### Abstract

We propose two simple and efficient deterministic extractors for J(Fq), the Jacobian of a genus 2 hyperelliptic curve H defined over Fq, for some odd q. Our first extractor, SEJ, called sum extractor, for a given point D on J(Fq), outputs the sum of abscissas of rational points on H in the support of D, considering D as a reduced divisor. Similarly the second extractor, PEJ, called product extractor, for a given point D on the J(Fq), outputs the product of abscissas of rational points in the support of D. Provided that the point D is chosen uniformly at random in J(Fq), the element extracted from the point D is indistinguishable from a uniformly random variable in Fq. Thanks to the Kummer surface K, that is associated to the Jacobian of H over Fq, we propose the sum and product extractors, SEK and PEK, for K(Fq). These extractors are the modified versions of the extractors SEJ and PEJ. Provided a point K is chosen uniformly at random in K, the element extracted from the point K is statistically close to a uniformly random variable in Fq.

Original language | English |
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Title of host publication | Proceedings of the 11th IMA International Conference on Cryptography and Coding, 18-20 December 2007, Cirencester, United Kingdom |

Editors | S.D. Galbraith |

Place of Publication | Berlin, Germany |

Publisher | Springer |

Pages | 313-335 |

ISBN (Print) | 978-3-540-77271-2 |

DOIs | |

Publication status | Published - 2007 |

Event | conference; IMA International Conference on Cryptography and Coding 11, Cirencester, United Kingdom; 2007-12-18; 2007-12-20 - Duration: 18 Dec 2007 → 20 Dec 2007 |

### Publication series

Name | Lecture Notes in Computer Science |
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Volume | 4887 |

ISSN (Print) | 0302-9743 |

### Conference

Conference | conference; IMA International Conference on Cryptography and Coding 11, Cirencester, United Kingdom; 2007-12-18; 2007-12-20 |
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Period | 18/12/07 → 20/12/07 |

Other | IMA International Conference on Cryptography and Coding 11, Cirencester, United Kingdom |

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## Cite this

Rezaeian Farashahi, R. (2007). Extractors for Jacobian of hyperelliptic curves of genus 2 in odd characteristic. In S. D. Galbraith (Ed.),

*Proceedings of the 11th IMA International Conference on Cryptography and Coding, 18-20 December 2007, Cirencester, United Kingdom*(pp. 313-335). (Lecture Notes in Computer Science; Vol. 4887). Springer. https://doi.org/10.1007/978-3-540-77272-9_19