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\begin{center} {\large \textbf{Math 437 -- Homework 6}} \smallskip \\ 
Due 10:15am on Thursday, February 23, 2017  \\ \ \\
 \end{center}

Please indicate any sources you used for a given problem on the solution to that problem. 
For example, if you worked with another student to get the solution to a problem, please indicate who.
You are welcome to work together in small groups, but please try the problems on your own first and write up 
your own solutions. \\ \medskip 




\begin{prob} \ 
\begin{itemize}
\item[(a)] Decrypt the following message that was made with a shift cipher: \smallskip 
\begin{center} 
HFJXFW ZXJI XMNKY HNUMJWX \smallskip 
\end{center}
\item[(b)] Decrypt the following message that was encrypted with a $2\times 2$ Hill cipher \smallskip 
\begin{center}
TNHSEQWAYUFTHNCU \smallskip 
\end{center}
if you know that the message starts ``HI ALICE''
\end{itemize}
\end{prob} \ \\


\begin{prob}
(Ch. 6 \#18) Prove that $a\in \Z_m$ will have a multiplicative inverse in $\Z_m$ if and only if $\gcd(a,m) = 1$.   
\end{prob}\ \\


\begin{prob} 
Alice and Bob want to share a secret time to meet up and encrypt it using RSA. 
Bob chooses $p = 41$ and $q = 59$ and calculates $n = p\cdot q = 2419$ and $m = (p-1)(q-1) = 2320$.  \smallskip 
\begin{itemize}
\item[(a)] Check that $\gcd(211,m)=1$ and calculate a find multiplicative inverse for $211$ in $\Z_m$.  \smallskip 
\item[(b)] Bob posts $n=2419$, $a=211$ and Alice sends Bob the encrypted time $1187$ (mod $n$).
What is the secret meeting time?  
\end{itemize}

\end{prob}\ \\






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