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\noindent \textbf{Math 314 - Fall 2018 \hfill 
Name:} 
\hspace{2in} %Replace this line with your name!

\noindent \textbf{Mission 3} \hfill Due September 24, 2018 \vspace{-4mm}\\

\small \textit{Cryptography succeeds when it’s no longer the weakest link.}\\\vspace{1mm} \hfill--- Ron Rivest

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\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.
	\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): 
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\noindent\framebox(12,12){} I did not receive any help on this assignment.
\end{itemize}

\section{Graded Problems}

\begin{enumerate}[1.]

	

		\item Use the Euclidean Algorithm to find the gcd of 217 and 1078.
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	\item Use the Euclidean algorithm to find integers $x$ and $y$ such that $23x+77y=1$.  What is $23^{-1} \pmod{77}$?  Show all of your steps!
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	\item Use modular exponentiation to compute $5^{268} \pmod{17}$.  Make sure to show your steps.
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	\item Alice wants to send a message to Bob using the 3-pass protocol.  She decides to use the prime $p=43$, and picks her key, $a=11$.  Bob picks his key, $b=13$.
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\item What are Alice and Bob's decryption keys?  Work them out using the Euclidean algorithm. 
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\item Alice wants to send the message m=3.  Find the values of each of the messages that Alice and Bob send back and forth.  Does Bob recover Alice's plaintext at the end?
\end{enumerate}
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	%Type your answer here!
	
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