Pseudorandomness for Permutation and Regular Branching Programs

In this paper, we prove the following results about the INW pseudorandom generator : (1) It fools constant width permutation branching programs with error ε using a seed of length O(log n · log(1/ε)). (2) It fools constant width regular branching programs with error ε using a seed of length O(log n · (log log n + log(1/ε))). These results match the recent results of Koucky et al. (STOC 2011) and Braverman et al. and Brody and Verbin (FOCS 2010). However, our analysis gives a better dependence of the seed length on the width for permutation branching programs than the results of Koucky et al. (STOC 2011). Perhaps, more significantly, our proof method is entirely different and linear algebraic in nature as opposed to the group theoretic methods of and the information theoretic and probabilistic methods of. Along the way, we also obtain pseudorandom generators for the "small biased spaces" for group products with a seed length O(log n · (log |G| + log(1/ε))). Previously, it was possible to get O(log n · (|G|^O(1) + log(1/ε))) using the pseudorandom generator of.