Pairing - friendly Hyperelliptic Curves 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. Our methods are based on the closed formulae for the order of the Jacobian of a hyperelliptic curve of type y = x + ax over a finite prime field Fp which are given by E. Furukawa, M. Haneda, M. Kawazoe and T. Takahashi. We present two methods in this paper. One is an analogue of the Cocks-Pinch method and the other is a cyclotomic method. Our methods construct a pairing-friendly hyperelliptic curve y = x + ax over Fp whose Jacobian has a prescribed embedding degree with respect to some prime number `. Curves constructed by the analogue of the Cocks-Pinch method satisfy p ≈ `, whereas p ≈ ` in Freeman’s construction. Moreover, for the case of embedding degree 24, we can construct a cyclotomic family with p ≈ `.