Private Key Cryptography

One of the first and most famous private key cryptosystems was the shift code used by Julius Caesar. We first digitize the alphabet by letting \(\text{A} = 00, \text{B} = 01, \ldots, \text{Z} = 25\). The encoding function will be \[\begin{aligned}\end{aligned}\] that is, \(A \mapsto D, B \mapsto E, \ldots, Z \mapsto C\). The decoding function is then \[\begin{aligned}\end{aligned}\]. Suppose we receive the encoded message DOJHEUD. To decode this message, we first digitize it: \[\begin{aligned}\end{aligned}\]. Next we apply the inverse transformation to get \[\begin{aligned}\end{aligned}\], or ALGEBRA. Notice here that there is nothing special about either of the numbers \(3\) or \(26\). We could have used a larger alphabet or a different shift.

Suppose we receive a message that we know was encrypted by using a shift transformation on single letters of the \(26\)-letter alphabet. To find out exactly what the shift transformation was, we must compute \(b\) in the equation \(f(p) = p + b \bmod 26\). We can do this using frequency analysis. The letter \(\text{E} = 04\) is the most commonly occurring letter in the English language. Suppose that \(\text{S} = 18\) is the most commonly occurring letter in the ciphertext. Then we have good reason to suspect that \(18 = 4 + b \bmod 26\), or \(b= 14\). Therefore, the most likely encrypting function is \[\begin{aligned}\end{aligned}\]. The corresponding decrypting function is \[\begin{aligned}\end{aligned}\]. It is now easy to determine whether or not our guess is correct.