Pseudorandomness and Fourier Growth Bounds for Width-3 Branching Programs

We present an explicit pseudorandom generator for oblivious, read-once, width-3 branching programs, which can read their input bits in any order. The generator has seed length Õ(log3 n). The previously best known seed length for this model is n1/2+o(1) due to Impagliazzo, Meka, and Zuckerman (FOCS ’12). Our work generalizes a recent result of Reingold, Steinke, and Vadhan (RANDOM ’13) for permutation branching programs. The main technical novelty underlying our generator is a new bound on the Fourier growth of width-3, oblivious, read-once branching programs. Specifically, we show that for any f : {0, 1} → {0, 1} computed by such a branching program, and k ∈ [n], ∑ s⊆[n]:|s|=k ∣∣∣f̂ [s]∣∣∣ ≤ n2 · (O(logn)), where f̂ [s] = E U [ f [U ] · (−1)s·U ] is the standard Fourier transform over Z2 . The base O(logn) of the Fourier growth above is tight up to a factor of log logn. 1998 ACM Subject Classification F.1