# Pseudorandomness for regular branching programs via Fourier analysis

### Citation:

Reingold, Omer, Thomas Steinke, and Salil Vadhan. “Pseudorandomness for regular branching programs via Fourier analysis.” In Sofya Raskhodnikova and José Rolim, editors, Proceedings of the 17th International Workshop on Randomization and Computation (RANDOM ‘13), Lecture Notes in Computer Science, 8096:655-670. Springer-Verlag, 2013.
 RANDOM2013.pdf 272 KB ArXiv2013.pdf 0 bytes

### Abstract:

Version HistoryFull 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.

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Last updated on 06/30/2020