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Problem 4 (The Set Of Simple Shift Permutations)
Let $S = \{a,b,c,\ldots,z\}$ be the set of letters in the Roman alphabet. For each $n\in\mathbb{Z}$, let $\phi_{n}:S\to S$ represent the encryption key that shifts each letter in the alphabet right $n$, where we wrap around from $z$ to $a$ for letters that need to be shifted past $z$. So $\phi_3(a)=d$, $\phi_3(b)=e$, and $\phi_3(z)=c$. We've called this a simple shift permutation of $S$.
- For which $n$ does $\phi_n$ not change the message?
- Consider the encryption key $\phi_{9}$, which shifts each letter right 9. If a message had been encrypted using $\phi_9$, then clearly $\phi_{-9}$ would decrypt the message as $\phi_{-9}(\phi_9(s))=s$ for any letter $s$. Give a positive integer $n$ that we could also use to decode a message that has been encrypted using $\phi_9$.
- Does $\phi_{30}=\phi_7$? Does $\phi_{33}=\phi_7$? For which $n$ does $\phi_n=\phi_7$? Explain.
- Consider the set of all shifts of $S$, namely $H=\{\phi_n\mid n\in\mathbb{Z}\}$. How many different functions are in $H$? Prove your answer is correct.
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