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

TL;DR

This paper studies the Fourier growth of structured $ ext{F}_2$-polynomials, providing new bounds and composition theorems, which are then used to develop pseudorandom generators and correlation bounds.

Contribution

It introduces new bounds on Fourier growth for symmetric and read-$ riangle$ $ ext{F}_2$-polynomials and a composition theorem, advancing pseudorandomness and complexity theory.

Findings

Bound $L_{1,k}$ for symmetric degree-$d$ polynomials as $O(d)^k$

Bound $L_{1,k}$ for read-$ riangle$ degree-$d$ polynomials as $(k riangle d)^{O(k)}$

New pseudorandom generators and correlation bounds derived from structural results

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-$\Delta$ degree-$d$ $\mathbb{F}_2$-polynomial $p$ has $L_{1,k}(p) \le \Pr[p=1] \cdot (k \Delta d)^{O(k)}$. - 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 $\mathbb{F}_2$-polynomials.