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Pseudorandom Generators from Polarizing Random Walks

Authors Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, Shachar Lovett

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Author Details

Eshan Chattopadhyay
  • Cornell Univeristy and IAS, Princeton, USA
Pooya Hatami
  • University of Texas at Austin, USA
Kaave Hosseini
  • University of California, San Diego, USA
Shachar Lovett
  • University of California, San Diego, USA

Funding

  • Chattopadhyay, Eshan: Supported by NSF grant CCF-1412958 and the Simons foundation.
  • Hatami, Pooya: Supported by a Simons Investigator Award (#409864, David Zuckerman).
  • Hosseini, Kaave: Supported by NSF grant CCF-1614023.
  • Lovett, Shachar: Supported by NSF grant CCF-1614023.

Cite As Get BibTex

Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, and Shachar Lovett. Pseudorandom Generators from Polarizing Random Walks. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 1:1-1:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.CCC.2018.1

Abstract

We propose a new framework for constructing pseudorandom generators for n-variate Boolean functions. It is based on two new notions. First, we introduce fractional pseudorandom generators, which are pseudorandom distributions taking values in [-1,1]^n. Next, we use a fractional pseudorandom generator as steps of a random walk in [-1,1]^n that converges to {-1,1}^n. We prove that this random walk converges fast (in time logarithmic in n) due to polarization. As an application, we construct pseudorandom generators for Boolean functions with bounded Fourier tails. We use this to obtain a pseudorandom generator for functions with sensitivity s, whose seed length is polynomial in s. Other examples include functions computed by branching programs of various sorts or by bounded depth circuits.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • AC0
  • branching program
  • polarization
  • pseudorandom generators
  • random walks
  • Sensitivity

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