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Fourier Growth of Structured 𝔽₂-Polynomials and Applications

Authors Jarosław Błasiok, Peter Ivanov, Yaonan Jin, Chin Ho Lee, Rocco A. Servedio, Emanuele Viola



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Author Details

Jarosław Błasiok
  • Columbia University, New York, NY, USA
Peter Ivanov
  • Northeastern University, Boston, MA, USA
Yaonan Jin
  • Columbia University, New York, NY, USA
Chin Ho Lee
  • Columbia University, New York, NY, USA
Rocco A. Servedio
  • Columbia University, New York, NY, USA
Emanuele Viola
  • Northeastern University, Boston, MA, USA

Acknowledgements

We thank Shivam Nadimpalli for stimulating discussions at the early stage of the project.

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Jarosław Błasiok, Peter Ivanov, Yaonan Jin, Chin Ho Lee, Rocco A. Servedio, and Emanuele Viola. Fourier Growth of Structured 𝔽₂-Polynomials and Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 53:1-53:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.53

Abstract

We analyze the Fourier growth, i.e. the L₁ Fourier weight at level k (denoted L_{1,k}), of various well-studied classes of "structured" m F₂-polynomials. This study is motivated by applications in pseudorandomness, in particular recent results and conjectures due to [Chattopadhyay et al., 2019; Chattopadhyay et al., 2019; Eshan Chattopadhyay et al., 2020] which show that upper bounds on Fourier growth (even at level k = 2) give unconditional pseudorandom generators. Our main structural results on Fourier growth are as follows: - We show that any symmetric degree-d m F₂-polynomial p has L_{1,k}(p) ≤ Pr [p = 1] ⋅ O(d)^k. This quadratically strengthens an earlier bound that was implicit in [Omer Reingold et al., 2013]. - We show that any read-Δ degree-d m F₂-polynomial p has L_{1,k}(p) ≤ Pr [p = 1] ⋅ (k Δ d)^{O(k)}. - We establish a composition theorem which gives L_{1,k} bounds on disjoint compositions of functions that are closed under restrictions and admit L_{1,k} bounds. Finally, we apply the above structural results to obtain new unconditional pseudorandom generators and new correlation bounds for various classes of m F₂-polynomials.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Circuit complexity
Keywords
  • Fourier analysis
  • Pseudorandomness
  • Fourier growth

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