) 377 1356 y(correctly)36 b(rev)m(ealed.)55 b(It)36 b(w)m(ould)e(b)s(e)g (really)g(unfortunate)h(to)h(bid)d($100)k(for)e(a)h(lamp,)f(but)g(ha)m (v)m(e)377 1469 y(the)j(bid)d(rev)m(ealed)i(to)h(b)s(e)f($100,000!)64 b(This)35 b(is)i(similar)d(to)k(the)f(faithfulness)e(requiremen)m(t)h (for)377 1582 y(encryption)30 b(sc)m(hemes.)377 1823 y Fe(P)13 b(r)s Fm([)p Fe(u)25 b Fc( )h Fe(A)p Fm(\(1)859 1790 y Fd(k)902 1823 y Fe(;)15 b(b;)g Fm(\\Commit")q(\))26 b Fc($)f Fe(B)5 b Fm(\(1)1767 1790 y Fd(k)1810 1823 y Fe(;)15 b Fm(\\Commit"\))26 b Fc(!)f Fe(v)s Fm(;)582 1936 y Fe(A)p Fm(\(1)730 1903 y Fd(k)773 1936 y Fe(;)15 b(u;)g Fm(\\Rev)m(eal)s(\))25 b Fc($)g Fe(B)5 b Fm(\(1)1546 1903 y Fd(k)1589 1936 y Fe(;)15 b(v)s(;)g Fm(\\Rev)m(eal")s(\))26 b Fc(!)f Fe(b)2288 1903 y Fb(0)2337 1936 y Fm(:)g Fe(b)g Fm(=)g Fe(b)2586 1903 y Fb(0)2610 1936 y Fm(])g(=)g(1)150 2177 y Fn(Secrecy)47 b Fm(No)40 b(matter)g(what)g(Bob)g(do)s(es,)h(he)f (can't)g(kno)m(w)g(what's)g(in)e(the)i(en)m(v)m(elop)s(e.)68 b(This)38 b(migh)m(t)377 2290 y(cause)33 b(him)e(to)i(consisten)m(tly)f (bid)f($1)i(o)m(v)m(er)h(Alice,)e(rather)h(than)f(bidding)d(the)k (maxim)m(um)e(price)377 2402 y(he)39 b(is)g(willing)c(to)40 b(pa)m(y)g(for)f(an)g(item.)67 b(Again,)41 b(this)d(is)h(v)m(ery)g (similar)e(to)j(what)f(w)m(e)g(ha)m(v)m(e)i(seen)377 2515 y(b)s(efore)30 b(in)f(encryption)h(sc)m(hemes.)377 2756 y Fc(8)25 b(f)p Fe(C)563 2771 y Fd(k)606 2756 y Fc(g)h(2)f Fe(I)7 b(P)13 b(P)g(T)g(F)38 b Fc(9)p Fe(\027)6 b Fm(\()p Fe(k)s Fm(\))30 b(suc)m(h)g(that)582 2869 y(Pr)o([)p Fe(b)c Fc($)f Fm(0)p Fe(;)15 b Fm(1)q(;)g Fe(u)26 b Fc( )f Fe(A)p Fm(\(1)1398 2836 y Fd(k)1442 2869 y Fe(;)15 b(b;)g Fm(\\Commit")26 b Fc($)f Fe(C)2182 2884 y Fd(k)2225 2869 y Fm(\(1)2305 2836 y Fd(k)2348 2869 y Fe(;)15 b Fm(\\Commit")q(\))25 b Fc(!)h Fe(b)3019 2836 y Fb(0)3067 2869 y Fm(:)582 2982 y Fe(b)621 2949 y Fb(0)669 2982 y Fm(=)f Fe(b)p Fm(])h Fc(\024)f Fm(1)p Fe(=)p Fm(2)c(+)f Fe(\027)6 b Fm(\()p Fe(k)s Fm(\))377 3223 y(Note)32 b(that)f Fe(P)13 b(P)g(T)g(F)43 b Fm(is)29 b(no)m(w)i Fe(I)7 b(P)13 b(P)g(T)g(F)g Fm(,)30 b(since)g(the)h(proto)s(col)f(is)f(in)m(teractiv)m(e.)150 3408 y Fn(Binding)46 b Fm(W)-8 b(e)23 b(cannot)f(op)s(en)e(the)i(en)m (v)m(elop)s(e)f(di\013eren)m(t)g(w)m(a)m(ys)h(to)g(ha)m(v)m(e)h (di\013eren)m(t)d(outputs.)37 b(This)20 b(is)g(new.)377 3520 y(In)25 b(an)f(encryption)g(sc)m(heme,)j(w)m(e)f(do)f(not)g(care)h (if)d(it)i(is)f(p)s(ossible)e(to)k(decrypt)f(something)f(m)m(ultiple) 377 3633 y(w)m(a)m(ys.)41 b(In)26 b(fact,)i(that)g(migh)m(t)e(b)s(e)g (an)h(adv)-5 b(an)m(tage,)29 b(since)e(it)f(w)m(ould)f(k)m(eep)j(the)f (adv)m(ersary)g(guessing)377 3746 y(what)36 b(the)g(input)e(w)m(as.)57 b(In)35 b(a)h(commitmen)m(t)g(sc)m(heme,)i(ho)m(w)m(ev)m(er,)h(w)m(e)d (m)m(ust)f(b)s(e)h(unam)m(biguous.)377 3859 y(This)29 b(could)g(allo)m(w)h(Alice)g(to)h(e\013ectiv)m(ely)g(c)m(hange)h(her)e (bid)e(after)j(the)g(auction)f(closes.)377 4100 y Fc(8)25 b(f)p Fe(C)563 4115 y Fd(k)606 4100 y Fc(g)h(2)f Fe(I)7 b(P)13 b(P)g(T)g(F)38 b Fc(9)p Fe(\027)6 b Fm(\()p Fe(k)s Fm(\))30 b(suc)m(h)g(that)582 4213 y(Pr)o([)p Fe(u)c Fc( )f Fe(C)963 4228 y Fd(k)1006 4213 y Fm(\(1)1086 4180 y Fd(k)1129 4213 y Fm(\))h Fc($)f Fe(B)5 b Fm(\(1)1460 4180 y Fd(k)1503 4213 y Fe(;)15 b Fm(\\Commit"\))26 b Fc(!)f Fe(v)k Fm(:)582 4326 y Fe(C)647 4341 y Fd(k)689 4326 y Fm(\()p Fe(u;)15 b Fm(0\))27 b Fc($)e Fe(B)5 b Fm(\(1)1193 4293 y Fd(k)1236 4326 y Fe(;)15 b(v)s(;)g Fm(\\Rev)m(eal")s(\))26 b Fc(!)f Fe(b)1935 4293 y Fb(0)1935 4350 y Fi(0)1975 4326 y Fm(;)582 4439 y Fe(C)647 4454 y Fd(k)689 4439 y Fm(\()p Fe(u;)15 b Fm(1\))27 b Fc($)e Fe(B)5 b Fm(\(1)1193 4406 y Fd(k)1236 4439 y Fe(;)15 b(v)s(;)g Fm(\\Rev)m(eal")s(\))26 b Fc(!)f Fe(b)1935 4406 y Fb(0)1935 4463 y Fi(1)2000 4439 y Fm(:)g Fe(b)2089 4406 y Fb(0)2089 4463 y Fi(0)2154 4439 y Fc(6)p Fm(=)g Fe(b)2289 4406 y Fb(0)2289 4463 y Fi(1)2328 4439 y Fm(])h Fe(<)f(\027)6 b Fm(\()p Fe(k)s Fm(\))377 4680 y(This)27 b(prop)s(ert)m(y)i(is)f(the)h(one)g(that)h(sets)f(commitmen)m(t)g(sc)m (hemes)h(apart)f(from)g(encryption)f(and)g(is)377 4792 y(wh)m(y)f(w)m(e)h(m)m(ust)f(use)g(a)h(commitmen)m(t)g(sc)m(heme)g (rather)f(than)g(encryption)f(to)i(e\013ectiv)m(ely)g(p)s(erform)377 4905 y(the)j(coin)f(\015ip)e(o)m(v)m(er)k(the)f(telephone.)377 5054 y(As)26 b(an)f(exercise,)i(sho)m(w)f(that)g(the)g(ab)s(o)m(v)m(e)g (de\014nition)e(of)h(binding)e(is)h(equiv)-5 b(alen)m(t)25 b(to)i(the)e(follo)m(wing)377 5167 y(one:)1867 5416 y(22-3)p eop %%Page: 4 4 4 3 bop 377 159 a Fc(8)25 b(f)p Fe(C)563 174 y Fd(k)606 159 y Fc(g)h(2)f Fe(I)7 b(P)13 b(P)g(T)g(F)38 b Fc(9)p Fe(\027)6 b Fm(\()p Fe(k)s Fm(\))30 b(suc)m(h)g(that)582 271 y(Pr)o([)p Fe(u)c Fc( )f Fe(C)963 286 y Fd(k)1006 271 y Fm(\(1)1086 239 y Fd(k)1129 271 y Fm(\))h Fc($)f Fe(B)5 b Fm(\(1)1460 239 y Fd(k)1503 271 y Fe(;)15 b Fm(\\Commit"\))26 b Fc(!)f Fe(v)k Fm(:)582 384 y Fe(C)647 399 y Fd(k)689 384 y Fm(\()p Fe(u;)15 b Fm(0\))27 b Fc($)e Fe(B)5 b Fm(\(1)1193 351 y Fd(k)1236 384 y Fe(;)15 b(v)s(;)g Fm(\\Rev)m(eal")s(\))26 b Fc(!)f Fe(b)1935 351 y Fb(0)1935 409 y Fi(0)1975 384 y Fm(;)582 497 y Fe(C)647 512 y Fd(k)689 497 y Fm(\()p Fe(u;)15 b Fm(1\))27 b Fc($)e Fe(B)5 b Fm(\(1)1193 464 y Fd(k)1236 497 y Fe(;)15 b(v)s(;)g Fm(\\Rev)m(eal")s (\))26 b Fc(!)f Fe(b)1935 464 y Fb(0)1935 522 y Fi(1)2000 497 y Fm(:)g Fe(b)2089 464 y Fb(0)2089 522 y Fi(0)2154 497 y Fm(=)g(0)p Fe(andb)2484 464 y Fb(0)2484 522 y Fi(1)2549 497 y Fm(=)g(1])h Fe(<)f(\027)6 b Fm(\()p Fe(k)s Fm(\))150 784 y Fj(6)135 b(Constructions)45 b(of)g(Commitmen)l(t)i(Sc)l(hemes)150 990 y Ff(6.1)112 b(O)m(WP-based)150 1162 y Fm(Let)29 b Fe(p)f Fm(b)s(e)g(a)h(O)m(WP)-8 b(.)29 b(If)f Fe(p)h Fm(is)e(a)i(O)m(WP)-8 b(,)30 b(then)e(consider)f 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s(duce)e(the)h(idea)f(of)i(an)f(in)m(teractiv)m(e)g(pro)s(of)g(is)f(b)m (y)h(an)f(example,)i(and)e(only)g(then)150 2093 y(to)j(giv)m(e)g(a)g (formal)e(de\014nition.)39 b(That)30 b(is)f(what)i(w)m(e)g(do:)150 2336 y Fm(1.1)112 b(Example:)49 b(Graph)39 b(Non-Isomorphism)d(problem) 150 2508 y Fq(Giv)m(en)j(t)m(w)m(o)h(non-isomorphic)d(graphs,)j(G)g (and)e(H,)h(Non-Isomorphism)f(problem)f(is)h(to)i(pro)m(v)m(e)f(that) 150 2621 y(they)31 b(are)f(non-isomorphic.)39 b(This)28 b(problem)h(will)f(b)s(e)i(hereinafter)f(referred)h(to)h(as)g(NISO)e (problem.)291 2734 y(Recall)24 b(that)i(t)m(w)m(o)g(graphs)d(are)i (isomorphic)e(if)h(the)g(v)m(ertices)i(of)e(one)h(can)g(b)s(e)f(relab)s (elled)e(\(p)s(erm)m(uted\))150 2847 y(to)34 b(get)h(the)e(other)h (graph.)50 b(F)-8 b(or)34 b(example,)g(the)g(graphs)e Fl(f)p Fq(\(1)p Fk(;)15 b Fq(2\))p Fk(;)g Fq(\(2)p Fk(;)g Fq(4\))p Fk(;)g Fq(\(4)p Fk(;)h Fq(3\))q Fk(;)f Fq(\(3)q Fk(;)g Fq(1)q(\))p Fl(g)40 b Fq(and)33 b(the)g(graph)150 2960 y Fl(f)p Fq(\(2)p Fk(;)15 b Fq(1\))p Fk(;)g Fq(\(1)p Fk(;)g Fq(3\))p Fk(;)g Fq(\(3)p Fk(;)h Fq(4\))p Fk(;)g Fq(\(4)q Fk(;)f Fq(2\))q Fl(g)38 b Fq(are)32 b(isomorphic,)e(since)i (relab)s(elling)c(v)m(ertices)33 b(of)e(the)h(second)g(pro)s(duces)150 3073 y(the)f(\014rst.)40 b(W)-8 b(e)31 b(just)f(need)g(to)h(rename)g(3) f(to)h(b)s(e)f(2,)h(4)g(to)g(b)s(e)f(3)h(and)e(2)i(to)g(b)s(e)f(4.)291 3186 y(Ho)m(w)m(ev)m(er,)40 b(the)c(graph)f Fl(f)p Fq(\(1)p Fk(;)15 b Fq(2\))p Fk(;)g Fq(\(2)p Fk(;)g Fq(4\))p Fk(;)g Fq(\(3)p Fk(;)h Fq(4\))q Fk(;)f Fq(\(4)q Fk(;)g Fq(1)q(\))p Fl(g)42 b Fq(is)35 b(not)h(ismorphic)e(to)j(either)e(one)h(of)g(those.) 150 3298 y(In)29 b(this)f(case)i(the)g(pro)s(of)f(is)f(easy)i(-)g(in)e (the)h(last)h(graph)e(the)i(v)m(ertix)f(4)h(has)f(3)h(inciden)m(t)e (edges)i(on)f(it,)h(but)150 3411 y(in)f(previous)g(t)m(w)m(o)j(graphs,) e(ev)m(ery)h(v)m(ertix)f(has)g(only)g(t)m(w)m(o)h(edges)g(inciden)m(t)e (on)i(it.)291 3624 y(Recall)h(the)g(concept)i(of)e(a)h(classical)e(pro) s(of)h(system.)46 b(Giv)m(en)32 b(a)h(language)g Fk(L)p Fq(,)g(a)f(pro)s(of)g(system)g(for)150 3737 y Fk(L)i Fq(is)f(a)h(pair)e(of)i(algorithms)f Fk(P)13 b Fq(,)35 b Fk(V)54 b Fq(\(the)35 b(pro)m(v)m(er)f(and)f(the)h(v)m(eri\014er\),)h (suc)m(h)e(that)i(on)f(input)d Fk(x)h Fl(2)e Fk(L)p Fq(,)35 b Fk(P)150 3850 y Fq(pro)s(duces)c Fk(p)p Fq(,)h(and)f(on)h(input)e(\() p Fk(x;)15 b(p)p Fq(\),)33 b Fk(V)52 b Fq(accepts,)34 b(and)e(on)f(input)f Fk(x)e Fl(62)g Fk(L)p Fq(,)k(there)g(is)f(no)h Fk(p)f Fq(for)h(whic)m(h)f Fk(V)150 3963 y Fq(accepts)25 b(\()p Fk(x;)15 b(p)p Fq(\).)39 b(W)-8 b(e)25 b(generally)e(w)m(an)m(t) i Fk(p)e Fq(to)h(b)s(e)f(p)s(olynomial)f(length)h(in)f Fk(x)i Fq(and)f Fk(V)44 b Fq(to)24 b(run)e(in)h(p)s(olynomial)150 4076 y(time.)40 b(Of)30 b(course,)h(languages)g(for)f(whic)m(h)f(this)g (is)h(p)s(ossible)d(are)k(exactly)g(the)g(collection)f Fk(N)10 b(P)j Fq(.)291 4188 y(Unfortunately)-8 b(,)34 b Fk(N)10 b(P)46 b Fq(is)33 b(not,)h(as)g(far)f(as)h(w)m(e)f(kno)m(w,)i (big)d(enough)h(to)h(con)m(tain)g(some)g(quite)f(useful)150 4301 y(languages.)54 b(F)-8 b(or)36 b(example,)g(if)e(one)h(w)m(an)m (ts)h(to)f(pro)m(v)m(e)h(that)f Fk(G)f Fl(6)2361 4276 y(\030)2361 4305 y Fq(=)2464 4301 y Fk(H)7 b Fq(,)36 b(one)f(could)f(write)g(do)m(wn)h(as)g(the)150 4414 y(pro)s(of)i(all)f (p)s(erm)m(utations)h(of)h Fk(G)p Fq(,)i(and)d(to)h(v)m(erify)f(the)h (pro)s(of)f(one)h(w)m(ould)e(c)m(hec)m(k)j(that)f(none)g(of)f(these)150 4527 y(p)s(erm)m(utations)20 b(is)h Fk(H)7 b Fq(.)37 b(Ho)m(w)m(ev)m(er,)26 b(this)20 b(pro)s(of)h(is)g(exp)s(onen)m(tially) e(long.)38 b(It)21 b(is)g(not)g(kno)m(wn)g(ho)m(w)h(to)g(pro)m(vide)150 4640 y(a)30 b(pro)s(of)f(of)i(length)e(p)s(olynomial)e(in)i Fk(max)p Fq(\()p Fl(j)p Fk(G)p Fl(j)p Fk(;)15 b Fl(j)p Fk(H)7 b Fl(j)p Fq(\),)32 b(with)d(the)h(v)m(eri\014er)f(also)h (running)d(in)i(p)s(olynomial)150 4753 y(time.)40 b(T)-8 b(o)31 b(help)e(us)h(with)f(this)g(problem,)g(w)m(e)i(turn)e(to)j(the)e (concept)h(of)g(in)m(teractiv)m(e)g(pro)s(ofs.)1838 5416 y(L23-1)p eop %%Page: 2 2 2 1 bop 150 159 a Fm(1.2)112 b(In)m(teractiv)m(e)36 b(pro)s(of)i(for)g (NISO)150 330 y Fq(In)h(the)i(classical)e(concept)i(of)f(the)g(pro)s (of)f(system,)k(the)d(pro)m(v)m(er)h(pro)s(duces)e(a)h(pro)s(of)f(of)i (a)f(fact,)k(and)150 443 y(the)36 b(v)m(eri\014er)f(than)h(c)m(hec)m (ks)h(it)e(and)h(accepts)h(or)f(rejects)g(the)g(pro)s(of.)57 b(Ho)m(w)m(ev)m(er,)39 b(in)c(real)g(life,)h(a)h(pro)s(of)150 556 y(is)i(an)m(ything)h(that)g(is)f(con)m(vincing.)69 b(So,)43 b(wh)m(y)d(no)m(w)g(allo)m(w)f(in)m(teractiv)m(e)i(pro)s(ofs,) h(where)e(the)g(v)m(eri\014er)150 669 y(b)s(ecomes)f(con)m(vinced)f(b)m (y)h(talking)e(to)j(the)e(pro)m(v)m(er,)k(not)c(b)m(y)h(just)f (examining)f(an)h(already)g(complete)150 782 y(pro)s(of)7 b(?)40 b(That)30 b(is)f(what)i(w)m(e)g(do.)291 895 y(Consider)c(the)h (NISO)g(problem.)39 b(Supp)s(ose)26 b(that)j(the)g(v)m(eri\014er)e(and) h(the)h(pro)m(v)m(er)g(run)e(the)h(follo)m(wing)150 1008 y(algorithm,)i(hereinafter)f(referred)h(to)h(as)g(Dialogue)f(A.)286 1218 y Fl(\017)46 b Fq(V)-8 b(eri\014er)24 b(pic)m(ks)g Fk(G)g Fq(or)g Fk(H)32 b Fq(at)25 b(random,)g(and)e(randomly)g(p)s(erm) m(utes)h(the)g(v)m(ertices;)j(sends)d(the)g(result)377 1331 y Fk(R)31 b Fq(to)g(the)g(pro)m(v)m(er.)286 1518 y Fl(\017)46 b Fq(Pro)m(v)m(er)31 b(\014nds)e(whic)m(h)g(of)i Fk(H)7 b Fq(,)30 b Fk(G)h Fq(is)e(isomorphic)g(to)i Fk(R)g Fq(and)f(sends)f(that)i(to)g(the)g(v)m(eri\014er.)286 1705 y Fl(\017)46 b Fq(If)32 b(the)g(pro)m(v)m(er)g(sends)f(the)i (wrong)e(graph)h(\(not)g(the)g(one)h(pic)m(k)m(ed)e(in)g(\014rst)g (step\))i(to)f(the)h(v)m(eri\014er,)377 1818 y(v)m(eri\014er)g (rejects;)j(otherwise)c(v)m(eri\014er)h(rep)s(eats)g(the)h(previous)e (steps)h(un)m(til)f(he)h(is)f(satis\014ed,)i(and)377 1931 y(then)c(accepts.)291 2142 y(It)38 b(should)e(b)s(e)i(clear)g(in)m (tuitiv)m(ely)e(that)i(Dialogue)h(A)f(is)f(con)m(vincing)g(with)g (small)f(probabilit)m(y)g(of)150 2255 y(error,)44 b(since)c(if)g Fk(G)i Fq(and)e Fk(H)48 b Fq(are)42 b(isomorphic,)g(the)f(probabilit)m (y)e(that)j(an)m(y)f(pro)m(v)m(er)h(will)c(b)s(e)j(able)f(to)150 2368 y(return)29 b(alw)m(a)m(ys)h(the)h(same)f(one)g(as)g(the)h(v)m (eri\014er)e(pic)m(k)m(ed)h(is)f(v)m(ery)h(small.)39 b(T)-8 b(o)30 b(mak)m(e)h(this)e(more)h(precise,)150 2480 y(w)m(e)h(need)f(to)h(de\014ne)f(formally)f(the)h(concept)h(of)g (an)f(in)m(teractiv)m(e)h(pro)s(of)f(system.)150 2724 y Fm(1.3)112 b(In)m(teractiv)m(e)36 b(pro)s(of)i(system)150 2895 y Fq(.)291 3008 y Fo(Defn:)48 b Fq(An)32 b(in)m(teractiv)m(e)h (pro)s(of)e(system)h(for)g(a)g(language)h Fk(L)f Fq(is)f(a)h(pair)f(of) h(in)m(teractiv)m(e)h(algorithms,)150 3121 y Fk(P)13 b Fq(,)31 b Fk(V)50 b Fq(suc)m(h)30 b(that:)286 3307 y Fl(\017)46 b Fq(Completeness:)40 b(All)29 b(true)i(statemen)m(ts)h (are)e(con)m(vincing.)719 3509 y Fl(9)p Fk(\027)g Fl(2)25 b Fk(neg)s(l)33 b Fl(8)p Fk(x)24 b Fl(2)h Fk(L)30 b(P)13 b(r)s Fq([\()p Fk(P)g Fq(\()p Fk(x)p Fq(\))27 b Fl($)e Fk(V)20 b Fq(\()p Fk(x)p Fq(\)\))26 b Fl(!)f Fk(d)h Fq(:)f Fk(d)h Fq(=)f Fk(Accept)p Fq(])h Fj(>)f Fq(1)c Fl(\000)f Fk(\027)6 b Fq(\()p Fl(j)p Fk(x)p Fl(j)p Fq(\))377 3712 y(The)37 b(probabilit)m(y)e(1)25 b Fl(\000)g Fk(\027)43 b Fq(ab)s(o)m(v)m(e)38 b(is)f(kno)m(wn)f(as)i(completeness)f (probabilit)m(y)-8 b(.)60 b(It)37 b(is)g(a)h(theorem)377 3825 y(that)24 b(an)m(y)g(in)m(teractiv)m(e)h(pro)s(of)d(system)i(can)g (b)s(e)f(c)m(hanged)h(to)g(ha)m(v)m(e)h(the)f(completeness)f (probabilit)m(y)377 3938 y(equal)30 b(1.)286 4125 y Fl(\017)46 b Fq(Soundness:)37 b(no)27 b(false)g(statemen)m(t)h(is)e(con)m(vincing) g(\(no)h(adv)m(ersary)g(can)h(con)m(vince)f(the)g(v)m(eri\014er)f(of) 377 4238 y(an)m(y)31 b(false)f(statemen)m(t\).)600 4440 y Fl(8)p Fk(x)25 b Fl(62)g Fk(L)30 b Fl(8)p Fk(A)25 b Fl(2)g Fk(Al)r(g)s(s)31 b Fl(9)p Fk(\027)f Fl(2)25 b Fk(neg)s(l)33 b(P)13 b(r)s Fq([\()p Fk(A)p Fq(\()p Fk(x)p Fq(\))26 b Fl($)f Fk(V)20 b Fq(\()p Fk(x)p Fq(\)\))27 b Fl(!)e Fk(d)g Fq(:)h Fk(d)f Fq(=)g Fk(Accept)p Fq(])i Fj(6)e Fk(\027)6 b Fq(\()p Fk(x)p Fq(\))377 4642 y(The)31 b(probabilit)m(y)f Fk(\027)37 b Fq(ab)s(o)m(v)m(e)c(is)d(kno)m(wn)h(as) h(soundness)e(probabilit)m(y)-8 b(.)43 b(Note)33 b(that)f(the)g(adv)m (ersary)377 4755 y Fk(A)f Fq(is)e(allo)m(w)m(ed)h(to)i(b)s(e)d (computationally)g(un)m(b)s(ounded)f(in)h(the)i(ab)s(o)m(v)m(e)g (de\014nition.)291 4941 y(In)23 b(addition,)g(w)m(e)g(demand)g(that)h Fk(V)43 b Fq(b)s(e)23 b(a)g(PPT,)g(since)g(w)m(e)g(w)m(an)m(t)h(pro)s (of)f(v)m(eri\014cation)g(to)h(b)s(e)e(e\016cien)m(t.)150 5054 y(W)-8 b(e)42 b(call)f(an)g(in)m(teractiv)m(e)h(pro)s(of)e(system) h(in)f(whic)m(h)g Fk(V)61 b Fq(is)40 b(e\016cien)m(t)i(an)f(e\016cien)m (t)h(in)m(teractiv)m(e)g(pro)s(of)150 5167 y(system.)1838 5416 y(L23-2)p eop %%Page: 3 3 3 2 bop 291 159 a Fq(No)m(w)37 b(that)f(w)m(e)g(ha)m(v)m(e)h(the)f (de\014nition,)f(let's)h(ask)g(whether)f(Dialogue)h(A)g(is)e(an)i(in)m (teractiv)m(e)h(pro)s(of)150 271 y(system.)k(The)30 b(answ)m(er)g(is)f (y)m(es.)291 478 y Fo(Claim:)47 b Fq(Dialogue)30 b(A)h(is)e(an)h (e\016cien)m(t)h(in)m(teractiv)m(e)g(pro)s(of)f(system)h(for)f(NISO.) 291 684 y Fo(Pr)-5 b(o)g(of:)47 b Fq(It)31 b(is)e(clear)i(that)g Fk(V)50 b Fq(is)29 b(e\016cien)m(t.)42 b(W)-8 b(e)31 b(v)m(erify)f(the)h(other)f(t)m(w)m(o)i(conditions.)291 797 y(1.)44 b(Completeness:)d(If)31 b Fk(G)c Fl(6)1203 772 y(\030)1203 801 y Fq(=)1300 797 y Fk(H)7 b Fq(,)32 b(then)f Fk(R)h Fq(is)e(isomorphic)f(to)j(at)g(most)f(one)h(of)f(them)g (\(and)g(exactly)150 910 y(one,)j(if)e(w)m(e)h(assume,)h(as)f(w)m(e)g (should,)f(that)h Fk(V)53 b Fq(is)32 b(pla)m(ying)g(b)m(y)h(the)g (rules\).)47 b(So)32 b(the)h(pro)m(v)m(er's)h(resp)s(onse)150 1022 y(is)29 b(alw)m(a)m(ys)i(the)g(same)g(as)f(what)h(v)m(eri\014er)e (exp)s(ects,)i(hence)g(completeness)f(is)f(one.)291 1135 y(2.)39 b(Soundness:)c(If)24 b Fk(G)1053 1110 y Fl(\030)1053 1140 y Fq(=)1149 1135 y Fk(H)31 b Fq(then)23 b Fk(R)h Fq(is)f(isomorphic)f(to)i(b)s(oth,)h(and)e(whic)m(h)f(one)i(the)g(v)m (eri\014er)f(exp)s(ects)150 1248 y(is)33 b(indep)s(enden)m(t)e(of)i Fk(R)q Fq(,)i(hence)e(indep)s(enden)m(t)f(of)h(the)h(adv)m(ersary's)g (c)m(hoice.)51 b(Th)m(us,)33 b(soundness)f(is)h(1)p Fk(=)p Fq(2)150 1361 y(for)e(a)g(single)e(round)g(of)i(the)g(algorithm;)f(to)i (ac)m(hiev)m(e)g(negligible)c(soundness,)i(let)h Fk(k)i Fq(b)s(e)e Fk(max)p Fq(\()p Fl(j)p Fk(G)p Fl(j)p Fk(;)15 b Fl(j)p Fk(H)7 b Fl(j)p Fq(\))150 1474 y(and)30 b(do)g Fk(k)k Fq(rounds)28 b(in)h(Dialogue)i(A.)3682 1654 y Fi(2)291 1833 y Fq(It)d(should)e(b)s(e)i(noted,)g(that)h(just)e(as)i (in)d(the)i(ab)s(o)m(v)m(e)i(pro)s(of,)e(for)f(a)i(general)f(pro)s(of)f (system)h(if)f(the)h(dif-)150 1946 y(ference)e(b)s(et)m(w)m(een)g (completeness)f(and)g(soundenss)f(can)h(b)s(e)g(made)g(non-negligible,) f(than)h(b)m(y)h(rep)s(eating)150 2059 y(the)31 b(algorithm)f(man)m(y)h (times)f(w)m(e)i(can)f(get)h(negligible)c(soundness)h(and)i(negligibly) d(smaller)h(than)i(one)150 2172 y(completeness.)150 2457 y Fn(2)135 b(Zero)45 b(Kno)l(wledge)i(Pro)t(ofs)150 2660 y Fq(It)d(is)f(often)i(useful)d(to)j(b)s(e)e(able)h(to)g(con)m(vince)h (someone)f(of)h(something)e(without)g(rev)m(ealing)h(more)150 2773 y(information)e(than)h(y)m(ou)h(w)m(an)m(t.)80 b(This)42 b(is)g(the)i(concept)g(that)g(is)f(formalized)f(as)i(Zero)f(Kno)m (wledge)150 2886 y(pro)s(ofs.)55 b(As)36 b(usual,)f(w)m(e)i(do)e(an)g (example)h(\014rst,)g(to)g(gain)f(in)m(tuition,)g(and)g(pro)s(ceed)h (to)g(formalize)f(the)150 2999 y(in)m(tuition)28 b(afterw)m(ards.)150 3241 y Fm(2.1)112 b(Example:)49 b(Quadratic)38 b(residuosit)m(y)e(pro)s (of)i(system)150 3412 y Fq(The)j(goal)g(is,)j(giv)m(en)d(an)g(RSA)g(mo) s(dulus)d Fk(n)j Fq(and)f(a)i(quadratic)f(residue)e Fk(a)i Fq(mo)s(d)g Fk(n)p Fq(,)i(to)f(pro)m(v)m(e)g(that)150 3525 y Fk(a)25 b Fl(2)g Fk(QR)q Fq(\()p Fk(n)p Fq(\).)41 b(The)30 b(pro)s(of)f(system)i(is)e(as)i(follo)m(ws:)286 3705 y Fl(\017)46 b Fq(Pro)m(v)m(er)31 b(pic)m(ks)f Fk(r)e Fl( )d Fh(Z)1147 3672 y Fg(\003)1147 3727 y Ff(n)1190 3705 y Fq(,)30 b(then)g(computes,)h Fk(s)25 b Fq(=)g Fk(r)2090 3672 y Fe(2)2159 3705 y Fq(and)30 b Fk(\013)h Fq(suc)m(h)f(that)h Fk(a)25 b Fq(=)g Fk(\013)3054 3672 y Fe(2)3094 3705 y Fq(.)286 3889 y Fl(\017)46 b Fq(Pro)m(v)m(er)31 b(sends)f Fk(s)p Fq(.)286 4074 y Fl(\017)46 b Fq(V)-8 b(eri\014er)30 b(pic)m(ks)g Fk(b)25 b Fl( )g(f)p Fq(0)p Fk(;)15 b Fq(1)p Fl(g)33 b Fq(and)c(sends)h(it.)286 4258 y Fl(\017)46 b Fq(Pro)m(v)m(er)31 b(computes)g Fk(t)25 b Fq(=)g Fk(r)s(\013)1330 4225 y Ff(b)1364 4258 y Fq(,)31 b(sends)e Fk(t)p Fq(.)286 4443 y Fl(\017)46 b Fq(V)-8 b(eri\014er)34 b(c)m(hec)m(ks)h(that)g Fk(t)1218 4410 y Fe(2)1289 4443 y Fq(=)c Fk(sa)1482 4410 y Ff(b)1516 4443 y Fq(.)51 b(If)34 b(not,)i(rejects,)g(if)d(y)m(es,)i(rep)s(eat)g Fl(j)p Fk(n)p Fl(j)f Fq(times,)h(and)e(if)g(all)g(pass,)377 4556 y(accept.)291 4735 y(W)-8 b(e)32 b(\014rst)d(sho)m(w)i(that)g (this)e(is)g(an)i(in)m(teractiv)m(e)g(pro)s(of)e(system)i(for)f (quadratic)g(residuosit)m(y)-8 b(.)291 4941 y Fo(Pr)j(o)g(of:)291 5054 y Fq(1.)45 b(Completeness:)e(This)29 b(is)i(one)h(for)f(eac)m(h)i (round,)e(since)g(the)h(pro)m(v)m(er)g(alw)m(a)m(ys)g(mak)m(es)h(an)e (answ)m(er)150 5167 y(that)g(v)m(eri\014es.)1838 5416 y(L23-3)p eop %%Page: 4 4 4 3 bop 291 159 a Fq(2.)57 b(Soundness:)50 b(Supp)s(ose)33 b Fk(a)i Fl(62)e Fk(QR)q Fq(\()p Fk(n)p Fq(\),)k(then)f(for)f(an)m(y)h Fk(s)g Fq(either)f Fk(s)e Fl(62)h Fk(QR)q Fq(\()p Fk(n)p Fq(\))i(or)f Fk(sa)f Fl(62)g Fk(QR)q Fq(\()p Fk(n)p Fq(\).)150 271 y Fk(b)j Fq(is)f(c)m(hosen)i(indep)s(enden)m(tly)c(of)j Fk(s)p Fq(,)h(so)f(v)m(eri\014er)g(rejects)g(with)f(probabilit)m(y)f(1) p Fk(=)p Fq(2.)61 b(F)-8 b(or)38 b Fk(k)i Fq(rounds)c(the)150 384 y(acception)31 b(probabilit)m(y)d(is)h(\(1)p Fk(=)p Fq(2\))1315 351 y Ff(k)1360 384 y Fq(,)i(so)f(soundness)f(holds)g(if)g (there)i(are)g Fl(j)p Fk(n)p Fl(j)f Fq(rounds.)3682 571 y Fi(2)291 757 y Fq(The)d(next)h(question)e(is,)i(what)f(do)g(w)m(e)h (mean)f(that)h(the)g(v)m(eri\014er)f(learns)f(nothing)g(from)h(this)g (pro)s(of)7 b(?)150 870 y(W)-8 b(e)34 b(recall)f(the)g(in)m(tuition)f (with)f(encryption:)46 b(whatev)m(er)34 b(an)f(adv)m(ersary)g(can)h (compute)g(after)f(ha)m(ving)150 983 y(seen)21 b(the)g(cyphertext)h (should)d(ha)m(v)m(e)j(b)s(een)e(computable)h(without)f(the)h (cyphertext.)38 b(In)21 b(the)g(case)h(of)f(zero)150 1096 y(kno)m(wledge)33 b(pro)s(ofs)f(w)m(e)h(w)m(an)m(t)h(to)f(mak)m(e) h(sure)e(that)h(whatev)m(er)h(an)f(adv)m(ersary)g Fk(V)2985 1063 y Fg(\003)3057 1096 y Fq(learns)f(in)m(teracting)150 1209 y(with)d Fk(P)43 b Fq(can)31 b(b)s(e)f(computed)g(without)f (talking)h(to)h Fk(P)13 b Fq(.)291 1322 y(Ho)m(w)m(ev)m(er,)47 b(this)40 b(creates)j(a)f(parado)m(x)g(-)g(if)e(the)i(v)m(eri\014er)e (learns)h(nothing,)j(ho)m(w)d(can)h(he)f(b)s(e)g(ev)m(er)150 1435 y(con)m(vinced)33 b(of)g(an)m(ything?)49 b(The)33 b(answ)m(er)g(is)f(that,)j(since)d(the)i(only)e(w)m(a)m(y)i Fk(P)46 b Fq(could)33 b(ha)m(v)m(e)h(c)m(heated)h(is)d(if)150 1548 y(he)26 b(knew)f(the)h(v)m(eri\014er's)g(random)f(tap)s(e,)i(v)m (eri\014er)e(is)g(con)m(vinced)h(b)s(ecause)g(he)g(kno)m(ws)g(his)e (random)i(tap)s(e.)150 1661 y(So)k(in)f(our)h(in)m(tuition)f(w)m(e)h (will)e(allo)m(w)i(the)h(randomness)e(that)i Fk(V)2379 1628 y Fg(\003)2449 1661 y Fq(had)f(to)h(b)s(e)f(used)f(in)g(computing) h(the)150 1774 y(same)35 b(things)d(that)j Fk(V)932 1741 y Fg(\003)1006 1774 y Fq(is)e(computing,)h(but)g(without)f(talking)g (to)i Fk(P)13 b Fq(.)52 b(This)32 b(brings)g(us)i(to)h(formalize)150 1887 y(ZK)30 b(pro)s(ofs)f(as)i(follo)m(ws:)150 2130 y Fm(2.2)112 b(F)-9 b(ormalization)35 b(of)j(ZK)f(pro)s(ofs)150 2301 y Fq(In)29 b(this)g(section,)h(w)m(e)h(will)c(write)i Fk(V)1378 2268 y Fg(\003)1418 2301 y Fq(\()p Fk(x;)15 b(R)q Fq(\))30 b(to)h(inicate)f(that)g(the)g(algorithm)f Fk(V)2923 2268 y Fg(\003)2993 2301 y Fq(ran)g(on)h(input)e Fk(x)i Fq(and)150 2414 y(its)g(random)f(tap)s(e)i(w)m(as)g Fk(R)q Fq(.)40 b Fk(R)31 b Fq(can)g(then)f(b)s(e)g(used)f(b)m(y)h (other)h(algorithms)e(as)i(a)g(usual)e(input.)291 2527 y Fo(Defn)7 b Fq(:)40 b(A)31 b(ZK)e(pro)s(of)h(system)g(for)h(a)f (language)h Fk(L)f Fq(is)g(an)g(in)m(teractiv)m(e)h(pro)s(of)f(system)g (suc)m(h)g(that)575 2730 y Fl(8)p Fk(V)698 2693 y Fg(\003)763 2730 y Fl(2)25 b Fk(I)7 b(P)13 b(P)g(T)g(F)43 b Fl(9)p Fk(M)35 b Fl(2)25 b Fk(P)13 b(P)g(T)g(F)43 b(s:t:)30 b Fl(8)p Fk(D)e Fl(2)d Fk(P)13 b(P)g(T)g(F)43 b Fl(9)p Fk(\027)30 b Fl(2)25 b Fk(neg)s(l)33 b(s:t:)p Fl(8)p Fk(x)24 b Fl(2)h Fk(L)1073 2933 y(P)13 b(r)s Fq([)p Fk(u)1265 2947 y Fe(0)1330 2933 y Fl( )25 b Fq(\()p Fk(V)1555 2896 y Fg(\003)1594 2933 y Fq(\()p Fk(x;)15 b(R)q Fq(\))26 b Fl($)f Fk(P)13 b Fq(\()p Fk(x)p Fq(\)\))p Fk(;)i(u)2288 2947 y Fe(1)2354 2933 y Fl( )26 b Fk(M)10 b Fq(\()p Fk(x;)15 b(R)q Fq(\))p Fk(;)950 3099 y(b)25 b Fl( )g(f)p Fq(0)p Fk(;)15 b Fq(1)p Fl(g)p Fk(;)g(b)1429 3062 y Fg(0)1480 3099 y Fl( )26 b Fk(D)s Fq(\()p Fk(x;)15 b(R)q(;)g(u)1964 3114 y Ff(b)1999 3099 y Fq(\))25 b(:)h Fk(b)f Fq(=)g Fk(b)2309 3062 y Fg(0)2332 3099 y Fq(])h Fj(6)f Fq(1)p Fk(=)p Fq(2)c(+)f Fk(\027)6 b Fq(\()p Fl(j)p Fk(x)p Fl(j)p Fq(\))291 3265 y(In)30 b(w)m(ords,)g(no)g(distinguisher)d Fk(D)33 b Fq(can)e(distinguish)26 b(output)k(of)h Fk(V)2564 3232 y Fg(\003)2634 3265 y Fq(from)f(that)g(of)h Fk(M)10 b Fq(.)291 3378 y(The)31 b(question)f(no)m(w)h(is)f(whether)g(the)i (example)e(ab)s(o)m(v)m(e)j(satis\014es)d(this)g(de\014nition.)40 b(The)31 b(answ)m(er)g(is)150 3491 y(y)m(es.)291 3703 y Fo(Pr)-5 b(o)g(of:)54 b Fq(The)33 b(k)m(ey)h(idea)g(to)g(construct)g Fk(M)44 b Fq(is)33 b(this:)46 b Fk(M)d Fq(has)34 b(access)h(to)f Fk(V)2855 3670 y Fg(\003)2895 3703 y Fq('s)g(randomness)e(and)h(to)150 3816 y(its)k(co)s(de,)k(so)d(he)g(can)g(sim)m(ulate)f(running)e(of)j Fk(V)1851 3783 y Fg(\003)1890 3816 y Fq(.)63 b(By)39 b(putting)d(things)h(on)h Fk(V)2947 3783 y Fg(\003)2987 3816 y Fq('s)g(receiv)m(e)g(tap)s(e)h(and)150 3929 y(reading)29 b(things)f(o\013)i(of)g(its)f(send)g(tap)s(e,)h Fk(M)39 b Fq(can)30 b(talk)g(to)g Fk(V)2198 3896 y Fg(\003)2238 3929 y Fq(.)40 b(If)29 b Fk(M)40 b Fq(can)30 b(do)f(it)g(so)h(that)g Fk(V)3282 3896 y Fg(\003)3351 3929 y Fq(notices)g(no)150 4042 y(di\013erence)h(b)s(et)m(w)m(een)i(this)e(and)g(talking)h(to)g Fk(P)13 b Fq(,)33 b(than)f Fk(M)42 b Fq(just)31 b(outputs)h(whatev)m (er)g Fk(V)3142 4009 y Fg(\003)3213 4042 y Fq(do)s(es,)g(and)g(the)150 4155 y(output)e(will)e(b)s(e)h(indistinguishable)c(from)30 b(that)h(of)g Fk(V)2030 4122 y Fg(\003)2100 4155 y Fq(talking)e(to)i Fk(P)13 b Fq(.)291 4268 y(So)30 b(ho)m(w)h(do)f(w)m(e)h(arrange)g(for)f Fk(M)41 b Fq(to)31 b(sim)m(ulate)f Fk(P)13 b Fq(?)40 b(W)-8 b(ell,)31 b(the)f(k)m(ey)i(is)d(to)i(go)g(bac)m(kw)m(ards.)42 b Fk(M)e Fq(do)s(es)150 4380 y(the)31 b(follo)m(wing:)286 4567 y Fl(\017)46 b Fq(Cho)s(ose)30 b Fk(t)c Fl( )f Fh(Z)933 4534 y Fg(\003)933 4589 y Ff(n)975 4567 y Fq(,)31 b Fk(b)1070 4534 y Fg(0)1118 4567 y Fl( )26 b(f)p Fq(0)p Fk(;)15 b Fq(1)p Fl(g)p Fq(,)32 b(set)f Fk(s)25 b Fq(=)g Fk(t)1851 4534 y Fe(2)1890 4567 y Fk(=a)1983 4534 y Ff(b)2013 4511 y Fd(0)2041 4567 y Fq(.)40 b(Send)29 b Fk(s)h Fq(to)h Fk(V)2586 4534 y Fg(\003)2625 4567 y Fq(.)286 4754 y Fl(\017)46 b Fq(Receiv)m(e)d Fk(b)f Fq(from)f Fk(V)1098 4721 y Fg(\003)1138 4754 y Fq(;)47 b(if)41 b Fk(b)j Fl(6)p Fq(=)f Fk(b)1541 4721 y Fg(0)1565 4754 y Fq(,)h('rewind')c Fk(V)2068 4721 y Fg(\003)2149 4754 y Fq(to)i(the)g(state)h(it)e(w)m(as) h(b)s(efore)f(it)g(receiv)m(ed)h Fk(s)377 4867 y Fq(\(this)36 b(can)h(b)s(e)g(done,)h(since)e Fk(M)47 b Fq(con)m(tains)37 b Fk(V)1950 4834 y Fg(\003)2026 4867 y Fq(inside)e(of)i(itself,)g(and)g (it)f(kno)m(ws)h(the)g(necessary)377 4980 y(randomness\).)286 5167 y Fl(\017)46 b Fq(If)30 b Fk(b)25 b Fq(=)g Fk(b)667 5134 y Fg(0)691 5167 y Fq(,)30 b(send)g Fk(t)p Fq(.)1838 5416 y(L23-4)p eop %%Page: 5 5 5 4 bop 291 159 a Fq(Note)36 b(that)f Fk(b)755 126 y Fg(0)812 159 y Fq(is)f(distributed)d(indep)s(enden)m(tly)h(on)i Fk(s)p Fq(,)h(and)f(hence,)h(since)f Fk(V)2966 126 y Fg(\003)3040 159 y Fq(only)g(sees)g Fk(s)g Fq(b)s(efore)150 271 y(pro)s(ducing)d Fk(b)p Fq(,)j(it)f(is)g(indep)s(enden)m(t)e(of)i Fk(b)p Fq(.)50 b(Hence)34 b(the)g(probabilit)m(y)d(of)i Fk(b)e Fq(=)f Fk(b)2822 238 y Fg(0)2878 271 y Fq(is)j(1)p Fk(=)p Fq(2)h(for)g(eac)m(h)g(round,)150 384 y(and)e(is)h(indep)s (enden)m(t)e(from)h(round)g(to)i(round.)47 b(Th)m(us)32 b(for)h(eac)m(h)h(round)e(of)h(con)m(v)m(ersation)h(b)s(et)m(w)m(een)g Fk(P)150 497 y Fq(and)c Fk(V)401 464 y Fg(\003)440 497 y Fq(,)h Fk(M)41 b Fq(will)29 b(do)i(on)g(a)m(v)m(erage)i(2)f(trials)d (in)h(the)h(con)m(v)m(ersation)h(with)e Fk(V)2757 464 y Fg(\003)2827 497 y Fq(it)h(con)m(tains)g(inside)e(itself.)150 610 y(Therefore,)h(the)h(exp)s(ected)g(running)c(time)j(is)g(2)p Fk(k)k Fq(trials,)29 b(and)h(is)f(p)s(olynomial.)291 723 y(W)-8 b(e)31 b(m)m(ust)e(sho)m(w)h(that)g Fk(V)1165 690 y Fg(\003)1234 723 y Fq(sees)g(the)f(same)h(con)m(v)m(ersation)h (when)d(it)i(is)e(inside)f Fk(M)40 b Fq(as)30 b(when)e(it)h(talks)150 836 y(to)i Fk(P)13 b Fq(.)291 1049 y Fo(Claim:)50 b Fq(F)-8 b(or)32 b(an)m(y)g(sequence)g Fk(q)j Fq(of)d Fk(k)j Fq(rounds,)c Fk(P)13 b(r)s Fq([)p Fk(q)34 b Fq(is)d(a)h(sequence)g(in)f(a)h(talk)g (with)e Fk(P)13 b Fq(])28 b(=)g Fk(P)13 b(r)s Fq([)p Fk(q)34 b Fq(is)150 1161 y(generated)d(b)m(y)g Fk(m)p Fq(].)291 1374 y Fo(Pr)-5 b(o)g(of:)45 b Fq(If)25 b Fk(q)j Fq(is)d(not)g(a)h(sequence)g(with)e Fk(i)p Fq(th)i(elemen)m(t,)h(for)e (eac)m(h)i Fk(i)p Fq(,)g(b)s(eing)d(\()p Fk(s)2852 1388 y Ff(i)2880 1374 y Fk(;)15 b(b)2959 1388 y Ff(i)2987 1374 y Fk(;)g(t)3060 1388 y Ff(i)3089 1374 y Fq(\))26 b(with)e Fk(t)3385 1341 y Fe(2)3385 1399 y Ff(i)3449 1374 y Fq(=)h Fk(s)3588 1341 y Ff(i)3616 1374 y Fk(a)3664 1341 y Ff(b)3694 1351 y Fc(i)3725 1374 y Fq(,)150 1487 y(the)h(probabilit)m(y)d(is)i(0)h(in)e(b)s(oth)h(cases.)40 b(So)25 b(supp)s(ose)f(that)j(the)f(ab)s(o)m(v)m(e)g(relation)f(holds)f (for)i(eac)m(h)g(elemen)m(t)150 1600 y(of)i(the)h(sequence)g Fk(q)s Fq(.)40 b(Then)27 b Fk(P)13 b(r)s Fq([\()p Fk(s)1343 1614 y Fe(1)1382 1600 y Fk(;)i(b)1461 1614 y Fe(1)1501 1600 y Fk(;)g(t)1574 1614 y Fe(1)1614 1600 y Fq(\))p Fk(;)g(:)g(:)g(:)i(;)e Fq(\()p Fk(s)1929 1615 y Ff(k)1972 1600 y Fk(;)g(b)2051 1615 y Ff(k)2094 1600 y Fk(;)g(t)2167 1615 y Ff(k)2210 1600 y Fq(\))28 b(comes)h(from)f Fk(P)13 b Fq(])29 b(is)e(the)h(probabilit)m(y)e(that)150 1755 y Fk(P)43 b Fq(pic)m(k)m(ed)31 b Fk(s)575 1769 y Fe(1)629 1755 y Fk(:)15 b(:)g(:)h(s)793 1770 y Ff(k)835 1755 y Fq(,)31 b(i.e.)1047 1654 y Fb(\020)1157 1719 y Fe(1)p 1112 1734 126 4 v 1112 1788 a Fg(j)p Fa(Z)1182 1769 y Fd(\003)1182 1804 y Fc(n)1218 1788 y Fg(j)1248 1654 y Fb(\021)1302 1677 y Ff(k)1345 1755 y Fq(.)291 1891 y(T)-8 b(o)31 b(compute)f Fk(P)13 b(r)s Fq([\()p Fk(s)1011 1905 y Fe(1)1050 1891 y Fk(;)i(b)1129 1905 y Fe(1)1169 1891 y Fk(;)g(t)1242 1905 y Fe(1)1282 1891 y Fq(\))p Fk(;)g(:)g(:)g(:)i(;)e Fq(\()p Fk(s)1597 1906 y Ff(k)1640 1891 y Fk(;)g(b)1719 1906 y Ff(k)1762 1891 y Fk(;)g(t)1835 1906 y Ff(k)1878 1891 y Fq(\))30 b(comes)h(from)e Fk(M)10 b Fq(],)31 b(w)m(e)f(note)h (that)f Fk(M)40 b Fq(w)m(ould)29 b(do)h(an)150 2004 y(equiv)-5 b(alen)m(t)35 b(thing)f(if)h(it)g(just)f(prepic)m(k)m(ed)h(man)m(y)h (\(sa)m(y)g Fk(k)2110 1971 y Fe(2)2150 2004 y Fq(\))g(triples)d(\()p Fk(s)2582 1971 y Fg(0)2582 2029 y Ff(i)2610 2004 y Fk(;)15 b(b)2689 1971 y Fg(0)2689 2029 y Ff(i)2718 2004 y Fk(;)g(t)2791 1971 y Fg(0)2791 2029 y Ff(i)2819 2004 y Fq(\),)37 b(and)e(then)g(go)i (through)150 2117 y(them)49 b(and)f(junk)g(an)m(y)h(where)f Fk(b)1332 2084 y Fg(0)1332 2142 y Ff(i)1409 2117 y Fq(do)s(es)h(not)g (matc)m(h)h(the)f Fk(b)g Fq(sen)m(t)g(b)m(y)g Fk(V)2796 2084 y Fg(\003)2836 2117 y Fq(.)96 b(Since)48 b(the)h(c)m(hoice)h(of) 150 2229 y(all)44 b(these)i(triples)d(is)i(indep)s(enden)m(t)e(from)i (the)g(c)m(hoice)i(of)e(whic)m(h)f(triples)g(to)i(junk,)i(the)d (probabil-)150 2342 y(it)m(y)32 b Fk(P)13 b(r)s Fq([\()p Fk(s)506 2356 y Fe(1)546 2342 y Fk(;)i(b)625 2356 y Fe(1)664 2342 y Fk(;)g(t)737 2356 y Fe(1)777 2342 y Fq(\))p Fk(;)g(:)g(:)g(:)i (;)e Fq(\()p Fk(s)1092 2357 y Ff(k)1135 2342 y Fk(;)g(b)1214 2357 y Ff(k)1257 2342 y Fk(;)g(t)1330 2357 y Ff(k)1373 2342 y Fq(\))33 b(comes)g(from)f Fk(P)13 b Fq(])33 b(reduces)f(just)g (to)h Fk(P)13 b(r)s Fq([)p Fk(M)42 b Fq(pic)m(ks)32 b Fk(t)3211 2356 y Fe(1)3265 2342 y Fk(:)15 b(:)g(:)i(t)3420 2357 y Ff(k)3462 2342 y Fq(,)33 b(whic)m(h)150 2498 y(again)d(is)484 2397 y Fb(\020)593 2462 y Fe(1)p 548 2477 V 548 2530 a Fg(j)p Fa(Z)618 2511 y Fd(\003)618 2547 y Fc(n)654 2530 y Fg(j)684 2397 y Fb(\021)738 2420 y Ff(k)781 2498 y Fq(.)3682 2685 y Fi(2)291 2873 y Fq(The)i(ab)s(o)m(v)m(e)h(claim)e (\014nishes)f(the)i(pro)s(of)f(that)i(this)e(is)g(a)h(ZK)f(system,)i (since)e(the)i(only)e(part)h(of)g(the)150 2986 y(de\014nition)c(not)j (y)m(et)g(pro)m(v)m(ed,)g(the)g(e\016ciency)f(of)h Fk(V)20 b Fq(,)30 b(is)g(clear.)3682 3174 y Fi(2)291 3361 y Fq(In)e(fact,)j(w)m (e)f(ha)m(v)m(e)g(sho)m(wn)e(more)i(than)e(what)h(the)h(de\014nition)c (of)k(ZK)e(system)h(demands.)39 b(W)-8 b(e)31 b(ha)m(v)m(e)150 3474 y(sho)m(wn)i(that)h Fk(M)44 b Fq(and)32 b Fk(V)1014 3441 y Fg(\003)1087 3474 y Fq(talking)g(to)i Fk(P)47 b Fq(pro)s(duce)32 b(equal)h(probabilit)m(y)e(distributions,)g(so)j(ev) m(en)g(if)e(w)m(e)150 3587 y(allo)m(w)m(ed)k Fk(D)j Fq(to)e(b)s(e)f (computationally)f(un)m(b)s(ounded)f(and)i(to)h(run)d(either)i Fk(M)47 b Fq(or)36 b Fk(V)3057 3554 y Fg(\003)3133 3587 y Fq(talking)f(to)i Fk(P)50 b Fq(as)150 3700 y(man)m(y)30 b(times)g(as)h(it)f(w)m(an)m(ts,)h(it)f(still)e(w)m(ould)i(not)g(b)s(e) g(able)g(to)h(distinguish)c(b)s(et)m(w)m(een)k(the)f(t)m(w)m(o.)291 3813 y(But)37 b(in)d(general,)k(ZK)e(systems)g(cannot)h(necessarily)e (b)s(e)h(ran)g(more)g(than)g(once)h(without)e(giving)150 3926 y(kno)m(wledge)e(a)m(w)m(a)m(y)-8 b(.)49 b(F)-8 b(or)33 b(example,)g(supp)s(ose)e(that)i Fk(P)46 b Fq(in)31 b(the)i(previous)e(example)h(w)m(ere)h(really)e(stupid)150 4039 y(and)j(published,)e(for)i(a)h(prime)d Fk(p)i Fq(dividing)d Fk(n)j Fq(and)g(a)h(random)e(string)h Fk(r)j Fq(it)c(\014xed)h(in)f (adv)-5 b(ance,)36 b(either)150 4152 y Fk(p)20 b Fl(\010)g Fk(r)34 b Fq(or)d Fk(r)i Fq(eac)m(h)f(time)e(it)h(ran)f(\(c)m(ho)s (osing)h(an)g(option)f(randomly\).)41 b(Then)30 b(with)g(one)h(run,)e (this)h(is)g(still)150 4265 y(a)g(ZK)f(system)h(\(since)g(the)g (additional)d(published)f(information)j(lo)s(oks)g(random\),)h(but)f (with)f(m)m(ultiple)150 4377 y(runs,)35 b(after)h(an)g(a)m(v)m(erage)i (of)e(t)m(w)m(o)g(runs,)g(the)f(adv)m(ersary)h(w)m(ould)f(ha)m(v)m(e)h (seen)g(b)s(oth)f Fk(r)j Fq(and)c Fk(r)27 b Fl(\010)c Fk(p)p Fq(,)37 b(and)150 4490 y(hence)31 b(w)m(ould)e(learn)g Fk(n)p Fq('s)h(factorization!)1838 5416 y(L23-5)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF tex.html0100664003325600237220000001237107352125250011771 0ustar mliskovta6875The Very Basics of TeX

The Very Basics of TeX

TeX is a typesetting programming language. The most important thing to know if you want to write your problem sets in TeX is that it runs in two modes: normal mode and math mode. In normal mode, only text is handled. You can alter the text (for example, making it bold-face or italic) but generally you can't use any fancy characters or typsetting in normal mode. A new paragraph is started when you leave a blank line in the TeX file.

Some normal-mode commands to know:

In math mode, letters are written in italics (they are assumed to be variables or functions, not words), and more advanced typsetting is allowed. In order to switch back and forth between math mode and normal mode, you use the $ character. That is, if you add "$x^2$" in the middle of a bunch of normal mode text, the first '$' changes the mode to math mode, the 'x^2' is interpreted and typset correctly, and then the second '$' changes the mode back to normal mode.

Some math-mode commands to know:

There are many other commands you can use in math mode. For a more extensive LaTeX reference online, try The Not So Short Introduction to LaTeX 2e, especially the List of Mathematical Symbols.

One final word about math mode: When you use the single '$' to go into math mode and then back out, this inserts your mathematical type in the middle of the paragraph. If what you want is to give an equation or mathematical phrase on its own, go into math mode with a double '$$' and back out the same way. This will put that incident of math mode on its own line, centered. Some commands will come out looking different if you put them on their own line. For instance, if you use $\sum_{i=0}^{n} i$, the limits on the sum will be to the right of the sum symbol. However, if you use $$\sum_{i=0}^{n} i$$, the limits will be on the top and the bottom.

Debugging

When you have finished writing your .tex file, run "latex ps1.tex" from the Athena (or whatever) prompt. If you had no bugs in your TeX code it'll go right through and give you an Athena prompt again. However, you will probably have some errors. Most errors fit into one of these few categories:
  1. You forgot to close something you opened. You forgot to put enough close parentheses or curly braces, or you left off the closing $ to go out of math mode. You may also have forgotten to put "\end{enumerate}" after your numbered list was finished. Remember that if you use $$ to put a math mode block on its own line, you must close with a $$. If you close with a single $, it will cause an error.
  2. Similar to the above, you forgot to open something you closed.
  3. You used a non-existant command.
  4. You used a command in normal mode that is only supposed to be used in math mode. Sometimes this can look like a #3-type problem.
Feel free to ask me questions about TeX over email; I'll help you if I can. I'm not particularly good at things like importing graphics or matrix layouts, but I get by pretty well generally.

.Dvi? What's that?

When your "latex file.tex" command goes through, it creates a file called "file.dvi". This can now be printed by typing "dvips file.dvi" - on Athena this will print to the default printer. DVI files can also be used to produce postscript files. To make file.dvi into file.ps, run "dvips file.dvi -o file.ps"

If you want to check over your file to make sure it looks right (without printing it), you have two options. First of all, you can use the above instructions to create a postscript file, and use ghostview ("gv file.ps &") to view it. Or, more simply, you can just run "xdvi file.dvi &" to view your .dvi file without changing it to postscript.

One Final Hint

If you have a TeX bug that you can't figure out, if worse comes to worse, just delete the problematic section and put a bunch of "\bigskip"s in its place. Then, once you print out your file, just write the solution in.

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