Pseudorandomness for Regular Branching Programs via Fourier Analysis

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.