Improving the Hadamard extractor

In this paper we construct a strong randomness extractor with two independent @?-bit input distributions with min entropies b"X,b"Y,b"X+b"Y>@? (the probability of any particular output is upper bounded by 2^-^b^"^X and 2^-^b^"^Y, respectively). For b"X,b"Y@?@?-1, our extractor produces one bit which is by the factor of 2 closer to the uniform distribution, when compared to the Hadamard extractor. What is more, this distance drops to zero if at least one of the min entropies raises to @?. This is in sharp contrast to the Hadamard extractor which fails to produce even a single unbiased bit, even if one of the input distributions is uniform. We also extend our construction to produce k bits of output with a bias that is by the factor of 3/2 smaller than that of the corresponding Hadamard extractor and retains the ability to produce unbiased bits if one of the input distributions is uniform. The strongness property of the extractor is maintained in both cases, however, in the multi-bit scenario the bias is increased by the factor of 3/2.

[1]  Xin Li,et al.  Improved Constructions of Three Source Extractors , 2011, 2011 IEEE 26th Annual Conference on Computational Complexity.

[2]  Miklos Santha,et al.  Generating Quasi-random Sequences from Semi-random Sources , 1986, J. Comput. Syst. Sci..

[3]  Avi Wigderson,et al.  Extracting Randomness Using Few Independent Sources , 2006, SIAM J. Comput..

[4]  Nathan Linial,et al.  Collective Coin Flipping , 1989, Adv. Comput. Res..

[5]  Noam Nisan,et al.  Extracting Randomness: A Survey and New Constructions , 1999, J. Comput. Syst. Sci..

[6]  Ran Raz,et al.  Improved Randomness Extraction from Two Independent Sources , 2004, APPROX-RANDOM.

[7]  Umesh V. Vazirani,et al.  Strong communication complexity or generating quasi-random sequences from two communicating semi-random sources , 1987, Comb..

[8]  David Zuckerman,et al.  Deterministic extractors for small-space sources , 2011, J. Comput. Syst. Sci..

[9]  Anup Rao,et al.  Extractors for Low-Weight Affine Sources , 2009, 2009 24th Annual IEEE Conference on Computational Complexity.

[10]  L. Fortnow,et al.  Recent Developments in Explicit Constructions of Extractors , 2002, Bull. EATCS.

[11]  Avi Wigderson,et al.  2-source dispersers for sub-polynomial entropy and Ramsey graphs beating the Frankl-Wilson construction , 2006, STOC '06.

[12]  Ran Raz,et al.  Deterministic extractors for affine sources over large fields , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[13]  Ronen Shaltiel,et al.  Increasing the output length of zero‐error dispersers , 2012, Random Struct. Algorithms.

[14]  Oded Goldreich,et al.  Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity , 1988, SIAM J. Comput..

[15]  Ran Raz,et al.  Deterministic extractors for bit-fixing sources by obtaining an independent seed , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[16]  Guy Kindler,et al.  Simulating independence: new constructions of condensers, ramsey graphs, dispersers, and extractors , 2005, STOC '05.

[17]  Anup Rao,et al.  Extractors for a constant number of polynomially small min-entropy independent sources , 2006, STOC '06.

[18]  Manuel Blum.  Independent unbiased coin flips from a correlated biased source—A finite state markov chain , 1986, Comb..

[19]  Oded Goldreich,et al.  The bit extraction problem or t-resilient functions , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[20]  J. Bourgain,et al.  MORE ON THE SUM-PRODUCT PHENOMENON IN PRIME FIELDS AND ITS APPLICATIONS , 2005 .

[21]  Avi Wigderson,et al.  Dispersers, deterministic amplification, and weak random sources , 1989, 30th Annual Symposium on Foundations of Computer Science.

[22]  Ronen Shaltiel,et al.  How to Get More Mileage from Randomness Extractors , 2006, 21st Annual IEEE Conference on Computational Complexity (CCC'06).

[23]  Xin Li,et al.  A New Approach to Affine Extractors and Dispersers , 2011, 2011 IEEE 26th Annual Conference on Computational Complexity.

[24]  Amir Yehudayoff,et al.  Affine extractors over prime fields , 2011, Comb..

[25]  David Zuckerman,et al.  DETERMINISTIC EXTRACTORS FOR BIT-FIXING SOURCES AND EXPOSURE-RESILIENT CRYPTOGRAPHY , 2003 .

[26]  Emanuele Viola.  Extractors for Circuit Sources , 2011, FOCS.

[27]  Oded Goldreich,et al.  Three XOR-Lemmas - An Exposition , 1995, Electron. Colloquium Comput. Complex..

[28]  Yevgeniy Dodis,et al.  On Extracting Private Randomness over a Public Channel , 2003, RANDOM-APPROX.

[29]  Anup Rao,et al.  An Exposition of Bourgain's 2-Source Extractor , 2007, Electron. Colloquium Comput. Complex..

[30]  Ran Raz,et al.  Extractors with weak random seeds , 2005, STOC '05.

[31]  L. Trevisan,et al.  Extracting Randomness from Samplable Distributions EXTENDED AESTRACT , 2000 .

[32]  Jean Bourgain,et al.  On the Construction of Affine Extractors , 2007 .