, Volume 6, Issue 2, pp 97–108 | Cite as

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

  • Manuel Blum


Von Neumann’s trick for simulating anabsolutely unbiased coin by a biased one is this:
  1. 1.

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

  2. 2.

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

  3. 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.

AMS subject classification (1980)

60 J 10 68 C 05 


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Copyright information

© Akadémiai Kiadó 1986

Authors and Affiliations

  • Manuel Blum
    • 1
  1. 1.Department of Electrical Eng. and Comp. Sci.University of California at BerkeleyBerkeleyUSA

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