Improved pseudorandom generators from pseudorandom multi-switching lemmas

TL;DR

This paper introduces the best known pseudorandom generators for certain circuit classes, achieving optimal seed lengths aligned with circuit lower bounds, by developing a new pseudorandom multi-switching lemma.

Contribution

The paper presents a novel pseudorandom multi-switching lemma that derandomizes multi-switching lemmas, leading to optimal seed-length PRGs for  C^0 circuits and sparse _2 polynomials.

Findings

PRG for  C^0 circuits with seed length log(M)^{d+O(1)} nd psilon

PRG for S-sparse _2 polynomials with seed length 2^{O((ig))}nd psilon

Achieved optimality of PRGs given current circuit lower bounds.

Abstract

We give the best known pseudorandom generators for two touchstone classes in unconditional derandomization: an $\varepsilon$-PRG for the class of size-$M$ depth-$d$ $\mathsf{AC}^0$ circuits with seed length $\log(M)^{d+O(1)}\cdot \log(1/\varepsilon)$, and an $\varepsilon$-PRG for the class of $S$-sparse $\mathbb{F}_2$ polynomials with seed length $2^{O(\sqrt{\log S})}\cdot \log(1/\varepsilon)$. These results bring the state of the art for unconditional derandomization of these classes into sharp alignment with the state of the art for computational hardness for all parameter settings: improving on the seed lengths of either PRG would require breakthrough progress on longstanding and notorious circuit lower bounds. The key enabling ingredient in our approach is a new \emph{pseudorandom multi-switching lemma}. We derandomize recently-developed \emph{multi}-switching lemmas, which are powerful generalizations of H{\aa}stad's switching lemma that deal with \emph{families} of depth-two circuits. Our pseudorandom multi-switching lemma---a randomness-efficient algorithm for sampling restrictions that simultaneously simplify all circuits in a family---achieves the parameters obtained by the (full randomness) multi-switching lemmas of Impagliazzo, Matthews, and Paturi [IMP12] and H{\aa}stad [H{\aa}s14]. This optimality of our derandomization translates into the optimality (given current circuit lower bounds) of our PRGs for $\mathsf{AC}^0$ and sparse $\mathbb{F}_2$ polynomials.