From Affine to Two-Source Extractors via Approximate Duality

Abstract

We establish a new connection between affine and two-source extractors by presenting black-box constructions of two-source extractors for min-entropy rate below half from any affine extractor for min-entropy rate below half. Two such constructions are presented, and one of our constructions can reach arbitrarily small min-entropy rate assuming that the affine extractor has sufficiently good parameters. The first part of our analysis shows that our constructions are two-source dispersers which are weak (but nontrivial) kinds of two-source extractors, also known as “bipartite Ramsey graphs.” To strengthen this result and obtain two-source extractors we introduce the approximate duality conjecture (ADC) and initiate its study. The ADC leads to a rather general result that can be used to convert a natural class of two-source dispersers---``low-rank dispersers''---into two-source extractors. More specifically, we first prove a special case of ADC that implies that the constructions mentioned above are two-source extractors with large (but nontrivial) constant error. In an attempt to reduce the error in our constructions we show that the polynomial Freiman--Ruzsa conjecture (PFR) in additive combinatorics implies a stronger “approximate duality” statement (and that this stronger statement also implies a weak but as-of-yet-unknown version of PFR). This stronger statement implies in turn that our constructions are two-source extractors with exponentially small error.