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\newcommand{\lat}{\mathcal{L}}
\newcommand{\basis}{\mathbf{B}}

\newcommand{\vol}{\mathsf{vol}}
\newcommand{\sh}{\lambda_1}
%\newcommand{\det}{\mathsf{det}}

\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
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%%%%%%%%% Vectors and Matrices

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\def\rot{\mathsf{Rot}}
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\title{MAT 301 Problem Set 3\\{\large [Posted: February 08, 2012. Due: February 17, 2012. Worth: 100 points]}}
\author{}
\date{}

\begin{document}
\maketitle
%\input{pset-preamble.tex}

%\lecture{1}{Vinod Vaikuntanathan}{September 13, 2011}{October 3, 2011}


\def\sslwe{\mathsf{ssLWE}}
\def\lwe{\mathsf{LWE}}

% Macros for vectors
\def\veca{\mathbf{a}}
\def\vecs{\mathbf{s}}
\def\vecc{\mathbf{c}}

\vspace*{-0.7in}

\medskip\noindent
{\bf Note:} I value {\em succinct} and {\em clearly written} solutions {\em without unnecessary verbiage}. Such solutions will be rewarded with bonus points.

\medskip \noindent
\textbf{Note the Friday deadline. The problem sets are due in the beginning of the tutorial, at 10am on Friday.}

\begin{enumerate}

\item \textbf{Exponentiation and Finding Roots (30 points)}

\begin{itemize}
\item (5 points) Find $2^{65} \pmod{97}$.  Show your work.

\item (15 points) Find the $65^{th}$ root of $2$ mod $97$. Show your work.
    
\end{itemize}

%\item \textbf{RSA (10 points)} Let the RSA parameters be $

\item \textbf{Square Roots and Factoring (40 points)}

\begin{itemize}
\item (1 point) How many solutions does the equation $x^2 = 1 \pmod{7}$ have? What are they? 

\item (1 point) How many solutions does the equation $x^2 = 3 \pmod{7}$ have? What are they? 

\item (2 points) How many solutions does the equation $x^2 = 1 \pmod{8}$ have? What are they?

\item (11 points) If $N$ is prime and $a \in \mathbb{Z}_N^*$, how many solutions does the equation $x^2 = a \pmod{N}$ have? 

\item (30 points) I am going to hand over to you a number $N$ which is a product of two distinct prime numbers. I will
 also give you two numbers $x_1$ and $x_2$ such that
\begin{eqnarray*}
x_1^2 & = & x_2^2 \pmod{N} \\
x_1 & \neq & x_2 \pmod{N} \\ 
x_1 & \neq & -x_2 \pmod{N}
\end{eqnarray*}
How will you find the prime factors of $N$ using this information?
\end{itemize}

\item \textbf{$\phi(N)$ and Factoring (30 points)}
I am going to hand over to you a number $N$ which is a product of two distinct prime numbers. I will also give you
$\phi(N)$, the Euler Totient function of $N$. 

\begin{itemize}
\item How will you find the prime factors of $N$ using this information?
\item How many basic computational steps does your algorithm take (you can express your answer in terms of the Big-Oh 
$O(\cdot)$ notation)?
\end{itemize}
(Note: Since you can easily compute $\phi(N)$ given the factorization of $N$, this problem is asking you to prove
that finding $\phi(N)$ is computationally as hard as factoring $N$).  


\end{enumerate}


\end{document}
