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Fractional Pseudorandom Generators from Any Fourier Level

Authors Eshan Chattopadhyay, Jason Gaitonde, Chin Ho Lee, Shachar Lovett, Abhishek Shetty

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ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Derandomization
  • pseudorandomness
  • pseudorandom generators
  • Fourier analysis

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Abstract

We prove new results on the polarizing random walk framework introduced in recent works of Chattopadhyay et al. [Chattopadhyay et al., 2019; Eshan Chattopadhyay et al., 2019] that exploit L₁ Fourier tail bounds for classes of Boolean functions to construct pseudorandom generators (PRGs). We show that given a bound on the k-th level of the Fourier spectrum, one can construct a PRG with a seed length whose quality scales with k. This interpolates previous works, which either require Fourier bounds on all levels [Chattopadhyay et al., 2019], or have polynomial dependence on the error parameter in the seed length [Eshan Chattopadhyay et al., 2019], and thus answers an open question in [Eshan Chattopadhyay et al., 2019]. As an example, we show that for polynomial error, Fourier bounds on the first O(log n) levels is sufficient to recover the seed length in [Chattopadhyay et al., 2019], which requires bounds on the entire tail.
We obtain our results by an alternate analysis of fractional PRGs using Taylor’s theorem and bounding the degree-k Lagrange remainder term using multilinearity and random restrictions. Interestingly, our analysis relies only on the level-k unsigned Fourier sum, which is potentially a much smaller quantity than the L₁ notion in previous works. By generalizing a connection established in [Chattopadhyay et al., 2020], we give a new reduction from constructing PRGs to proving correlation bounds. Finally, using these improvements we show how to obtain a PRG for 𝔽₂ polynomials with seed length close to the state-of-the-art construction due to Viola [Emanuele Viola, 2009].

Cite As Get BibTex

Eshan Chattopadhyay, Jason Gaitonde, Chin Ho Lee, Shachar Lovett, and Abhishek Shetty. Fractional Pseudorandom Generators from Any Fourier Level. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 10:1-10:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.CCC.2021.10

Author Details

Eshan Chattopadhyay
  • Department of Computer Science, Cornell University, Ithaca, NY, USA
Jason Gaitonde
  • Department of Computer Science, Cornell University, Ithaca, NY, USA
Chin Ho Lee
  • Department of Computer Science, Columbia University, New York City, NY, USA
Shachar Lovett
  • Department of Computer Science, University of California, San Diego, CA, USA
Abhishek Shetty
  • Department of Computer Science, University of California, Berkeley, CA, USA

Funding

  • Chattopadhyay, Eshan: Supported by NSF grant CCF-1849899 and NSF CAREER award 2045576.
  • Gaitonde, Jason: Supported by NSF grant CCF-1408673 and AFOSR grant FA9550-19-1-0183.
  • Lee, Chin Ho: Supported by the Croucher Foundation and the Simons Collaboration on Algorithms and Geometry.
  • Lovett, Shachar: Supprted by NSF grants CCF-2006443 and DMS-1953928.
  • Shetty, Abhishek: Supported by a JP Morgan Chase Faculty Fellowship.

Acknowledgements

We thank the anonymous reviewers for their helpful comments and suggestions.

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