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

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 $L_{1,k}(p)\le \mathrm{Pr}[p=1]\cdot O(d{)}^k$, 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 $L_{1,k}(p)\le \mathrm{Pr}[p=1]\cdot (k\Delta d{)}^{O(k)}$. - 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.

Corrected a mistake in Lemma 27 in the previous version of the paper

References in corpus (1)

• Quantum versus Randomized Communication Complexity, with Efficient Players