On Optimal Algorithms for Generating Random Bits from Loaded Dice

The problem of generating random bits from an arbitrary biased coin, dating back to von Neumann’s 1951 work, has been extensively studied. Specifically, given a fixed number of biased coin tosses with unknown probability, it is well known how to generate random bits with an asymptotically optimal efficiency. In this paper we address two basic questions related to the generation of random bits. The first question considers an m-sided die as the source – generalizing the known case of a 2-sided coin. The second question considers variable to fixedlength random number generators – as opposed to the existing fixed to variable-length generators. We are interested in generating random bits from a more general source – an m-sided die with unknown bias (called a loaded die). The question is can we use the existing (or any) optimal algorithms that generate random bits from a biased (2-sided) coin? We provide a positive answer to this question, specifically; we present a universal scheme for transforming an arbitrary algorithm for a biased 2-sided coin to generate random bits from the general source of an m-sided die, hence enabling the application of arbitrary existing algorithms to general sources. We notice that all existing optimal algorithms for biased coins receive as input a fixed number of coin tosses and generate a variable number of random bits. Since the number of random bits generated cannot be accurately controlled, these algorithms are not feasible (or not efficient) for many practical applications. We study this new paradigm and present an optimal universal variable to fixed-length random number generator.