Independent unbiased coin flips from a correlated biased source—A finite state markov chain

Abstract

Von Neumann’s trick for simulating anabsolutely unbiased coin by a biased one is this:

• 1. Toss the biased coin twice, getting 00, 01, 10, or 11.

• 2. If 00 or 11 occur, go back to step 1; else

• 3. Call 10 aH, 01 aT.

Since Pr[H]=Pr[1]Pr[0]=Pr[T], the output is unbiased. Example: 00 10 11 01 01 →HTT.

Peter Elias gives an algorithm to generate an independent unbiased sequence ofHs andTs that nearly achieves the Entropy of the one-coin source. His algorithm is excellent, but certain difficulties arise in trying to use it (or the original von Neumann scheme) to generate bits in expected linear time from a Markov chain.

In this paper, we return to the original one-coin von Neumann scheme, and show how to extend it to generate an independent unbiased sequence ofHs andTs from any Markov chain in expected linear time. We give a wrong and a right way to do this. Two algorithms A and B use the simple von Neumann trick on every state of the Markov chain. They differ in the time they choose to announce the coin flip. This timing is crucial.