Improved pseudorandom generators from pseudorandom multi-switching lemmas

Abstract

We give the best known pseudorandom generators for two touchstone classes in unconditional derandomization: an $\epsilon$-PRG for the class of size-$M$ depth-$d$ ${\mathsf{AC}}^0$ circuits with seed length $\log (M{)}^{d+O(1)}\cdot \log (1/\epsilon )$, and an $\epsilon$-PRG for the class of $S$-sparse ${\mathbb{F}}_2$ polynomials with seed length $2^{O(\sqrt{\log S})}\cdot \log (1/\epsilon )$. 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 Håstad'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 Håstad [Hås14]. 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.