Deterministic extractors for small-space sources

  • Authors:

  • Jesse Kamp;Anup Rao;Salil Vadhan;David Zuckerman

  • Affiliations:

  • University of Texas, Austin, TX;University of Texas, Austin, TX;Harvard University, Cambridge, MA;University of Texas, Austin, TX

  • Venue:
  • Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
  • Year:
  • 2006

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Abstract

We give polynomial-time, deterministic randomness extractors for sources generated in small space, where we model space s sources on (0,1)n as sources generated by width 2s branching programs: For every constant δ0, we can extract .99 δ n bits that are exponentially close to uniform (in variation distance) from space s sources of min-entropy δ n, where s=Ω(n). In addition, assuming an efficient deterministic algorithm for finding large primes, there is a constant η 0 such that for any δn-η, we can extract m=(δ-δ)n bits that are exponentially close to uniform from space s sources with min-entropy δ n, where s=Ω(β3 n). Previously, nothing was known for δ ≤ 1/2, even for space 0.Our results are obtained by a reduction to a new class of sources that we call independent-symbol sources, which generalize both the well-studied models of independent sources and symbol-fixing sources. These sources consist of a string of n independent symbols over a d symbol alphabet with min-entropy k. We give deterministic extractors for such sources when k is as small as polylog(n), for small enough d.