From affine to two-source extractors via approximate duality

Two-source and affine extractors and dispersers are fundamental objects studied in the context of derandomization. This paper shows how to construct two-source extractors and dispersers for arbitrarily small min-entropy rate in a black-box manner given affine extractors with sufficiently good parameters. Our analysis relies on the study of approximate duality, a concept related to the polynomial Freiman-Ruzsa conjecture (PFR) from additive combinatorics. Two black-box constructions of two-source extractors from affine ones are presented. Both constructions work for min-entropy rate ρ < 1/2. One of them can potentially reach arbitrarily small min-entropy rate provided the the affine extractor used to construct it outputs, on affine sources of min-entropy rate 1/2, a relatively large number of output bits, and has sufficiently small error. Our results are obtained by first showing that each of our constructions yields a two-source disperser for a certain min-entropy rate ρ < 1/2 and then using a general extractor-to-disperser reduction that applies to a large family of constructions. This reduction says that any two-source disperser for min-entropy rate ρ coming from this family is also a two-source extractor with constant error for min-entropy rate ρ+ε for arbitrarily small ε > 0. We show that assuming the PFR conjecture, the error of this two-source extractor is exponentially small. The extractor-to-disperser reduction arises from studying approximate duality, a notion related to additive combinatorics. The duality measure of two sets A, B ⊆ double-struck F ₂ⁿ aims to quantify how "close" these sets are to being dual and is defined as μ^⊥(A, B) = |double-struck E_a∈A,b∈B [(-1)^∑in=1aibi]|. Notice that μ^⊥(A,B)=1 implies that A is contained in an affine shift of B^⊥ - the space dual to the double-struck F₂-span of B. We study what can be said of A,B when their duality measure is large but strictly smaller than 1 and show that A,B contain subsets A′,B′ of nontrivial size for which μ^⊥(A′,B′) = 1 and consequently A′ is contained in an affine shift of (B′) ^⊥. This implies that our constructions are two-source extractors with constant error. Surprisingly, the PFR implies that such A′,B′ exist exist when A,B are large, even if the duality measure is exponentially small in n, and this implication leads to two-source extractors with exponentially small error.