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Counting Points on Hyperelliptic Curves over Finite Fields

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Algorithmic Number Theory (ANTS 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1838))

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Abstract

We describe some algorithms for computing the cardinality of hyperelliptic curves and their Jacobians over finite fields. They include several methods for obtaining the result modulo small primes and prime powers, in particular an algorithm à la Schoof for genus 2 using Cantor’s division polynomials. These are combined with a birthday paradox algorithm to calculate the cardinality. Our methods are practical and we give actual results computed using our current implementation. The Jacobian groups we handle are larger than those previously reported in the literature.

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Author information

Authors and Affiliations

  1. LIX, École Polytechnique, 91128, Palaiseau Cedex, France

    Pierrick Gaudry

  2. Projet Cristal, INRIA, Domaine de Voluceau – Rocquencourt, 78153, Le Chesnay, France

    Robert Harley

Authors
  1. Pierrick Gaudry
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  2. Robert Harley
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Editor information

Editors and Affiliations

  1. Mathematisch Instituut, Universiteit Nijmegen, Postbus 9010, 6500 GL, Nijmegen, The Netherlands

    Wieb Bosma

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© 2000 Springer-Verlag Berlin Heidelberg

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Gaudry, P., Harley, R. (2000). Counting Points on Hyperelliptic Curves over Finite Fields. In: Bosma, W. (eds) Algorithmic Number Theory. ANTS 2000. Lecture Notes in Computer Science, vol 1838. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10722028_18

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  • DOI: https://doi.org/10.1007/10722028_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67695-9

  • Online ISBN: 978-3-540-44994-2

  • eBook Packages: Springer Book Archive

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