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.
@InProceedings{steinke_et_al:LIPIcs.APPROX-RANDOM.2014.885, author = {Steinke, Thomas and Vadhan, Salil and Wan, Andrew}, title = {{Pseudorandomness and Fourier Growth Bounds for Width-3 Branching Programs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {885--899}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.885}, URN = {urn:nbn:de:0030-drops-47456}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.885}, annote = {Keywords: Pseudorandomness, Branching Programs, Discrete Fourier Analysis} }
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