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Pseudorandom Bit Generators That Fool Modular Sums

  • Conference paper
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX 2009, RANDOM 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5687))

Abstract

We consider the following problem: for given n,M, produce a sequence X 1,X 2,…,X n of bits that fools every linear test modulo M. We present two constructions of generators for such sequences. For every constant prime power M, the first construction has seed length O M (log(n/ε)), which is optimal up to the hidden constant. (A similar construction was independently discovered by Meka and Zuckerman [MZ]). The second construction works for every M,n, and has seed length O(logn + log(M/ε)log(Mlog(1/ε))).

The problem we study is a generalization of the problem of constructing small bias distributions [NN], which are solutions to the M = 2 case. We note that even for the case M = 3 the best previously known constructions were generators fooling general bounded-space computations, and required O(log2 n) seed length.

For our first construction, we show how to employ recently constructed generators for sequences of elements of ℤ M that fool small-degree polynomials (modulo M). The most interesting technical component of our second construction is a variant of the derandomized graph squaring operation of [RV]. Our generalization handles a product of two distinct graphs with distinct bounds on their expansion. This is then used to produce pseudorandom-walks where each step is taken on a different regular directed graph (rather than pseudorandom walks on a single regular directed graph as in [RTV, RV]).

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Author information

Authors and Affiliations

  1. Department of Computer Science, Weizmann Institute of Science, Rehovot, 76100, Israel

    Shachar Lovett & Omer Reingold

  2. Computer Science Division, University of California, Berkeley, CA, USA

    Luca Trevisan

  3. School of Engineering and Applied Science, Harvard University, Cambridge, MA, USA, 02138

    Salil Vadhan

Authors
  1. Shachar Lovett
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  2. Omer Reingold
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  3. Luca Trevisan
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  4. Salil Vadhan
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Editor information

Editors and Affiliations

  1. Dept. of Applied Math and Computer Science, The Weizmann Institute of Science, 76100, Rehovot, Israel

    Irit Dinur

  2. Institute for Computer Science and Applied Mathematics, University of Kiel, Olshausenstr. 40, 24098, Kiel, Germany

    Klaus Jansen

  3. Technion, Computer Science Department, 32000, Haifa, Israel

    Joseph Naor

  4. University of Geneva, Centre Universitaire d’Informatique, Battelle Bat. A, 7 rte de Drize, 1227, Carouge, Switzerland

    José Rolim

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© 2009 Springer-Verlag Berlin Heidelberg

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Lovett, S., Reingold, O., Trevisan, L., Vadhan, S. (2009). Pseudorandom Bit Generators That Fool Modular Sums. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2009 2009. Lecture Notes in Computer Science, vol 5687. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03685-9_46

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  • DOI: https://doi.org/10.1007/978-3-642-03685-9_46

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03684-2

  • Online ISBN: 978-3-642-03685-9

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