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\centerline{\textbf{Honors HW03. Morally DUE Mon Feb 28}}

In this HW Alice and Bob are doing Diffie Hellman with $p=47$
and $g=5$. You can verify that $g$ IS a generator (you do not have to, but you could). 

\begin{enumerate}
\item
(35 points)
Alice uses $a=10$ and Bob uses $b=11$. What is the shared secret key?
Express as a number in $\{0,\ldots,46\}$

\item
(35 points)
Alice uses $a=11$ and Bob uses $b=10$. What is the shared secret key?
Express as a number in $\{0,\ldots,46\}$

\item
(30 points)
If you did the problem correctly the last two answers were the same.
Prove the following theorem:

{\bf Thm} Let $p$ be a prime and $g$ be a generator. 
Let $a,b\in \{0,\ldots,p-1\}$. 
Let $s_{a,b}$ be the shared secret key if Alice uses $a$ and
Bob uses $b$. Show that $s_{a,b} = s_{b,a}$. 
\end{enumerate}

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