New Pseudorandom Generators and Correlation Bounds Using Extractors

TL;DR

This paper introduces new pseudorandom generators and correlation bounds for generalized low-degree polynomial models, significantly improving seed length efficiency and extending bounds to larger circuit classes using extractor-based fortification techniques.

Contribution

It presents the first PRGs for width-2 branching programs with near-optimal seed length, extends correlation bounds to larger circuit classes, and generalizes bounds for degree-$F_2$-polynomials using extractor fortification.

Findings

PRG for width-2 branching programs with seed length close to optimal

Correlation bounds for larger AC0 circuit classes with symmetric and threshold gates

Exponential correlation bounds for degree-$F_2$-polynomials set-multilinear over many parts

Abstract

We establish new correlation bounds and pseudorandom generators for a collection of computation models. These models are all natural generalizations of structured low-degree $F_2$-polynomials that we did not have correlation bounds for before. In particular: 1. We construct a PRG for width-2 $poly(n)$-length branching programs which read $d$ bits at a time with seed length $2^{O(\sqrt{\log n})}\cdot d^2\log^2(1/\epsilon)$. This comes quadratically close to optimal dependence in $d$ and $\log(1/\epsilon)$. The previous PRG by Bogdanov, Dvir, Verbin, and Yehudayoff had an exponentially worse dependence on $d$ with seed length of $O(d\log n + d2^d\log(1/\epsilon))$. 2. We provide correlation bounds and PRGs against size-$n^{\Omega(\log n)}$ AC0 circuits with either $n^{.99}$ SYM gates (computing an arbitrary symmetric function) or $n^{.49}$ THR gates (computing an arbitrary linear threshold function). Previous work of Servedio and Tan only handled $n^{.49}$ SYM gates or $n^{.24}$ THR gates, and previous work of Lovett and Srinivasan only handled polysize circuits. 3. We give exponentially small correlation bounds against degree-$n^{O(1)}$ $F_2$-polynomials set-multilinear over some partition of the input into $n^{.99}$ parts (noting that at $n$ parts, we recover all low-degree polynomials). This generalizes correlation bounds against degree-$(d-1)$ polynomials which are set-multilinear over a fixed partition into $d$ blocks, which were established by Bhrushundi, Harsha, Hatami, Kopparty and Kumar. The common technique behind all of these results is to fortify a hard function with the right type of extractor to obtain stronger correlation bounds. Although this technique has been used in previous work, it relies on the model shrinking to a very small class under random restrictions. Our results show such fortification can be done even for classes that do not enjoy such behavior.