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Paper: Pairing-friendly Hyperelliptic Curves with Ordinary Jacobians of Type $y^2=x^5+ax$

Authors:
Mitsuru Kawazoe
Tetsuya Takahashi
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URL: http://eprint.iacr.org/2008/026
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Abstract: An explicit construction of pairing-friendly hyperelliptic curves with ordinary Jacobians was firstly given by D.~Freeman. In this paper, we give other explicit constructions of pairing-friendly hyperelliptic curves with ordinary Jacobians based on the closed formulae for the order of the Jacobian of a hyperelliptic curve of type $y^2=x^5+ax$. We present two methods in this paper. One is an analogue of the Cocks-Pinch method and the other is a cyclotomic method. By using these methods, we construct a pairing-friendly hyperelliptic curve $y^2=x^5+ax$ over a finite prime field ${¥mathbb F}_p$ whose Jacobian is ordinary and simple over ${¥mathbb F}_p$ with a prescribed embedding degree. Moreover, the analogue of the Cocks-Pinch produces curves with $¥rho¥approx 4$ and the cyclotomic method produces curves with $3¥le ¥rho¥le 4$.
BibTeX
@misc{eprint-2008-17703,
  title={Pairing-friendly Hyperelliptic Curves with Ordinary Jacobians of Type $y^2=x^5+ax$},
  booktitle={IACR Eprint archive},
  keywords={foundations / number theory, pairing based cryptography},
  url={http://eprint.iacr.org/2008/026},
  note={ kawazoe@las.osakafu-u.ac.jp 14032 received 21 Jan 2008, last revised 2 Jun 2008},
  author={Mitsuru Kawazoe and Tetsuya Takahashi},
  year=2008
}