Extracting randomness via repeated condensing

On an input probability distribution with some (min-)entropy an extractor outputs a distribution with a (near) maximum entropy rate (namely the uniform distribution). A natural weakening of this concept is a condenser, whose output distribution has a higher entropy rate than the input distribution (without losing much of the initial entropy). We construct efficient explicit condensers. The condenser constructions combine (variants or more efficient versions of) ideas from several works, including the block extraction scheme of Nisan and Zuckerman (1996), the observation made by Srinivasan and Zuckerman (1994) and Nisan and Ta-Schma (1999) that a failure of the block extraction scheme is also useful, the recursive "win-win" case analysis of Impagliazzo et al. (1999, 2000), and the error correction of random sources used by Trevisan (1999). As a natural byproduct, (via repeated iterating of condensers), we obtain new extractor constructions. The new extractors give significant qualitative improvements over previous ones for sources of arbitrary min-entropy; they are nearly optimal simultaneously in the main two parameters-seed length and output length. Specifically, our extractors can make any of these two parameters optimal (up to a constant factor), only at a poly-logarithmic loss in the other. Previous constructions require polynomial loss in both cases for general sources. We also give a simple reduction converting "standard" extractors (which are good for an average seed) to "strong " ones (which are good for mast seeds), with essentially the same parameters.

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

[2]  Noam Nisan,et al.  Randomness is Linear in Space , 1996, J. Comput. Syst. Sci..

[3]  Amnon Ta-Shma,et al.  Extractors from Reed-Muller Codes , 2001, Electron. Colloquium Comput. Complex..

[4]  Moni Naor,et al.  Small-bias probability spaces: efficient constructions and applications , 1990, STOC '90.

[5]  Jørn Justesen,et al.  Class of constructive asymptotically good algebraic codes , 1972, IEEE Trans. Inf. Theory.

[6]  Luca Trevisan,et al.  Extractors and pseudorandom generators , 2001, JACM.

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

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

[9]  Oded Goldreich,et al.  On the power of two-point based sampling , 1989, J. Complex..

[10]  Avi Wigderson,et al.  Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

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

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

[13]  Ran Raz,et al.  On recycling the randomness of states in space bounded computation , 1999, STOC '99.

[14]  Amnon Ta-Shma,et al.  Loss-less condensers, unbalanced expanders, and extractors , 2001, STOC '01.

[15]  Noam Nisan,et al.  Extracting randomness: how and why. A survey , 1996, Proceedings of Computational Complexity (Formerly Structure in Complexity Theory).

[16]  Amnon Ta-Shma,et al.  On extracting randomness from weak random sources (extended abstract) , 1996, STOC '96.

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

[18]  Amnon Ta-Shma,et al.  On Extracting Randomness From Weak Random Sources , 1995, Electron. Colloquium Comput. Complex..

[19]  David Zuckerman,et al.  General weak random sources , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[20]  Aravind Srinivasan,et al.  Computing with very weak random sources , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[21]  Manuel Blum,et al.  Independent unbiased coin flips from a correlated biased source—A finite state markov chain , 1984, Comb..

[22]  David Zuckerman,et al.  Randomness-optimal oblivious sampling , 1997, Random Struct. Algorithms.

[23]  Ran Raz,et al.  Extracting all the randomness and reducing the error in Trevisan's extractors , 1999, STOC '99.

[24]  Avi Wigderson,et al.  Near-optimal conversion of hardness into pseudo-randomness , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[25]  Avi Wigderson,et al.  Tiny Families of Functions with Random Properties: A Quality-Size Trade-off for Hashing , 1997, Electron. Colloquium Comput. Complex..

[26]  Jaikumar Radhakrishnan,et al.  Bounds for Dispersers, Extractors, and Depth-Two Superconcentrators , 2000, SIAM J. Discret. Math..

[27]  Avi Wigderson,et al.  Randomness conductors and constant-degree lossless expanders , 2002, STOC '02.

[28]  Avi Wigderson,et al.  Extractors and pseudo-random generators with optimal seed length , 2000, STOC '00.