Better Pseudorandom Generators from Milder Pseudorandom Restrictions

TL;DR

This paper introduces an iterative method for constructing pseudorandom generators using mild restrictions, achieving near-optimal seed-length for various classes, improving over previous constructions with larger seed-lengths.

Contribution

It presents a new iterative approach with milder restrictions for pseudorandom generators, enabling near-optimal seed-lengths for combinatorial rectangles, read-once CNFs, and width-3 branching programs.

Findings

Achieves seed-length O(log(n/epsilon)) for low-error regimes.

Constructs pseudorandom generators for multiple classes with improved seed-length.

Uses small-bias spaces to derandomize mild restrictions.

Abstract

We present an iterative approach to constructing pseudorandom generators, based on the repeated application of mild pseudorandom restrictions. We use this template to construct pseudorandom generators for combinatorial rectangles and read-once CNFs and a hitting set generator for width-3 branching programs, all of which achieve near-optimal seed-length even in the low-error regime: We get seed-length O(log (n/epsilon)) for error epsilon. Previously, only constructions with seed-length O(\log^{3/2} n) or O(\log^2 n) were known for these classes with polynomially small error. The (pseudo)random restrictions we use are milder than those typically used for proving circuit lower bounds in that we only set a constant fraction of the bits at a time. While such restrictions do not simplify the functions drastically, we show that they can be derandomized using small-bias spaces.