Reducing The Seed Length In The Nisan-Wigderson Generator*

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. * This paper is based on two conference papers 11,12] by the same authors. † Research Supported by NSF Award CCR-9734911, NSF Award CCR-0098197, Sloan Research Fellowship BR-3311, grant #93025 of the joint US-Czechoslovak Science and Technology Program, and USA-Israel BSF Grant 97-00188. ‡ Part of this work was done while at the Hebrew University and the Institute for advanced study. § This research was supported by grant number 69/96 of the Israel Science Foundation, founded by the Israel Academy for Sciences and Humanities and USA-Israel BSF Grant 97-00188.