Construction of Pairing-Friendly Hyperelliptic Curves Based on the Closed Formulae of the Order of the Jacobian Group

An explicit construction of pairing-friendly hyperelliptic curves with ordinary Jacobians was firstly given by D. Freeman for the genus two case. In this paper, we give an explicit construction of pairing-friendly hyperelliptic curves of genus two and four with ordinary Jacobians based on the closed formulae for the order of the Jacobian of special hyperelliptic curves. For the case of genus two, we prove the closed formula for curves of type y2 = x5 + c. By using the formula, we develop an analogue of the Cocks-Pinch method for curves of type y2 = x5 + c. For the case of genus four, we also develop an analogue of the Cocks-Pinch method for curves of type y2 = x9 + cx. In particular, we construct the first examples of pairing-friendly hyperelliptic curves of genus four with ordinary Jacobians.