Deterministic extractors for small-space sources

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 2 s branching programs: For every constant δ>0, 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 δ>n -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(β 3 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.

Authors

• Jesse Kamp
• Anup Rao 0001
• Salil P. Vadhan
• David Zuckerman

Keywords

• pseudorandomness
• randomness extractors