Crypto edits

Advanced/Cryptography/parts/1 mod.tex

@@ -3,6 +3,12 @@
 \definition{}
 $\mathbb{Z}_n$ is the set of integers mod $n$. For example, $\mathbb{Z}_5 = \{0, 1, 2, 3, 4\}$. \par
 
+\vspace{2mm}
+
+Multiplication in $\mathbb{Z}_n$ works much like multiplication in $\mathbb{Z}$: \par
+If $a, b$ are elements of $\mathbb{Z}_n$, $a \times b$ is the remainder of $a \times b$ when divided by $n$. \par
+\note{For example, $2 \times 2 = 4$ and $3 \times 4 = 12 = 2$ in $\mathbb{Z}_5$}
+
 \problem{}
 Create a multiplication table for $\mathbb{Z}_4$:
 
@@ -37,12 +43,25 @@ $a$ has an inverse in $\mathbb{Z}_n$ iff $\gcd(a, n) = 1$ \par
 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 $4$ in $\mathbb{Z}_7$, if one exists.
+
+\begin{solution}
+	\begin{itemize}
+		\item $3^{-1}$ in $\mathbb{Z}_{4}$ is $3$
+		\item $20^{-1}$ in $\mathbb{Z}_{14}$ doesn't exist.
+		\item $4^{-1}$ in $\mathbb{Z}_{7}$ is $2$
+	\end{itemize}
+\end{solution}
+
 \vfill
 
 
 \problem{}
-Today, we will often assume that $n$ is prime. \par
-Why? What is special about $\mathbb{Z}_n$ with a prime $n$?
+Show that if $n$ is not prime, $\mathbb{Z}_n$ has at least one element with no inverse.
+
+\vfill
+
+\problem{}
+Is this true if $n$ is prime?
 
 \vfill
 \pagebreak