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Computer Security and Cryptography by Alan G. Konheim

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3.7 THE HIDDEN MARKOV MODEL (HMM)

A class of stochastic processes now referred to as Hidden Markov models (HMM) are described in the two important papers published by Petrie [1969] and Baum et al. [1969]. The application of HMM to automatic speech recognition (ASR) was quickly recognized, and is detailed in the survey papers by Levinson et al. [1983], Rabiner and Juang [1986] and Poritz [1988]. We outline the main ideas and show how HMM may be applied to cryptanalyze a monoalphabetic substitution.

A hidden Markov model (HMM) is a two-stage random process; both the input X = (X0, X1,…, Xn) and output states Y = (Y0, Y1,…, Yn) consists of integers in image. The HMM is constructed from

  1. A Markov chain with parameters (π, P) generating (hidden) states X

    image

    image

  2. An output probability distribution q(j/i) = Pr{Yt = j/Xt = i} for each hidden state i

    image

image

Figure 3.3 Observing the hidden states.

The evolution of the HMM may be described as follows:

  1. The initial hidden state X0 = x0 is chosen with probability ...

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