Cryptology ePrint Archive: Report 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 $3¥le ¥rho¥le 4$.

Category / Keywords: foundations / number theory, pairing based cryptography

Date: received 21 Jan 2008, last revised 2 Jun 2008

Contact author: kawazoe at las osakafu-u ac jp

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

Version: 20080602:185014 (All versions of this report)

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