Arithmetic on superelliptic curves

Abstract

In this paper we present an efficient, polynomial-time method to perform calculations in the divisor class group of a curve which has a single point on its normalization above infinity. In particular, we provide a unique representation of divisor classes and an algorithm for reducing a divisor on such a curve to its corresponding representative. Such curves include the case of elliptic, odd-degree hyperelliptic and superelliptic curves. In the case when the curve is defined over a finite field, the divisor class group is a finite group which can be used for implementing discrete logarithm based public key cryptosystems. This paper therefore provides a new class of groups for cryptography. On the other hand, we present a method to solve the discrete logarithm problem in these groups. This method is sub-exponential when the degree of the defining equation of the curve is large.