Generalizing the Blum-Elias method for generating random bits from Markov chains

The problem of random number generation from an uncorrelated random source (of unknown probability distribution) dates back to von Neumann's 1951 work. Elias (1972) generalized von Neumann's scheme and showed how to achieve optimal efficiency in unbiased random bits generation. Hence, a natural question is what if the sources are correlated? Both Elias and Samueleson proposed methods for generating unbiased random bits in the case of correlated sources (of unknown probability distribution), specifically, they considered finite Markov chains. However, their proposed methods are not efficient (Samueleson) or have implementation difficulties (Elias). Blum (1986) devised an algorithm for efficiently generating random bits from degree-2 finite Markov chains in expected linear time, however, his beautiful method is still far from optimality. In this paper, we generalize Blum's algorithm to arbitrary degree finite Markov chains and combine it with Elias's method for efficient generation of unbiased bits. As a result, we provide the first known algorithm that generates unbiased random bits from an arbitrary finite Markov chain, operates in expected linear time and achieves the information-theoretic upper bound on efficiency.