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A message is encrypted using an affine cryptosystem in which plaintext uses the 26 letters A through Z (all blanks are omitted), the letters are identified with the residue classes of integers (mod 26) in the natural way, and the encryption and decryption functions are defined by x -> f(x) = ax + b (mod 26) and y -> g(y) = cx + d (mod 26) respectively, where a; b; c; d are certain constants (unknown to you) and g(f(x)) is the identity function. Suppose you intercept the ciphertext UWWPUWKPPFWPWKPJWW: Assuming that E and then T are the two most commonly occurring letters in English text, find the decryption function g(y) and decrypt the message.

This is my homework question. I really dont know how to solve it . Could anyone give me a help ? Thanks in advance.

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Which letter do you think decrypts to E? And which to T? – Henning Makholm Oct 28 '12 at 16:38
Have you ever seen or heard of an affine cipher before? If not, make one up for yourself (choose some $a$ and $b$ as in the problem) and use it to encrypt a simple message, such as "iwillmeetyouunderourfavoritetree". Just to get a sense of whaat is going on :) – Hans Engler Oct 28 '12 at 18:17
up vote 2 down vote accepted

HINTS: First make a frequency table; this should be your first step in dealing any cipher.

$$\begin{array}{rcc} \text{Letter}:&\text{F}&\text{J}&\text{K}&\text{P}&\text{U}&\text{W}\\ \text{Frequency}:&1&1&2&5&2&7 \end{array}$$

You’re told to assume that ‘E and then T are the two most commonly occurring letters in English text’. In other words, you’re told to assume that the most frequent letter in the message represents E, and the second-most frequent letter represents T. What are those letters?

Let’s say that you decide that the ciphertext letters representing E and T are letters $m$ and $n$, respectively, modulo $26$. E and T are $5$ and $20$, respectively, so if $f(x)=ax+b$ and $g(x)=cx+d$ are your enciphering and deciphering functions, you now know that

$$\left\{\begin{align*} &5a+b\bmod 26=m\\ &20a+b\bmod 26=n \end{align*}\right.\tag{1}$$


$$\left\{\begin{align*} &cm+d\bmod 26=5\\ &cn+d\bmod 26=20 \end{align*}\right.\tag{2}$$

Your task, once you’ve identified $m$ and $n$, is to solve $(1)$ for $a$ and $b$ and $(2)$ for $c$ and $d$ to reconstruct the enciphering and deciphering functions.

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