Citation:
RANDOM2013.pdf | 272 KB | |
ArXiv2013.pdf | 0 bytes |
Abstract:
Version History: Full version posted as ECCC TR13-086 and arXiv:1306.3004 [cs.CC].
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 \(O(\log^2n)\), 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.