A class of stochastic processes now referred to as Hidden Markov models (HMM) are described in the two important papers published by Petrie  and Baum et al. . The application of HMM to automatic speech recognition (ASR) was quickly recognized, and is detailed in the survey papers by Levinson et al. , Rabiner and Juang  and Poritz . 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 . The HMM is constructed from
The evolution of the HMM may be described as follows: