Computing a Group Action from the Class Field Theory of Imaginary Hyperelliptic Function Fields

TL;DR

This paper presents an efficient algorithm for computing a group action derived from the class field theory of imaginary hyperelliptic function fields, with practical implementation and polynomial-time inversion methods.

Contribution

It introduces a novel algorithm for the group action on Drinfeld modules, including explicit implementation and complexity analysis, extending the analogy with classical class field theory.

Findings

The group action can be computed efficiently in practice.

Inverting the action reduces to finding isogenies, solvable in polynomial time.

Explicit asymptotic complexity bounds are provided for all algorithms.

Abstract

We explore algorithmic aspects of a simply transitive commutative group action coming from the class field theory of imaginary hyperelliptic function fields. Namely, the Jacobian of an imaginary hyperelliptic curve defined over $\mathbb F_q$ acts on a subset of isomorphism classes of Drinfeld modules. We describe an algorithm to compute the group action efficiently. This is a function field analog of the Couveignes-Rostovtsev-Stolbunov group action. We report on an explicit computation done with our proof-of-concept C++/NTL implementation; it took a fraction of a second on a standard computer. We prove that the problem of inverting the group action reduces to the problem of finding isogenies of fixed $\tau$-degree between Drinfeld $\mathbb F_q[X]$-modules, which is solvable in polynomial time thanks to an algorithm by Wesolowski. We give asymptotic complexity bounds for all algorithms presented in this paper.