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Title:
Pairing-friendly Hyperelliptic Curves with Ordinary Jacobians of Type y^2=x^5+ax
Authors: Mitsuru Kawazoe, 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.
ePrint: https://eprint.iacr.org/2008/026
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