Fractional Pseudorandom Generators from Any Fourier Level

TL;DR

This paper introduces a new approach to constructing pseudorandom generators based on Fourier spectrum bounds, improving seed length efficiency and answering open questions in the field.

Contribution

It provides a novel analysis of fractional PRGs using Taylor's theorem, relying only on level-k Fourier sums, and extends the framework to achieve near state-of-the-art seed lengths.

Findings

Constructed PRGs with seed length depending on Fourier level k

Showed sufficiency of bounds on first O(log n) levels for polynomial error

Achieved seed lengths close to Viola's for F2 polynomials

Abstract

We prove new results on the polarizing random walk framework introduced in recent works of Chattopadhyay {et al.} [CHHL19,CHLT19] that exploit $L_1$ 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 [CHHL19], or have polynomial dependence on the error parameter in the seed length [CHLT10], and thus answers an open question in [CHLT19]. 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 [CHHL19], 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 \emph{level-k unsigned Fourier sum}, which is potentially a much smaller quantity than the $L_1$ notion in previous works. By generalizing a connection established in [CHH+20], we give a new reduction from constructing PRGs to proving correlation bounds. Finally, using these improvements we show how to obtain a PRG for $\mathbb{F}_2$ polynomials with seed length close to the state-of-the-art construction due to Viola [Vio09], which was not known to be possible using this framework.