Reducing the seed length in the Nisan-Wigderson generator

Russell Impagliazzo, Ronen Shaltiel, Avi Wigderson

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)647-681
Number of pages35
Issue number6
StatePublished - Dec 2006

Bibliographical note

Funding Information:
* Th is paper is based on two conference papers [11,12] by th e same auth ors. † Research Supported by NSF Award CCR-9734911, NSF Award CCR-0098197, Sloan Research Fellowsh ip BR-3311, grant #93025 of th e joint US-Czech oslovak Science and Tech nology Program, and USA-Israel BSF Grant 97-00188. ‡ Part of th is work was done wh ile at th e Hebrew University and th e Institute for advanced study. § Th is research was supported by grant number 69/96 of th e Israel Science Foundation, founded by th e Israel Academy for Sciences and Humanities and USA-Israel BSF Grant 97-00188.


  • 68Q15

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics


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