### Abstract

We give explicit constructions of extractors which work for a source of any min-entropy on strings of length n. These extractors can extract any constant fraction of the min-entropy using O(log^{2} n) additional random bits, and can extract all the min-entropy using O(log^{3} n) additional random bits. Both of these constructions use fewer truly random bits than any previous construction which works for all min-entropies and extracts a constant fraction of the min-entropy. We then improve our second construction and show that we can reduce the entropy loss to 2 log(1/ε) + O(1) bits, while still using O(log^{3} n) truly random bits (where entropy loss is defined as [(source min - entropy) + (# truly random bits used) - (# output bits)], and ε is the statistical difference from uniform achieved). This entropy loss is optimal up to a constant additive term. Our extractors are obtained by observing that a weaker notion of "combinatorial design" suffices for the Nisan-Wigderson pseudorandom generator, which underlies the recent extractor of Trevisan. We give near-optimal constructions of such "weak designs" which achieve much better parameters than possible with the notion of designs used by Nisan-Wigderson and Trevisan. We also show how to improve our constructions (and Trevisan's construction) when the required statistical difference ε from the uniform distribution is relatively small. This improvement is obtained by using multilinear error-correcting codes over finite fields, rather than the arbitrary error-correcting codes used by Trevisan.

Original language | English (US) |
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Pages (from-to) | 97-128 |

Number of pages | 32 |

Journal | Journal of Computer and System Sciences |

Volume | 65 |

Issue number | 1 |

DOIs | |

State | Published - Aug 2002 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Networks and Communications
- Computational Theory and Mathematics
- Applied Mathematics

### Keywords

- Combinatorial designs
- Expander graphs
- Extractors
- Probabilistic method
- Pseudorandom generators

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## Cite this

*Journal of Computer and System Sciences*,

*65*(1), 97-128. https://doi.org/10.1006/jcss.2002.1824