Two-source condensers with low error and small entropy gap via entropy-resilient functions

Avraham Ben-Aroya, Gil Cohen, Dean Doron, Amnon Ta-Shma

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

3 Scopus citations

Abstract

In their seminal work, Chattopadhyay and Zuckerman (STOC’16) constructed a two-source extractor with error ε for n-bit sources having min-entropy polylog(n/ε). Unfortunately, the construction’s running-time is poly(n/ε), which means that with polynomial-time constructions, only polynomially-small errors are possible. Our main result is a poly(n, log(1/ε))-time computable two-source condenser. For any k ≥ polylog(n/ε), our condenser transforms two independent (n, k)-sources to a distribution over m = k − O(log(1/ε)) bits that is ε-close to having min-entropy m − o(log(1/ε)). Hence, achieving entropy gap of o(log(1/ε)). The bottleneck for obtaining low error in recent constructions of two-source extractors lies in the use of resilient functions. Informally, this is a function that receives input bits from r players with the property that the function’s output has small bias even if a bounded number of corrupted players feed adversarial inputs after seeing the inputs of the other players. The drawback of using resilient functions is that the error cannot be smaller than ln r/r. This, in return, forces the running time of the construction to be polynomial in 1/ε. A key component in our construction is a variant of resilient functions which we call entropy-resilient functions. This variant can be seen as playing the above game for several rounds, each round outputting one bit. The goal of the corrupted players is to reduce, with as high probability as they can, the min-entropy accumulated throughout the rounds. We show that while the bias decreases only polynomially with the number of players in a one-round game, their success probability decreases exponentially in the entropy gap they are attempting to incur in a repeated game.

Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019
EditorsDimitris Achlioptas, Laszlo A. Vegh
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771252
DOIs
StatePublished - 1 Sep 2019
Externally publishedYes
Event22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019 - Cambridge, United States
Duration: 20 Sep 201922 Sep 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume145
ISSN (Print)1868-8969

Conference

Conference22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019
Country/TerritoryUnited States
CityCambridge
Period20/09/1922/09/19

Keywords

  • Condensers
  • Explicit constructions
  • Extractors
  • Resilient functions

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