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Fourier growth of structured $\mathbb{F}_2$-polynomials and applications

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

TL;DR

This work analyzes the $L_{1,k}$ Fourier growth of structured $\,\mathbb{F}_2$-polynomials and links it to unconditional pseudorandom generators. It proves sharp upper bounds for two key structured classes: symmetric degree-$d$ polynomials satisfy $L_{1,k}(p) \le \Pr[p=1] \cdot O(d)^k$, and read-$\Delta$ polynomials satisfy $L_{1,k}(p) \le \Pr[p=1] \cdot (k\Delta d)^{O(k)}$, with a general composition theorem that preserves these bounds under disjoint composition. These structural results yield new PRGs fooling read-few polynomials and improved correlation bounds against parity-type targets, advancing the CHLT program for $\,\mathbb{F}_2$-polynomials. The methods combine weight-mod-$m$ analyses via Kravchuk polynomials for symmetric polynomials, a disjoint-decomposition bias framework for read-$\Delta$ polynomials, and a derivative/random-restriction approach to composition, enabling bounds that extend beyond the classes considered. Overall, the paper makes concrete progress toward unconditional pseudorandomness and correlation results for structured $\,\mathbb{F}_2$-polynomials and clarifies how composition interacts with Fourier growth.

Abstract

We analyze the Fourier growth, i.e. the $L_1$ Fourier weight at level $k$ (denoted $L_{1,k}$), of various well-studied classes of "structured" $\mathbb{F}_2$-polynomials. This study is motivated by applications in pseudorandomness, in particular recent results and conjectures due to [CHHL19,CHLT19,CGLSS20] 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-$d$ $\mathbb{F}_2$-polynomial $p$ has $L_{1,k}(p) \le \Pr[p=1] \cdot O(d)^k$, and this is tight for any constant $k$. This quadratically strengthens an earlier bound that was implicit in [RSV13]. - We show that any read-$Δ$ degree-$d$ $\mathbb{F}_2$-polynomial $p$ has $L_{1,k}(p) \le \Pr[p=1] \cdot (k Δd)^{O(k)}$. - 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 $\mathbb{F}_2$-polynomials.

Fourier growth of structured $\mathbb{F}_2$-polynomials and applications

TL;DR

This work analyzes the Fourier growth of structured -polynomials and links it to unconditional pseudorandom generators. It proves sharp upper bounds for two key structured classes: symmetric degree- polynomials satisfy , and read- polynomials satisfy , with a general composition theorem that preserves these bounds under disjoint composition. These structural results yield new PRGs fooling read-few polynomials and improved correlation bounds against parity-type targets, advancing the CHLT program for -polynomials. The methods combine weight-mod- analyses via Kravchuk polynomials for symmetric polynomials, a disjoint-decomposition bias framework for read- polynomials, and a derivative/random-restriction approach to composition, enabling bounds that extend beyond the classes considered. Overall, the paper makes concrete progress toward unconditional pseudorandomness and correlation results for structured -polynomials and clarifies how composition interacts with Fourier growth.

Abstract

We analyze the Fourier growth, i.e. the Fourier weight at level (denoted ), of various well-studied classes of "structured" -polynomials. This study is motivated by applications in pseudorandomness, in particular recent results and conjectures due to [CHHL19,CHLT19,CGLSS20] which show that upper bounds on Fourier growth (even at level ) give unconditional pseudorandom generators. Our main structural results on Fourier growth are as follows: - We show that any symmetric degree- -polynomial has , and this is tight for any constant . This quadratically strengthens an earlier bound that was implicit in [RSV13]. - We show that any read- degree- -polynomial has . - We establish a composition theorem which gives bounds on disjoint compositions of functions that are closed under restrictions and admit bounds. Finally, we apply the above structural results to obtain new unconditional pseudorandom generators and new correlation bounds for various classes of -polynomials.

Paper Structure

This paper contains 33 sections, 22 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Background: $L_1$ Fourier norms and Fourier growth
  3. Motivation for this work: Fourier growth, pseudorandomness, $\mathbbm F_2$-polynomials, and the CHLT conjecture
  4. Pseudorandom generators from Fourier growth bounds.
  5. $\mathbbm F_2$-polynomials and the CHLT conjecture.
  6. Extending the CHLT conjecture.
  7. Our results: Fourier bounds for structured $\mathbbm F_2$-polynomials
  8. Symmetric $\mathbbm F_2$-polynomials
  9. Read-$\Delta$ $\mathbbm F_2$-polynomials
  10. A composition theorem
  11. Applications of our results
  12. Pseudorandom generators
  13. Correlation bounds
  14. Related work
  15. Our techniques
  16. ...and 18 more sections

Key Result

Theorem 3

Let ${\cal F}$ be a family of $n$-variable Boolean functions that is closed under restrictions and has Fourier growth $L_1(a,b)$. Then there is an explicit pseudorandom generator that $\epsilon$-fools ${\cal F}$ with seed length $O(b^2\log(n/\epsilon)(\log \log n + \log(a/\epsilon))).$

Theorems & Definitions (52)

  • Definition 1: $L_1$ Fourier norm at level $k$
  • Definition 2: Fourier growth
  • Theorem 3: PRGs from Fourier growth: Theorem 23 of CHHL
  • Theorem 4: PRGs from $L_1$ Fourier norm bounds at level $k=2$: Theorem 2.1 of CHLT
  • Theorem 5: PRGs from $L_1$ Fourier norm bounds up to level $k$ for any $k$: Theorem 4.3 of CGLSS
  • Conjecture 6: CHLT
  • Conjecture 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • ...and 42 more