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Counting Points for Hyperelliptic Curves of Type y2=x5+ax over Finite Prime Fields

  • Eisaku Furukawa
  • Mitsuru Kawazoe
  • Tetsuya Takahashi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3006)

Abstract

Counting rational points on Jacobian varieties of hyperelliptic curves over finite fields is very important for constructing hyperelliptic curve cryptosystems (HCC), but known algorithms for general curves over given large prime fields need very long running time. In this article, we propose an extremely fast point counting algorithm for hyperelliptic curves of type y 2=x 5+ax over given large prime fields \(\mathbb{F}_{p}\), e.g. 80-bit fields. For these curves, we also determine the necessary condition to be suitable for HCC, that is, to satisfy that the order of the Jacobian group is of the form l· c where l is a prime number greater than about 2160 and c is a very small integer. We show some examples of suitable curves for HCC obtained by using our algorithm. We also treat curves of type y 2=x 5+a where a is not square in \(\mathbb{F}_{p}\).

Keywords

Prime Number Characteristic Polynomial Elliptic Curf Abelian Variety Hyperelliptic Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Eisaku Furukawa
    • 1
  • Mitsuru Kawazoe
    • 2
  • Tetsuya Takahashi
    • 2
  1. 1.Fujitsu Kansai-Chubu Net-Tech Limited 
  2. 2.Department of Mathematics and Information Sciences College of Integrated Arts and SciencesOsaka Prefecture UniversityOsakaJapan

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