## 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.

Original language | English |
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Pages (from-to) | 647-681 |

Number of pages | 35 |

Journal | Combinatorica |

Volume | 26 |

Issue number | 6 |

DOIs | |

State | Published - 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.

## Keywords

- 68Q15

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics