Fast Arithmetics in Artin-Schreier Towers over Finite Fields

TL;DR

This paper introduces efficient algorithms for arithmetic in Artin-Schreier towers over finite fields, enabling faster computations relevant to elliptic curve isogenies, with practical implementation demonstrating improved performance.

Contribution

It provides quasi-linear time algorithms for arithmetic in Artin-Schreier towers and applies them to compute elliptic curve isogenies more efficiently.

Findings

Algorithms achieve quasi-linear time complexity.

Implementation confirms practical efficiency gains.

Facilitates faster elliptic curve computations.

Abstract

An Artin-Schreier tower over the finite field F_p is a tower of field extensions generated by polynomials of the form X^p - X - a. Following Cantor and Couveignes, we give algorithms with quasi-linear time complexity for arithmetic operations in such towers. As an application, we present an implementation of Couveignes' algorithm for computing isogenies between elliptic curves using the p-torsion.