Two-source extractors for leaky sources

Abstract

A (worst-case) 2-source extractor is a deterministic algorithm that transforms pairwise independent weak random sources into almost uniform random strings. Despite non-constructive proofs that such objects exist with almost optimal parameters, it has been a longstanding open problem to construct `explicit' (aka efficient) functions for sources of `small' constant entropy rate. In particular, best known constructions either require entropy rate of at least 0.4999 (due to Bourgain), or one source must remain with constant entropy rate above half (due to Raz). Motivated by cryptographic applications, we observe that if one source is a leaky source (or it contains a few deterministically extractable entropy), then we will be able to efficiently extract almost all entropy from both sources with nearly optimal entropy loss. Further, our extractor (for leaky sources) does not suffer from the half entropy rate barrier, and it works for all linear (and even sub-linear) entropy sources. The extractor is constructed using the technique of alternating extraction by Dziembowski and Pietrzak (FOCS 2007). Finally, we show that the extractor is almost a worse-case extractor (for the same parameters) in the sense that it only fails for a negligible fraction of sources.