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Crypto edits

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Mark 2024-10-24 09:33:50 -07:00
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14
Advanced/Cryptography/parts/0 euclidean.tex
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@ -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 For any integers $a, b, c$, \par
$\gcd(ac + b, a) = \gcd(a, b)$ $\gcd(ac + b, a) = \gcd(a, b)$
\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> \problem{The Euclidean Algorithm}<euclid>
Using the two theorems above, detail an algorithm for finding $\gcd(a, b)$. \par Using the two theorems above, detail an algorithm for finding $\gcd(a, b)$. \par
Then, compute $\gcd(1610, 207)$ by hand. \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} \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: 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} \begin{multicols}{2}
@ -139,6 +129,10 @@ We now want to write the 2 in the last equation in terms of 20 and 14.
$\gcd(541, 34) = 541(11) + 34(-175)$ $\gcd(541, 34) = 541(11) + 34(-175)$
\end{solution} \end{solution}
\begin{solution}
\huge
This problem is too hard. Break it into many.
\end{solution}
\vfill \vfill
\pagebreak \pagebreak

11
Advanced/Cryptography/parts/1 mod.tex
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@ -28,7 +28,7 @@ Create a multiplication table for $\mathbb{Z}_4$:
\definition{} \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 $\mathbb{Z}_n$. If $a \times b = 1$, we say that $b$ is the \textit{inverse} of $a$ in $\mathbb{Z}_n$.
\vspace{2mm} \vspace{2mm}
@ -37,7 +37,7 @@ We usually write \say{$a$ inverse} as $a^{-1}$. \par
Inverses are \textbf{not} guaranteed to exist. Inverses are \textbf{not} guaranteed to exist.
\theorem{}<mod_has_inverse> \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{} \problem{}
Find the inverse of $3$ in $\mathbb{Z}_4$, if one exists. \par 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{} \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 \vfill
\problem{} \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 \vfill
\pagebreak \pagebreak
\problem{}<general_inverse> \problem{}<general_inverse>

4
Advanced/Cryptography/parts/5 Elgamal.tex
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@ -106,8 +106,8 @@ Run this algorithm and make sure it works.
\problem{} \problem{}
Is this secure? What information does Eve have? \par What information does Eve have? \par
What does Eve need to find $m$? What does Eve need to do to find $m$?
\vfill \vfill
\problem{} \problem{}