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Crypto edits

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Mark
2024-10-24 09:33:50 -07:00
parent 8b10780fbe
commit 1609ec919a
3 changed files with 11 additions and 18 deletions
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11
Advanced/Cryptography/parts/1 mod.tex
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@ -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>