Pseudorandomness for Regular Branching Programs via Fourier Analysis

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 n), where n is the length of the branching program. The previous best seed length known for this model was n, which follows as a special case of a generator due to Impagliazzo, Meka, and Zuckerman (FOCS 2012) (which gives a seed length of s for arbitrary branching programs of size s). Our techniques also give seed length n for general oblivious, read-once branching programs of width 2 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) at level k, independent of the length of the program.