Let $M$ be a message, which we will think of as a sequence of letters from the standard 26 letter Roman alphabet. We can write $M=(m_1,m_2,m_3, \ldots, m_n)$ where $m_i\in\{a, b, c, \ldots, z\}$. For simplicity, our messages are not case-sensitive and all spaces and punctuation have been removed. To send the phrase “Let's meet at dawn.” we'd just use the plain text message $M=(l,e,t,s,m,e,e,t,a,t,d,a,w,n)$.

We now define an encryption key to produce ciphertext from a message $M$. Given a message $M=(m_1,m_2,m_3, \ldots, m_n)$, let $\phi_3(m_i)$ be the letter in the alphabet that has been shifted three right from the current letter. When we hit the end of the alphabet, we'll wrap around so that shifting $z$ right one gets us to $a$. Let's call this the simple shift permutation $\phi_3$. So we now have $\phi_3(a)=d$, $\phi_3(m)=p$, and $\phi_3(z)=c$. In a similar manner, we can define the simple shift permutation $\phi_n$ which shifts each letter right $n$ units.

definition.simple_shift_permutation_of_the_alphabet.tex

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% DEPENDENCIES 
% RequiredMacros: \DeclareMathOperator{\syl}{Syl} 
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\begin{definition}
Let $M$ be a message, which we will think of as a sequence of letters from the standard 26 letter Roman alphabet.  We can write $M=(m_1,m_2,m_3, \ldots, m_n)$ where $m_i\in\{a, b, c, \ldots, z\}$. For simplicity, our messages are not case-sensitive and all spaces and punctuation have been removed. To send the phrase "Let's meet at dawn." we'd just use the plain text message $M=(l,e,t,s,m,e,e,t,a,t,d,a,w,n)$. 
 
We now define an encryption key to produce ciphertext from a message $M$.  Given a message $M=(m_1,m_2,m_3, \ldots, m_n)$, let $\phi_3(m_i)$ be the letter in the alphabet that has been shifted three right from the current letter. When we hit the end of the alphabet, we'll wrap around so that shifting $z$ right one gets us to $a$. Let's call this the simple shift permutation $\phi_3$. So we now have $\phi_3(a)=d$, $\phi_3(m)=p$, and $\phi_3(z)=c$. In a similar manner, we can define the simple shift permutation $\phi_n$ which shifts each letter right $n$ units.
\end{definition}

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definition ben