\documentclass[12pt]{amsart}
\pagestyle{plain}
\usepackage{amsthm, setspace, framed, hyperref}
\usepackage{enumerate}
\usepackage{fullpage}
\begin{document}
\vspace{-5mm}\noindent \textbf{Math 314 - Fall 2018 \hfill
Name:}
\hspace{2in} %Replace this line with your name!
\noindent \textbf{Mission 11} \hfill Due December 10th, 2018 \vspace{-4mm}\\
\small \textit{ It is possible to write endlessly on elliptic curves. (This is not a threat.)}\\
\vspace{1mm} \hfill--- Serge Lang
\vspace{-5mm}
\normalsize
\noindent \hrulefill
\section*{Guidelines}
\begin{itemize}
\item All work must be shown for full credit.
\item You can choose to use SageMath code to help you solve the problems. If you do, print out your code (or use the same folder as the latex code on SMC).
\item Either print out this assignment and write your answers on it, or edit the latex source on SMC and type your answers in the document. Make sure you still show your work!
There is one point of extra credit available on this assignment if you use \LaTeX
\item You may work with classmates, but be sure to turn in your own written solutions. Write down the name(s) of anyone who helps you.
\item Check one:\\
%You can put an x inside the framebox to "check" the box in latex for example: \framebox(12,12){x}
\framebox(12,12){} I worked with the following classmate(s):
\rule{7cm}{0.5pt}\\ %Replace this line with names of students.
\noindent\framebox(12,12){} I did not receive any help on this assignment.
\end{itemize}
\section{Graded Problems}
\begin{enumerate}[1.]
\item Suppose Alice uses a hash function that produces 40 bit digests. She uses this hash along with RSA to sign contracts. Eve wants to use a birthday attack to trick her into signing a bad contract. She drafts a contract and finds 10 places that she could make a change to the contract without changing its meaning. Is this likely to be enough to make the attack work? What if she finds 15 places? 25 places?
\begin{framed}
\vspace{2.9in}
%Type your answer here!
\end{framed}
\pagebreak
\item List all of the points on the elliptic curve $y^2 \equiv x^3 +x+4 \pmod{7}$. (Hint: Start by computing all the values of $y^2 \pmod{7}$.)
\begin{framed}
\vspace{3.5in}
%Type your answer here!
\end{framed}
\item Let $P=(4,3)$ and $Q=(5,1)$ be points on the elliptic curve $y^2 \equiv x^3 +x+4 \pmod{7}$. Compute $P+Q$ and $P+P$.
\begin{framed}
\vspace{3.5in}
%Type your answer here!
\end{framed}
\pagebreak
\item Suppose $E$ is an elliptic curve $\mod p$. Show that there cannot be more than $2p+1$ points on this curve. (Hint: use the fact that for a fixed value of $c$, $y^2\equiv c \pmod p$ can never have more than two solutions.)
\begin{framed}
\vspace{3.5in}
%Type your answer here!
\end{framed}
\end{enumerate}
\end{document}