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Computer Security and Cryptography by Alan G. Konheim

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3.6 PARTIAL MAXIMUM LIKELIHOOD ESTIMATION OF A MONOALPHABETIC SUBSTITUTION

Can we find the substitution without a crib? We suppose ciphertext y = (y0, y1,…, yn−1) results from a monoalphabetic substitution of plaintext x = (x0, x1,…, xn−1), both written with letters in the alphabet image with an unknown substitution θ.

We assume the substitution θ has been chosen randomly independent of x and according to the uniform distribution Pra priori{Θ = θ} = 1/m. The cryptanalysis problem

Given: y
Evaluate: the likelihood of the hypothesis H(τ) that Θ = τ

is solved by the maximum likelihood estimation (MLE). Computation of the MLE assumes the plaintext has been generated by a Markov language model with parameters (π, P). Knowledge of the ciphertext changes the likelihood of Θ:

image

Using Baye's Law

image

we have

image

The MLE of the substitution is any image which satisfies

image

Assuming and Pra posteriori{Y = y} does ...

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