Counting Points in Medium Characteristic Using Kedlaya's Algorithm

Abstract

Recently, many new results have been found concerning algorithms for counting points on curves over finite fields of characteristic p, mostly due to the use of p-adic liftings. The complexity of these new methods is exponential in {\small $\log p$}, therefore they behave well when p is small, the ideal case being p=2. When applicable, these new methods are usually faster than those based on SEA algorithms, and are more easily extended to nonelliptic curves. We investigate more precisely this dependence on the characteristic, and in particular, we show that after a few modifications using fast algorithms for radix-conversion, Kedlaya's algorithm works in time almost linear in p. As a consequence, this algorithm can also be applied to medium values of p. We give an example of a cryptographic size genus 3 hyperelliptic curve over a finite field of characteristic 251.