Deterministic extractors for small-space sources

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 2 s 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 ( ζ 3 n ) . 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.