Chapter 2.8, Problem 10E

Suppose that in a long ciphertext message the letter $D$ occurred most frequently, followed in frequency by $N$. Using the fact that in the $27$-letter alphabet $A$, described in Example $2$, "blank" occurs most frequently, followed in frequency by $e$, read the portion of the message

$\text{G}\text{E}\text{N}\text{D}\text{O}\text{C}\text{F}\text{A}\text{A}\text{D}\text{O}\text{Q}\text{N}\text{I}\text{D}\text{P}\text{G}\text{M}\text{D}\text{C}\text{F}\text{E}$ enciphered using an affine mapping on $A$. Write out the affine mapping $f$ and its inverse.

Example 2 Translation Cipher Associate the $n$ letters of the "alphabet" with the integers $0,1,2,3\mathrm{.....}n-1$. Let $A=\left\{0,1,2,3\mathrm{.....}n\text{ - }1\right\}$ and define the mapping

$f:A\to A$ by

$f\left(x\right)=x+kmodn$ where $k$ is the key, the number of positions from the plaintext to the ciphertext. If our alphabet consists of $a$ through $z$, in natural order, followed by a blank, then we have $27$ "letters" that we associate with the integers $0,1,2,\mathrm{...},26$ as follows:

$\begin{array}{cccccccccccccc}\text{Alphabet:}& \text{a}& \text{b}& \text{c}& \text{d}& \text{e}& \text{f}& \mathrm{...}& \text{v}& \text{w}& \text{x}& \text{y}& \text{z}& \text{"blank"}\\ \text{A:}& \text{0}& \text{1}& \text{2}& \text{3}& \text{4}& \text{5}& & \text{21}& \text{22}& \text{23}& \text{24}& \text{25}& \text{26}\end{array}$