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

Mitsuru Kawazoe; Tetsuya Takahashi

Paper 2008/026

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

Mitsuru Kawazoe and Tetsuya Takahashi

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 $3¥le ¥rho¥le 4$.

Note: Proofs of Theorem 2 and Theorem 3 have been added. Numerical results have been updated.

Metadata

Available format(s): PDF
Category: Foundations
Publication info: Published elsewhere. Unknown where it was published
Keywords: number theory pairing based cryptography
Contact author(s): kawazoe @ las osakafu-u ac jp
History: 2008-06-02: last of 3 revisions; 2008-01-22: received; See all versions
Short URL: https://ia.cr/2008/026
License: CC BY

BibTeX

@misc{cryptoeprint:2008/026,
      author = {Mitsuru Kawazoe and Tetsuya Takahashi},
      title = {Pairing-friendly Hyperelliptic Curves with Ordinary Jacobians of Type $y^2=x^5+ax$},
      howpublished = {Cryptology {ePrint} Archive, Paper 2008/026},
      year = {2008},
      url = {https://eprint.iacr.org/2008/026}
}