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 $\tilde{O}( \log³ n ).$ The previously best known seed length for this model is $n^{1/2+o(1)}$ due to Impagliazzo, Meka, and Zuckerman (FOCS '12). Our work generalizes a recent result of Reingold, Steinke, and Vadhan (RANDOM '13) for \textit{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 $f:{0,1}^n\rightarrow {0,1}$ computed by such a branching program, and $k\in [n],$ $$\sum{s\subseteq [n]: |s|=k} \left| \hat{f}[s] \right| \leq n² \cdot (O(\log n))^k,$$ where $\widehat{f}[s] = \mathbb{E}\left[f[U] \cdot (-1)^{s \cdot U}\right]$ is the standard Fourier transform over $\mathbb{Z}2^n$. The base $O(\log n)$ of the Fourier growth is tight up to a factor of $\log \log n$.