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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