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

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 = x + 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 = x + ax over a finite prime field Fp whose Jacobian is ordinary and simple over Fp with a prescribed embedding degree. Moreover, the analogue of the CocksPinch produces curves with ρ ≈ 4 and the cyclotomic method produces curves with 3 ≤ ρ ≤ 4.