Fast Jacobian Group Arithmetic on C ab Curves

Abstract

AbstractThe goal of this paper is to describe a practical and efficient algorithm for computing in the Jacobian of a large class of algebraic curves over a finite field. For elliptic and hyperelliptic curves, there exists an algorithm for performing Jacobian group arithmetic in O(g 2) operations in the base field, 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 field 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 efficient than the previous algorithm as a parameter a in C ab curves grows large.KeywordsDiscrete logarithm problemalgebraic curve cryptographyJacobian groupideal class groupsuperelliptic curves C ab curves