Paper 2008/026 Original: eprint.iacr.org/2008/026 ↗

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

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 $3¥le ¥rho¥le 4$.