Log-seed pseudorandom generators via iterated restrictions

Dean Doron, Pooya Hatami, William M. Hoza

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

4 Scopus citations

Abstract

There are only a few known general approaches for constructing explicit pseudorandom generators (PRGs). The “iterated restrictions” approach, pioneered by Ajtai and Wigderson [2], has provided PRGs with seed length polylog n or even Oe(log n) for several restricted models of computation. Can this approach ever achieve the optimal seed length of O(log n)? In this work, we answer this question in the affirmative. Using the iterated restrictions approach, we construct an explicit PRG for read-once depth-2 AC0[⊕] formulas with seed length O(log n) + Oe(log(1/ε)). In particular, we achieve optimal seed length O(log n) with near-optimal error ε = exp(−Ω(log e n)). Even for constant error, the best prior PRG for this model (which includes read-once CNFs and read-once F2-polynomials) has seed length Θ(log n · (log log n)2) [22]. A key step in the analysis of our PRG is a tail bound for subset-wise symmetric polynomials, a generalization of elementary symmetric polynomials. Like elementary symmetric polynomials, subset-wise symmetric polynomials provide a way to organize the expansion of Qmi=1(1 + yi). Elementary symmetric polynomials simply organize the terms by degree, i.e., they keep track of the number of variables participating in each monomial. Subset-wise symmetric polynomials keep track of more data: for a fixed partition of [m], they keep track of the number of variables from each subset participating in each monomial. Our tail bound extends prior work by Gopalan and Yehudayoff [17] on elementary symmetric polynomials.

Original languageEnglish
Title of host publication35th Computational Complexity Conference, CCC 2020
EditorsShubhangi Saraf
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771566
DOIs
StatePublished - 1 Jul 2020
Externally publishedYes
Event35th Computational Complexity Conference, CCC 2020 - Virtual, Online, Germany
Duration: 28 Jul 202031 Jul 2020

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume169
ISSN (Print)1868-8969

Conference

Conference35th Computational Complexity Conference, CCC 2020
Country/TerritoryGermany
CityVirtual, Online
Period28/07/2031/07/20

Keywords

  • Parity gates
  • Pseudorandom generators
  • Pseudorandom restrictions
  • Read-once depth-2 formulas

ASJC Scopus subject areas

  • Software

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