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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