An extension of kedlaya’s point-counting algorithm to superelliptic curves

Abstract

We present an algorithm for counting points on superelliptic curves yr = f(x) over a finite field (Formula Presented)q of small characteristic different from r. This is an extension of an algorithm for hyperelliptic curves due to Kedlaya. In this extension, the complexity, assuming r and the genus are fixed, is O(log3+ε q) in time and space, just like for hyperelliptic curves. We give some numerical examples obtained with our first implementation, thus provingthat cryptographic sizes are now reachable.