eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2014-09-04
885
899
10.4230/LIPIcs.APPROX-RANDOM.2014.885
article
Pseudorandomness and Fourier Growth Bounds for Width-3 Branching Programs
Steinke, Thomas
Vadhan, Salil
Wan, Andrew
We present an explicit pseudorandom generator for oblivious, read-once, width-3 branching programs, which can read their input bits in any order. The generator has seed length O~( log^3 n ).
The previously best known seed length for this model is n^{1/2+o(1)} due to Impagliazzo, Meka, and Zuckerman (FOCS'12). Our work generalizes a recent result of Reingold, Steinke, and Vadhan (RANDOM'13) for permutation branching programs. The main technical novelty underlying our generator is a new bound on the Fourier growth of width-3, oblivious, read-once branching programs. Specifically, we show that for any f : {0,1}^n -> {0,1} computed by such a branching program, and k in [n], sum_{|s|=k} |hat{f}(s)| < n^2 * (O(\log n))^k,
where f(x) = sum_s hat{f}(s) (-1)^<s,x> is the standard Fourier transform over Z_2^n. The base O(log n) of the Fourier growth is tight up to a factor of log log n.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol028-approx-random2014/LIPIcs.APPROX-RANDOM.2014.885/LIPIcs.APPROX-RANDOM.2014.885.pdf
Pseudorandomness
Branching Programs
Discrete Fourier Analysis