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Improved Extractors for Recognizable and Algebraic Sources

Authors Fu Li, David Zuckerman

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Author Details

Fu Li
  • Department of Computer Science, University of Texas at Austin, USA
David Zuckerman
  • Department of Computer Science, University of Texas at Austin, USA


We wish to thank Salil Vadhan, Ronen Shaltiel, Avishay Tal, and William Hoza for helpful discussions and comments.

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Fu Li and David Zuckerman. Improved Extractors for Recognizable and Algebraic Sources. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 72:1-72:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


We study the task of seedless randomness extraction from recognizable sources, which are uniform distributions over sets of the form {x : f(x) = 1} for functions f in some specified class C. We give two simple methods for constructing seedless extractors for C-recognizable sources. Our first method shows that if C admits XOR amplification, then we can construct a seedless extractor for C-recognizable sources by using a mildly hard function for C as a black box. By exploiting this reduction, we give polynomial-time, seedless randomness extractors for three natural recognizable sources: (1) constant-degree algebraic sources over any prime field, where constant-degree algebraic sources are uniform distributions over the set of zeros of a system of constant degree polynomials; (2) sources recognizable by randomized multiparty communication protocols of cn bits, where c>0 is a small enough constant; (3) halfspace sources, or sources recognizable by linear threshold functions. In particular, the new extractor for each of these three sources has linear output length and exponentially small error for min-entropy k >= (1-alpha)n, where alpha>0 is a small enough constant. Our second method shows that a seed-extending pseudorandom generator with exponentially small error for C yields an extractor with exponentially small error for C-recognizable sources, improving a reduction by Kinne, Melkebeek, and Shaltiel [Kinne et al., 2012]. Using the hardness of the parity function against AC^0 [Håstad, 1987], we significantly improve Shaltiel’s extractor [Shaltiel, 2011] for AC^0-recognizable sources. Finally, assuming sufficiently strong one-way permutations, we construct seedless extractors for sources recognizable by BPP algorithms, and these extractors run in quasi-polynomial time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation
  • Extractor
  • Pseudorandomness


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