Pseudorandom generators for regular branching programs

We give new pseudorandom generators for regular read-once branching programs of small width. A branching program is regular if the in-degree of every vertex in it is either 0 or 2. For every width d and length n, our pseudorandom generator uses a seed of length O((log d + log log n + log(1/ε)) log n) to produce n bits that cannot be distinguished from a uniformly random string by any regular width d length n read-once branching program, except with probability ε. We also give a result for general read-once branching programs, in the case that there are no vertices that are reached with small probability. We show that if a (possibly non-regular) branching program of length n and width d has the property that every vertex in the program is traversed with probability at least γ on a uniformly random input, then the error of the generator above is at most 2ε/γ ².