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 ⊆ F_2ⁿ aims to quantify how "close" these sets are to being dual and is defined as [u(A,B)=|E_{a ∈ A, b ∈ B}[(-1)^{∑_i=1n a_i b_i}]|] Notice that u(A,B)=1 implies that A is contained in an affine shift of B^⊥ --- the space dual to the F₂span 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 u(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.