Paper 2008/026

Pairing-friendly Hyperelliptic Curves with Ordinary Jacobians of Type y2=x5+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 y2=x5+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 y2=x5+ax over a finite prime field ¥mathbbFp whose Jacobian is ordinary and simple over with a prescribed embedding degree. Moreover, the analogue of the Cocks-Pinch produces curves with and the cyclotomic method produces curves with .

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 theorypairing 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
Creative Commons Attribution
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}
}
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