Extractors And Rank Extractors For Polynomial Sources

Abstract.In this paper we construct explicit deterministic extractors from polynomial sources, which are distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources (which are degree 1 polynomials). A direct consequence is a deterministic extractor for distributions sampled by polynomial size arithmetic circuits over exponentially large fields. The steps in our extractor construction, and the tools (mainly from algebraic geometry) that we use for them, are of independent interest:The first step is a construction of rank extractors, which are polynomial mappings which ‘extract’ the algebraic rank from any system of low degree polynomials. More precisely, for any n polynomials, k of which are algebraically independent, a rank extractor outputs k algebraically independent polynomials of slightly higher degree. The rank extractors we construct are applicable not only over finite fields but also over fields of characteristic zero.The next step is relating algebraic independence to min-entropy. We use a theorem of Wooley to show that these parameters are tightly connected. This allows replacing the algebraic assumption on the source (above) by the natural information theoretic one. It also shows that a rank extractor is already a high quality condenser for polynomial sources over polynomially large fields.Finally, to turn the condensers into extractors, we employ a theorem of Bombieri, giving a character sum estimate for polynomials defined over curves. It allows extracting all the randomness (up to a multiplicative constant) from polynomial sources over exponentially large prime fields.

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

[2]  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.

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

[4]  L. Fortnow Recent Developments in Explicit Constructions of Extractors , 2002 .

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

[6]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

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

[8]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[9]  Amnon Ta-Shma,et al.  Extractor codes , 2001, IEEE Transactions on Information Theory.

[10]  Noam Nisan,et al.  More deterministic simulation in logspace , 1993, STOC.

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

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

[13]  Ran Raz,et al.  Error reduction for extractors , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

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

[15]  Luca Trevisan,et al.  Extracting randomness from samplable distributions , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[16]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[17]  David A. Cox,et al.  Ideals, Varieties, and Algorithms , 1997 .

[18]  Trevor D. Wooley,et al.  A note on simultaneous congruences , 1996 .

[19]  M. S. L’vov,et al.  Calculation of invariants of programs interpreted over an integrality domain , 1984, Cybernetics.

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

[21]  I. Shafarevich Basic algebraic geometry , 1974 .

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

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

[24]  Piotr Indyk,et al.  Uncertainty principles, extractors, and explicit embeddings of l2 into l1 , 2007, STOC '07.

[25]  Ran Raz,et al.  Deterministic extractors for affine sources over large fields , 2008, Comb..

[26]  Oded Goldreich,et al.  The Bit Extraction Problem of t-Resilient Functions (Preliminary Version) , 1985, FOCS.

[27]  Gian-Carlo Rota,et al.  Apolarity and Canonical Forms for Homogeneous Polynomials , 1993, Eur. J. Comb..

[28]  Joe W. Harris,et al.  Algebraic Geometry: A First Course , 1995 .

[29]  Neeraj Kayal The Complexity of the Annihilating Polynomial , 2009, 2009 24th Annual IEEE Conference on Computational Complexity.

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

[31]  David Zuckerman,et al.  Deterministic extractors for bit-fixing sources and exposure-resilient cryptography , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

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

[33]  K. Conrad Finite Fields , 2004, Series and Products in the Development of Mathematics.

[34]  Avi Wigderson,et al.  Expanders That Beat the Eigenvalue Bound: Explicit Construction and Applications , 1999, Comb..