License:
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.45
URN: urn:nbn:de:0030-drops-112605
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/11260/
Servedio, Rocco A. ;
Tan, Li-Yang
Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas
Abstract
We give the best known pseudorandom generators for two touchstone classes in unconditional derandomization: small-depth circuits and sparse F_2 polynomials. Our main results are an epsilon-PRG for the class of size-M depth-d AC^0 circuits with seed length log(M)^{d+O(1)}* log(1/epsilon), and an epsilon-PRG for the class of S-sparse F_2 polynomials with seed length 2^{O(sqrt{log S})}* log(1/epsilon). These results bring the state of the art for unconditional derandomization of these classes into sharp alignment with the state of the art for computational hardness for all parameter settings: improving on the seed lengths of either PRG would require breakthrough progress on longstanding and notorious circuit lower bounds.
The key enabling ingredient in our approach is a new pseudorandom multi-switching lemma. We derandomize recently-developed multi-switching lemmas, which are powerful generalizations of Håstad's switching lemma that deal with families of depth-two circuits. Our pseudorandom multi-switching lemma - a randomness-efficient algorithm for sampling restrictions that simultaneously simplify all circuits in a family - achieves the parameters obtained by the (full randomness) multi-switching lemmas of Impagliazzo, Matthews, and Paturi [Impagliazzo et al., 2012] and Håstad [Johan Håstad, 2014]. This optimality of our derandomization translates into the optimality (given current circuit lower bounds) of our PRGs for AC^0 and sparse F_2 polynomials.
BibTeX - Entry
@InProceedings{servedio_et_al:LIPIcs:2019:11260,
author = {Rocco A. Servedio and Li-Yang Tan},
title = {{Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
pages = {45:1--45:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-125-2},
ISSN = {1868-8969},
year = {2019},
volume = {145},
editor = {Dimitris Achlioptas and L{\'a}szl{\'o} A. V{\'e}gh},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/11260},
URN = {urn:nbn:de:0030-drops-112605},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.45},
annote = {Keywords: pseudorandom generators, switching lemmas, circuit complexity, unconditional derandomization}
}
Keywords: |
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pseudorandom generators, switching lemmas, circuit complexity, unconditional derandomization |
Collection: |
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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019) |
Issue Date: |
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2019 |
Date of publication: |
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17.09.2019 |