Pseudorandomness for Regular Branching Programs via Fourier Analysis

TL;DR

This paper introduces an explicit pseudorandom generator for certain branching programs with seed length significantly shorter than previous methods, using Fourier analysis to analyze regular branching programs.

Contribution

The paper presents a novel Fourier-analytic approach to constructing pseudorandom generators for oblivious, read-once, regular branching programs with improved seed length.

Findings

Seed length is $O( ext{log}^2 n)$ for constant width models.

Fourier mass at level $k$ is bounded by $(2w^2)^k$, independent of program length.

Generator extends to broader classes with seed length $n^{1/2+o(1)}$.

Abstract

We present an explicit pseudorandom generator for oblivious, read-once, permutation branching programs of constant width that can read their input bits in any order. The seed length is $O(\log^2 n)$, where $n$ is the length of the branching program. The previous best seed length known for this model was $n^{1/2+o(1)}$, which follows as a special case of a generator due to Impagliazzo, Meka, and Zuckerman (FOCS 2012) (which gives a seed length of $s^{1/2+o(1)}$ for arbitrary branching programs of size $s$). Our techniques also give seed length $n^{1/2+o(1)}$ for general oblivious, read-once branching programs of width $2^{n^{o(1)}}$, which is incomparable to the results of Impagliazzo et al.Our pseudorandom generator is similar to the one used by Gopalan et al. (FOCS 2012) for read-once CNFs, but the analysis is quite different; ours is based on Fourier analysis of branching programs. In particular, we show that an oblivious, read-once, regular branching program of width $w$ has Fourier mass at most $(2w^2)^k$ at level $k$, independent of the length of the program.