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\begin{document}
\headers{Math373}{Homework 8}{Due Wed, Mar. 9, 2022}

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\noindent
\textbf{Directions:} Solve the following problems.  All written work must be your own.  See the course syllabus for detailed rules.

\begin{enumerate}

\item Solve the following systems of congruences.
\begin{enumerate}
	\item \begin{align*}
		x &\equiv 18 \pmod{25}\\
		x &\equiv 7 \pmod{11}\\
		x &\equiv 16 \pmod{32}
	\end{align*}
	
	\item \begin{align*}
		17x &\equiv 8 \pmod{43}\\
		6x &\equiv  41 \pmod{55}\\
		5x &\equiv 4 \pmod{9}
	\end{align*}

	\item \begin{align*}
		7x &\equiv 33 \pmod{145}\\
		11x &\equiv 44 \pmod{45}\\
		17x &\equiv 38 \pmod{75}
	\end{align*}
	Note: The given moduli are not pairwise relatively prime (for example, $3\mid 45$ and $3 \mid 75$), so CRT does not apply directly.
\end{enumerate}

\item Alice and Bob wish to use the ElGamal cryptosystem to communicate, and they are having difficulty deciding on a prime/base pair $(p,g)$.  The pairs that they are considering are $(345601, 71482)$ (option A), $(516163, 482305)$ (option B), and $(177007, 145014)$ (option C).  Which option do you recommend for Alice and Bob, and why?

\end{enumerate}
\end{document}
