Crypto edits

Advanced/Cryptography/main.tex

@@ -8,6 +8,7 @@
 \usepackage{../../resources/macros}
 
 \usepackage{multicol}
+\usepackage{mathtools}
 
 \uptitlel{Advanced 2}
 \uptitler{\smallurl{}}

Advanced/Cryptography/parts/0 euclidean.tex

@@ -17,20 +17,19 @@ Find $\gcd(20, 14)$ by hand.
 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
 $\gcd(ac + b, a) = \gcd(a, b)$
 
+\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

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

Advanced/Cryptography/parts/2 groups.tex

@@ -29,7 +29,8 @@ Is $(\mathbb{Z}_5, -)$ a group? \par
 
 
 \problem{}
-Show that $(\mathbb{R}, \times)$ is not a group, then make it one by modifying $\mathbb{R}$. \par
+Show that $(\mathbb{R}, \times)$ is not a group,
+then find a subset $S$ of $\mathbb{R}$ so that $(S, \times)$ is a group.
 
 \begin{solution}
 	$(\mathbb{R}, \times)$ is not a group because $0$ has no inverse. \par
@@ -58,8 +59,8 @@ What is the smallest group we can create?
 
 \problem{}
 Let $(G, \ast)$ be a group with finitely many elements, and let $a \in G$. \par
-Show that $\exists n \in \mathbb{Z}^+$ so that $a^n = e$ \par
-\hint{$a^n = a \ast a \ast ... \ast a$ repeated $n$ times.}
+Show that there exists an $n$ in $\mathbb{Z}^+$ so that $a^n = e$ \par
+\hint{$a^n \coloneqq a \ast a \ast ... \ast a$, with $a$ repeated $n$ times.}
 
 \vspace{2mm}
 
@@ -77,8 +78,9 @@ What is the order of 2 in $(\mathbb{Z}_{17}^\times, \times)$? \par
 \theorem{}
 Let $p$ be a prime number. \par
 In any group $(\mathbb{Z}_p^\times, \ast)$ there exists a $g \in \mathbb{Z}_p^\times$ where...
-\begin{itemize}
-	\item The order of $g$ is $p - 1$
+
+\begin{itemize}[itemsep=1mm]
+	\item The order of $g$ is $p - 1$, and
 	\item $\{a^0,~ a^1,~ ...,~ a^{p - 2}\} = \mathbb{Z}_n^\times$
 \end{itemize}
 We call such a $g$ a \textit{generator}, since its powers generate every other element in the group.

Advanced/Cryptography/parts/3 DLP.tex

@@ -21,7 +21,7 @@ Show that $\exp$ is a bijection, which will guarantee the existence of $\log$. \
 \vfill
 
 \problem{}
-What's the simplest (but not the most efficient) way to calculate $\log_g(a)$?
+Find a simple (but perhaps inefficient) way to calculate $\log_g(a)$
 
 \vfill
 

Advanced/Cryptography/parts/4 DiffieHellman.tex

@@ -86,7 +86,10 @@ Eve can read all public values, but she cannot change them in any way.
 
 \problem{}
 Complete the algorithm. What should Alice and Bob compute? \par
-What is their shared secret?
+\hint{
+	The goal of this process is to arrive at a \textit{shared secret} \par
+	That is, Alice and Bob should arrive at the same value without exposing it to Eve.
+}
 
 \vfill
 

Advanced/Cryptography/parts/challenge.tex

@@ -182,5 +182,6 @@ $\gcd(ac + b, a) = \gcd(a, b)$ \par
 [Note on \ref{eua_runtime}] This proof can be used to show that the Euclidean
 algorithm finishes in logarithmic time, and it is the first practical application
 of the Fibonacci numbers. If you have finished all challenge problems,
-finish the proof: show that the Euclidean algorithm runs in $O(\log{n})$
+finish the proof: find how many steps the Euclidean algorithm needs to arrive at
+a solution for a given $a$ and $b$.
 \pagebreak