Pairing-Friendly Hyperelliptic Curves with Ordinary Jacobians of Type y2 = x5 + 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 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 ρ ≈ 4 and the cyclotomic method produces curves with 3 ≤ ρ ≤ 4.