  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.
