Deterministic extractors for small-space sources

摘要

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. Specifically, there is a constant η>0 such that for any ζ>n−η, our algorithm extracts m=(δ−ζ)n bits that are exponentially close to uniform (in variation distance) from space s sources with min-entropy δn, where s=Ω(ζ3n). Previously, nothing was known for δ⩽1/2, even for space 0. Our results are obtained by a reduction to the class of total-entropy independent sources. This model generalizes both the well-studied models of independent sources and symbol-fixing sources. These sources consist of a set of r independent smaller sources over {0,1}ℓ, where the total min-entropy over all the smaller sources is k. We give deterministic extractors for such sources when k is as small as polylog(r), for small enough ℓ.