# Independent Unbiased Coin Flips From a Correlated Biased Source: a Finite State Markov Chain

```@inproceedings{Blum1984IndependentUC,
title={Independent Unbiased Coin Flips From a Correlated Biased Source: a Finite State Markov Chain},
author={Manuel Blum},
booktitle={IEEE Annual Symposium on Foundations of Computer Science},
year={1984}
}```
• M. Blum
• Published in
IEEE Annual Symposium on…
24 October 1984
• Mathematics
von Neumann's trick for generating an absolutely unbiased coin from 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 a H, 01 a T. Since p[H] = p[1]*p[0] = p[T], the output is unbiased. Example: 00 10 11 01 01 /spl I.oarr/ H T T. Peter Elias gives an algorithm to generate an independent unbiased sequence of Hs and Ts that nearly achieves the Entropy of the one-coin source. His algorithm is excellent, but…
139 Citations

### How to turn loaded dice into fair coins

• Computer Science
IEEE Trans. Inf. Theory
• 2000
A new generalization of von Neumann's algorithm distinguished by its high level of practicality and amenability to analysis is described, and it is able to prove that in an asymptotic sense the algorithm extracts the full entropy of its input.

### Efficient Generation of Random Bits From Finite State Markov Chains

• Mathematics, Computer Science
IEEE Transactions on Information Theory
• 2012
This paper generalize Blum's algorithm to arbitrary degree finite Markov chains and combine it with Elias's method for efficient generation of unbiased bits, providing the first known algorithm that generates unbiased random bits from an arbitrary finiteMarkov chain, operates in expected linear time and achieves the information-theoretic upper bound on efficiency.

### Generalizing the Blum-Elias method for generating random bits from Markov chains

• Mathematics, Computer Science
2010 IEEE International Symposium on Information Theory
• 2010
Blum's algorithm is generalized to arbitrary degree finite Markov chains and combined with Elias's method for efficient generation of unbiased bits, providing 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.

### Blind-friendly von Neumann's Heads or Tails

• Mathematics
Am. Math. Mon.
• 2014
This paper addresses how to extract uniformly distributed bits of information from a nonuniform source and studies some probabilities related to biased dice and coins, culminating in an interesting variation of von Neumann's mechanism that can be employed in a more restricted setting where the actual results of the coin tosses are not known to the contestants.

### Blind-friendly von Neumann's Heads or Tails

• Mathematics
• 2014
Abstract The toss of a coin is usually regarded as the epitome of randomness, and has been used for ages as a means to resolve disputes in a simple, fair way. Perhaps as ancient as consulting objects

### Streaming Algorithms for Optimal Generation of Random Bits

• Computer Science
ArXiv
• 2012
This paper presents an algorithm that generates random bit streams from biased coins, uses bounded space and runs in expected linear time, and approaches the information-theoretic upper bound on efficiency.

### Randomness-optimal oblivious sampling

• D. Zuckerman
• Computer Science, Mathematics
Random Struct. Algorithms
• 1997
This work presents the first efficient oblivious sampler that uses an optimal number of random bits, up to an arbitrary constant factor bigger than 1, and gives applications to constructive leader election and reducing randomness in interactive proofs.

### New Imperfect Random Source with Applications to Coin-Flipping

• Y. Dodis
• Computer Science, Mathematics
ICALP
• 2001
A new imperfect random source that realistically generalizes the SV-source of Santha and Vazirani and the bit-fixing source of Lichtenstein, Linial and Saks is introduced, and there exists no black-box transformation from a non-adaptively secure coin-flipping protocol resulting in an adaptively secure protocol tolerating ω(√n) faulty players.

### Von Neumann Normalisation of a Quantum Random Number Generator

• Mathematics
Comput.
• 2012
A successful application of von Neumann normalisation does exactly what it promises, un-biasing, one (among infinitely many) symptoms of randomness; it will not produce "true" randomness.

### Optimal random number generation from a biased coin

• Computer Science, Mathematics
SODA '05
• 2005
The model of computation is sufficiently general to encompass virtually all previously known algorithms for this problem, and it is proved that it is impossible to construct an optimal tree algorithm recursively using a model based on the algebraic decision tree.

## References

SHOWING 1-6 OF 6 REFERENCES

### Towards a strong communication complexity theory or generating quasi-random sequences from two communicating slightly-random sources

Santha and Vazirani consider a very general model for such imperfect sources of randomness: the slightly random source.

### Unbiased bits from sources of weak randomness and probabilistic communication complexity

• Computer Science, Mathematics
26th Annual Symposium on Foundations of Computer Science (sfcs 1985)
• 1985
It is shown that most Boolean functions have linear communication complexity in a very strong sense when used to extract almost unbiased and independent bits from the output of any two independent "probability-bounded" sources.

### Information Theory and Reliable Communication

This chapter discusses Coding for Discrete Sources, Techniques for Coding and Decoding, and Source Coding with a Fidelity Criterion.

### Various Techniques Use in Connection with Random Digits,

• National Bureau of Standards, Applied Math Series,
• 1195