An Extension of Kedlaya's Point-Counting Algorithm to Superelliptic Curves

We present an algorithm for counting points on superelliptic curves yr = f(x) over a finite field Fq 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 proving that cryptographic sizes are now reachable.

• Type: Conference
• Year: 2001
• Where: ASIACRYPT
• Authors: Pierrick Gaudry, Nicolas Gürel
• Keywords: ASIACRYPT 2001, Cryptology, Hyperelliptic Curves, Superelliptic Curves Yr, finite field Fq