Rewrote first section of crypto handout

Advanced/Cryptography/parts/1 mod.tex

@@ -0,0 +1,74 @@
+\section{Modular Arithmetic}
+
+\definition{}
+$\mathbb{Z}_n$ is the set of integers mod $n$. For example, $\mathbb{Z}_5 = \{0, 1, 2, 3, 4\}$. \par
+
+\problem{}
+Create a multiplication addition table for $\mathbb{Z}_4$:
+
+\begin{center}
+\begin{tabular}{c | c c c c}
+	+ & 0 & 1 & 2 & 3 \\
+	\hline
+	0 & ? & ? & ? & ? \\
+	1 & ? & ? & ? & ? \\
+	2 & ? & ? & ? & ? \\
+	3 & ? & ? & ? & ? \\
+\end{tabular}
+\end{center}
+
+
+
+
+
+\definition{}
+Let $a, b \in \mathbb{Z}_n$. \par
+If $a \times b = 1$, we say that $b$ is the \textit{inverse} of $a$ in $\mathbb{Z}_n$.
+
+\vspace{2mm}
+
+We usually write \say{$a$ inverse} as $a^{-1}$. \par
+Inverses are \textbf{not} guaranteed to exist.
+
+\theorem{}<mod_has_inverse>
+$a$ has an inverse in $\mathbb{Z}_n$ iff $\gcd(a, n) = 1$ \par
+
+\problem{}
+Find the inverse of $3$ in $\mathbb{Z}_4$, if one exists. \par
+Find the inverse of $20$ in $\mathbb{Z}_{14}$, if one exists. \par
+Find the inverse of $2$ in $\mathbb{Z}_5$, if one exists.
+%$34^\star \equiv -175 \equiv 366 \pmod{541}$.
+\vfill
+
+
+\problem{}
+Today, we will often assume that $n$ is prime. \par
+Why? What is special about $\mathbb{Z}_n$ with a prime $n$?
+
+\vfill
+\pagebreak
+
+\problem{}<general_inverse>
+In general, how can we find the inverse of $a$ in $\mathbb{Z}_n$? Assume $a$ and $n$ are coprime.\par
+\hint{You can find that $34^{-1}$ is $-175$ in $\mathbb{Z}_{541}$ by looking at a previous problem.}
+
+\begin{solution}
+	We need an $a^{-1}$ so that $a \times a^{-1} = 1$. \par
+	This means that $aa^{-1} - mk = 1$. \par
+	Since $a$ and $m$ are coprime, $\gcd(a, m) = 1$ and $aa^{-1} - mk = \gcd(a, m)$ \par
+	Now use the extended Euclidean algorithm from \ref{extendedeuclid} to find $a^\star$.
+\end{solution}
+
+\vfill
+
+\definition{}
+Elements in $\mathbb{Z}_n$ that have an inverse are called \textit{units}. \par
+The set of units in $\mathbb{Z}_n$ is called $\mathbb{Z}_n^\times$, which is read \say{$\mathbb{Z}$ mod $n$ cross}.
+
+\problem{}
+What is $\mathbb{Z}_5^\times$? \par
+What is $\mathbb{Z}_{12}^\times$? \par
+
+\vfill
+\pagebreak
+