Fast Jacobian Group Arithmetic on CabCurves

The goal of this paper is to describe a practical and eecient algorithm for computing in the Jacobian of a large class of algebraic curves over a nite eld. For elliptic and hyperelliptic curves, there exists an algorithm for performing Jaco-bian group arithmetic in O(g 2) operations in the base eld, where g is the genus of a curve. The main problem in this paper is whether there exists a method to perform the arithmetic in more general curves. Galbraith, Paulus, and Smart proposed an algorithm to complete the arithmetic in O(g 2) operations in the base eld for the so-called superelliptic curves. We generalize the algorithm to the class of C ab curves, which includes superelliptic curves as a special case. Furthermore, in the case of C ab curves, we show that the proposed algorithm is not just general but more eecient than the previous algorithm as a parameter a in C ab curves grows large.