From affine to two-source extractors via approximate duality

Eli Ben-Sasson, Noga Zewi

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

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 Ea∈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.

Original languageEnglish
Title of host publicationSTOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing
Pages177-186
Number of pages10
DOIs
StatePublished - 2011
Externally publishedYes
Event43rd ACM Symposium on Theory of Computing, STOC'11 - San Jose, CA, United States
Duration: 6 Jun 20118 Jun 2011

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference43rd ACM Symposium on Theory of Computing, STOC'11
Country/TerritoryUnited States
CitySan Jose, CA
Period6/06/118/06/11

Keywords

  • Ramsey graphs
  • affine sources
  • approximate duality
  • discrepancy
  • dispersers
  • extractors
  • independent sources
  • polynomial Freiman-Ruzsa conjecture

ASJC Scopus subject areas

  • Software

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