A computational approach to Drinfeld modules

TL;DR

This paper surveys the computational aspects of Drinfeld modules over finite fields, including their construction, properties, and applications, with practical SageMath implementations for researchers in number theory and cryptography.

Contribution

It provides a comprehensive algorithmic framework for Drinfeld modules, highlighting their arithmetic properties, computational techniques, and applications in cryptography and coding theory.

Findings

Explicit SageMath implementations for Drinfeld modules

Applications to polynomial factorization and cryptography

Insights into the arithmetic and structural properties of Drinfeld modules

Abstract

This survey provides a practical and algorithmic perspective on Drinfeld modules over $\mathbb F_q[T]$. Starting with the construction of the Carlitz module, we present Drinfeld modules in any rank and some of their arithmetic properties. We emphasise the analogies with elliptic curves, and in the meantime, we also highlight key differences such as their rank structure and their associated Anderson motives. This document is designed for researchers in number theory, arithmetic geometry, algorithmic number theory, cryptography, or computer algebra, offering tools and insights to navigate the computational aspects of Drinfeld modules effectively. We include detailed SageMath implementations to illustrate explicit computations and facilitate experimentation. Applications to polynomial factorisation, isogeny computations, cryptographic constructions, and coding theory are also presented.