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Two-Source Condensers with Low Error and Small Entropy Gap via Entropy-Resilient Functions

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

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Author Details

Avraham Ben-Aroya
  • The Blavatnik School of Computer Science, Tel-Aviv University, Tel Aviv, Israel
Gil Cohen
  • The Blavatnik School of Computer Science, Tel-Aviv University, Tel Aviv, Israel
Dean Doron
  • Department of Computer Science, University of Texas at Austin, USA
Amnon Ta-Shma
  • The Blavatnik School of Computer Science, Tel-Aviv University, Tel Aviv, Israel

Funding

  • Ben-Aroya, Avraham: Israel Science Foundation Grant 994/14 and by Len Blavatnik and the Blavatnik Family Foundation.
  • Doron, Dean: Israel Science Foundation Grant 994/14 and by Len Blavatnik and the Blavatnik Family Foundation. This work was done while being at Tel-Aviv University.
  • Ta-Shma, Amnon: Israel Science Foundation Grant 994/14.

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Avraham Ben-Aroya, Gil Cohen, Dean Doron, and Amnon Ta-Shma. Two-Source Condensers with Low Error and Small Entropy Gap via Entropy-Resilient Functions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 43:1-43:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.43

Abstract

In their seminal work, Chattopadhyay and Zuckerman (STOC'16) constructed a two-source extractor with error epsilon for n-bit sources having min-entropy {polylog}(n/epsilon). Unfortunately, the construction’s running-time is {poly}(n/epsilon), which means that with polynomial-time constructions, only polynomially-small errors are possible. Our main result is a {poly}(n,log(1/epsilon))-time computable two-source condenser. For any k >= {polylog}(n/epsilon), our condenser transforms two independent (n,k)-sources to a distribution over m = k-O(log(1/epsilon)) bits that is epsilon-close to having min-entropy m - o(log(1/epsilon)). Hence, achieving entropy gap of o(log(1/epsilon)).
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/epsilon.
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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Condensers
  • Extractors
  • Resilient functions
  • Explicit constructions

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