Cryptography edits

Advanced/Cryptography/main.tex

@@ -2,7 +2,8 @@
 % use [solutions] flag to show solutions.
 \documentclass[
 	solutions,
-	singlenumbering
+	singlenumbering,
+	shortwarning
 ]{../../resources/ormc_handout}
 
 \usepackage{multicol}

Advanced/Cryptography/parts/0 euclidean.tex

@@ -13,10 +13,20 @@ Find $\gcd(20, 14)$ by hand.
 
 \vfill
 
-\theorem{The Division Algorithm}
+\theorem{The Division Algorithm}<divalgo>
 Given two integers $a, b$, we can find two integers $q, r$, where $0 \leq r < b$ and $a = qb + r$. \par
 In other words, we can divide $a$ by $b$ to get $q$ remainder $r$.
 
+\begin{instructornote}
+	\ref{divalgo} looks scary on paper, but it's quite simple. \par
+	Doing a small example on the board (like $14 \div 3$) may be a good idea. \par
+
+	\vspace{2mm}
+
+	For those that are new to modular arithmetic, you may want to explain how remainders,
+	clock-face counting, division algorithm, and modular arithmetic are all the same.
+\end{instructornote}
+
 \theorem{}<gcd_abc>
 For any integers $a, b, c$, \par
 $\gcd(ac + b, a) = \gcd(a, b)$
@@ -64,7 +74,12 @@ Using the output of the Euclidean algorithm,
 	% gcd = 1
 	% u = 11; v = -175
 \end{itemize}
-This is called the \textit{extended Euclidean algorithm}.
+This is called the \textit{extended Euclidean algorithm}. \par
+\hint{
+	You don't need to fully solve the last part of this question. \\
+	Understand how you \textit{would} do it, then move on.
+	Don't spend too much time on arithmetic.
+}
 
 %For which numbers $c$ can we find a $(u, v)$ so that $541u + 34v = c$? \\
 %For every such $c$, what are $u$ and $v$?

Advanced/Cryptography/parts/1 mod.tex

@@ -4,11 +4,11 @@
 $\mathbb{Z}_n$ 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 $\mathbb{Z}_4$:
+Create a multiplication table for $\mathbb{Z}_4$:
 
 \begin{center}
 \begin{tabular}{c | c c c c}
-	+ & 0 & 1 & 2 & 3 \\
+	\times & 0 & 1 & 2 & 3 \\
 	\hline
 	0 & ? & ? & ? & ? \\
 	1 & ? & ? & ? & ? \\
@@ -36,8 +36,7 @@ $a$ 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.
-%$34^\star \equiv -175 \equiv 366 \pmod{541}$.
+Find the inverse of $4$ in $\mathbb{Z}_7$, if one exists.
 \vfill