Arithmetic on superelliptic curves

This paper is concerned with algorithms for computing in the divisor class group of a nonsingular plane curve of the form y(n) = c(x) which has only o ne point at infinity. Divisors are represented as ideals, and an ideal redu ction algorithm based on lattice reduction is given. We obtain a unique rep resentative for each divisor class and the algorithms for addition and redu ction of divisors run in polynomial time. An algorithm is also given for so lving the discrete logarithm problem when the curve is defined over a finit e field.