### Abstract

In this paper we construct explicit deterministic extractors from polynomial sources, namely from distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources. A direct consequence is a deterministic extractor for distributions sampled by polynomial size arithmetic circuits over exponentially large fields. The first step towards extraction is a construction of rank extractors, which are polynomial mappings that "extract" the algebraic rank from any system of low degree polynomials. More precisely, for any n polynomials, k of which are algebraically independent, a rank extractor outputs k algebraically independent polynomials of slightly higher degree. A result of Wooley allows us to relate algebraic rank and min-entropy and to show that a rank extractor is also a high quality condenser for polynomial sources overpolynomially large fields. Finally, to turn this condenser into an extractor, we employ a theorem of Bombieri, giving a character sum estimate for polynomials defined over curves. It allows extracting all the randomness (up to a multiplicative constant) from polynomial sources over exponentially large fields. " 2007 IEEE.

Original language | English (US) |
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Title of host publication | Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007 |

Pages | 52-62 |

Number of pages | 11 |

DOIs | |

State | Published - Dec 1 2007 |

Externally published | Yes |

Event | 48th Annual Symposium on Foundations of Computer Science, FOCS 2007 - Providence, RI, United States Duration: Oct 20 2007 → Oct 23 2007 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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ISSN (Print) | 0272-5428 |

### Other

Other | 48th Annual Symposium on Foundations of Computer Science, FOCS 2007 |
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Country | United States |

City | Providence, RI |

Period | 10/20/07 → 10/23/07 |

### All Science Journal Classification (ASJC) codes

- Engineering(all)

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## Cite this

*Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007*(pp. 52-62). [4389479] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS). https://doi.org/10.1109/FOCS.2007.4389479