Crypto edits

Advanced/Cryptography/parts/0 euclidean.tex

@@ -21,15 +21,6 @@ In other words, we can divide $a$ by $b$ to get $q$ remainder $r$.
 For any integers $a, b, c$, \par
 $\gcd(ac + b, a) = \gcd(a, b)$
 
-\problem{}
-Compute the gcd of 12 and 976.
-\begin{solution}
-	$976 = 3 \times 324 + 4 = 3 \times 4 \times 81 + 4$
-	So, $\gcd(a, b) = 4$
-\end{solution}
-
-\vfill
-
 \problem{The Euclidean Algorithm}<euclid>
 Using the two theorems above, detail an algorithm for finding $\gcd(a, b)$. \par
 Then, compute $\gcd(1610, 207)$ by hand. \par
@@ -109,7 +100,6 @@ We now want to write the 2 in the last equation in terms of 20 and 14.
 
 
 \begin{solution}
-
 	Using the output of the Euclidean Algorithm, we can use substitution and a bit of algebra to solve such problems. Consider the following example:
 
 	\begin{multicols}{2}
@@ -139,6 +129,10 @@ We now want to write the 2 in the last equation in terms of 20 and 14.
 	$\gcd(541, 34) = 541(11) + 34(-175)$
 \end{solution}
 
+\begin{solution}
+	\huge
+	This problem is too hard. Break it into many.
+\end{solution}
 
 \vfill
 \pagebreak

Advanced/Cryptography/parts/1 mod.tex

@@ -28,7 +28,7 @@ Create a multiplication table for $\mathbb{Z}_4$:
 
 
 \definition{}
-Let $a, b \in \mathbb{Z}_n$. \par
+Let $a, b$ be elements of %\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}
@@ -37,7 +37,7 @@ 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
+$a$ has an inverse in $\mathbb{Z}_n$ if and only if $\gcd(a, n) = 1$ \par
 
 \problem{}
 Find the inverse of $3$ in $\mathbb{Z}_4$, if one exists. \par
@@ -56,14 +56,13 @@ Find the inverse of $4$ in $\mathbb{Z}_7$, if one exists.
 
 
 \problem{}
-Show that if $n$ is not prime, $\mathbb{Z}_n$ has at least one element with no inverse.
-
+Show that if $n$ is prime, every element of $\mathbb{Z}_n$ (except 0) has an inverse.
 \vfill
 
 \problem{}
-Is this true if $n$ is prime?
-
+Show that if $n$ is not prime, $\mathbb{Z}_n$ has at least one element with no inverse.
 \vfill
+
 \pagebreak
 
 \problem{}<general_inverse>

Advanced/Cryptography/parts/5 Elgamal.tex

@@ -106,8 +106,8 @@ Run this algorithm and make sure it works.
 
 
 \problem{}
-Is this secure? What information does Eve have? \par
-What does Eve need to find $m$?
+What information does Eve have? \par
+What does Eve need to do to find $m$?
 \vfill
 
 \problem{}