### Abstract

We study the task of randomness extraction from sources which are distributed uniformly on an unknown algebraic variety. In other words, we are interested in constructing a function (an extractor) whose output is close to uniform even if the input is drawn uniformly from the set of solutions of an unknown system of low degree polynomials. This problem generalizes the problem of extraction from affine sources which has drawn a considerable amount of interest lately. We present two constructions of explicit extractors for varieties. The first works for varieties of any size (including one dimensional varieties, or curves) and requires field size which is exponential in the overall dimension of the space. Our second extractor allows the field size to be polynomial in the degree of the equations defining the variety, but works only for varieties whose size is at least the square root of the total size of the space.

Original language | English (US) |
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Title of host publication | Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity, CCC 2009 |

Pages | 102-113 |

Number of pages | 12 |

DOIs | |

State | Published - Nov 9 2009 |

Externally published | Yes |

Event | 2009 24th Annual IEEE Conference on Computational Complexity, CCC 2009 - Paris, France Duration: Jul 15 2009 → Jul 18 2009 |

### Publication series

Name | Proceedings of the Annual IEEE Conference on Computational Complexity |
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ISSN (Print) | 1093-0159 |

### Other

Other | 2009 24th Annual IEEE Conference on Computational Complexity, CCC 2009 |
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Country | France |

City | Paris |

Period | 7/15/09 → 7/18/09 |

### All Science Journal Classification (ASJC) codes

- Software
- Theoretical Computer Science
- Computational Mathematics

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## Cite this

*Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity, CCC 2009*(pp. 102-113). [5231245] (Proceedings of the Annual IEEE Conference on Computational Complexity). https://doi.org/10.1109/CCC.2009.7