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Reducing The Seed Length In The Nisan-Wigderson Generator*

Content Provider	Springer Nature Link
Author	Impagliazzo, Russell Wigderson§, Avi Shaltiel‡, Ronen
Copyright Year	2006
Abstract	The Nisan–Wigderson pseudo-random generator [19] was constructed to derandomize probabilistic algorithms under the assumption that there exist explicit functions which are hard for small circuits. We give the first explicit construction of a pseudo-random generator with asymptotically optimal seed length even when given a function which is hard for relatively small circuits. Generators with optimal seed length were previously known only assuming hardness for exponential size circuits [13,26].We also give the first explicit construction of an extractor which uses asymptotically optimal seed length for random sources of arbitrary min-entropy. Our construction is the first to use the optimal seed length for sub-polynomial entropy levels. It builds on the fundamental connection between extractors and pseudo-random generators discovered by Trevisan [29], combined with the construction above.The key is a new analysis of the NW-generator [19]. We show that it fails to be pseudorandom only if a much harder function can be efficiently constructed from the given hard function. By repeatedly using this idea we get a new recursive generator, which may be viewed as a reduction from the general case of arbitrary hardness to the solved case of exponential hardness.
Ending Page	681
Page Count	35
Starting Page	647
File Format	PDF
ISSN	02099683
e-ISSN	14396912
Journal	Combinatorica
Issue Number	6
Volume Number	26
Language	English
Publisher	Springer-Verlag
Publisher Date	2006-01-01
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Complexity classes (hierarchies, relations among complexity classes, etc.) Mathematics Combinatorics
Content Type	Text
Resource Type	Article
Subject	Discrete Mathematics and Combinatorics Computational Mathematics

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