Combinatorica

, Volume 6, Issue 2, pp 97–108

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

• Manuel Blum
Article

## Abstract

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.

60 J 10 68 C 05

## Bibliography

1. [1]
Josh D. Cohen, Fairing of biased Coins in Bounded Time,Yale Computer Science Technical Report 372 (1985).Google Scholar
2. [2]
Peter Elias, The Efficient Construction of an Unbiased Random Sequence, Ann. Math. Statist.43 (1972), 865–870.Google Scholar
3. [3]
Benny Chor andOded Goldreich, Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity,Proc. 26 th IEEE FOCS (1985), 429–442.Google Scholar
4. [4]
R. G. Gallager,Information Theory and Reliable Communication, Wiley, New York (1968).
5. [5]
Miklós Sántha andUmesh V. Vazirani, Generating Quasi-Random Sequences from Slightly-Random Sources,Proc. 25 th IEEE FOCS (1985), 434–440. The following paper has a comprehensive bibliography:Google Scholar
6. [6]
Quentin F. Stout andBette Warren, Tree Algorithms for Unbiased Coin Tossing with a Baised Coin,Ann Prob. 12 (1984), 212–222.
7. [7]
Umesh V. Vazirani, Towards a Strong Communication Complexity Theory or Generating Quasi-Random Sequences from Two Communicating Slightly-Random Sources,Proc. 17 th ACM STOC (1985), 366–378.Google Scholar
8. [8]
John Von Neumann, Various Techniques Used in Connection with Random Digits, Notes by G. E. Forsythe, National Bureau of Standards,Applied Math Series, Vol.12 (1951), 36–38.Reprinted in: von Neumann’s Collected Works, Vol.5, Pergamon Press (1963), 768–770.Google Scholar