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

A Point Counting Algorithm for Cyclic Covers of the Projective Line

We present a Kedlaya-style point counting algorithm for cyclic covers y = f(x) over a finite field Fpn with p not dividing r, and r and deg f not necessarily coprime. This algorithm generalizes the Gaudry–Gürel algorithm for superelliptic curves to a more general class of curves, and has essentially the same complexity. Our practical improvements include a simplified algorithm exploiting the au...

Arithmetic on superelliptic curves

superelliptic curves, jacobians, cryptography, discrete logarithm problem 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 corre...

An Extension of Kedlaya's Algorithm to Artin-Schreier Curves in Characteristic 2

In this paper we present an extension of Kedlaya’s algorithm for computing the zeta function of an Artin-Schreier curve over a finite field Fq of characteristic 2. The algorithm has running time O(g log q) and needs O(g log q) storage space for a genus g curve. Our first implementation in MAGMA shows that one can now generate hyperelliptic curves suitable for cryptography in reasonable time. We...

Counting Points in Medium Characteristic Using Kedlaya's Algorithm

Fast Jacobian Group Arithmetic on CabCurves

The goal of this paper is to describe a practical and eecient algorithm for computing in the Jacobian of a large class of algebraic curves over a nite eld. For elliptic and hyperelliptic curves, there exists an algorithm for performing Jaco-bian group arithmetic in O(g 2) operations in the base eld, where g is the genus of a curve. The main problem in this paper is whether there exists a method...