Theory of Computing ------------------- Title : Pseudorandomness and Fourier-Growth Bounds for Width-3 Branching Programs Authors : Thomas Steinke, Salil Vadhan, and Andrew Wan Volume : 13 Number : 12 Pages : 1-50 URL : https://theoryofcomputing.org/articles/v013a012 Abstract -------- 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 $O~(\log^3 n).$ The best previously known seed length for this model is $n^{1/2+o(1)}$ due to Impagliazzo, Meka, and Zuckerman (FOCS'12). Our result 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 $f:\{0,1\}^n\rightarrow \{0,1\}$ computed by such a branching program, and $k\in [n],$ $$\sum_{s\subseteq [n]: |s|=k}|\hat{f}[s]|\leq n^2 (O(\log n))^k,$$ where $\hat{f}[s] = E_{U}(f[U] (-1)^{sU})$ is the standard Fourier transform over $Z_2^n$. The base $O(\log n)$ of the Fourier growth is tight up to a factor of $\log\log n$. A conference version of this paper appeared in the Proceedings of the 18th International Workshop on Randomization and Computation (RANDOM’14).