### 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.

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

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},
note = {\url{https://eprint.iacr.org/2008/026}},
url = {https://eprint.iacr.org/2008/026}
}

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