Orders occurring as endomorphism ring of a Drinfeld module in some isogeny classes of Drinfeld modules of higher ranks

TL;DR

This paper investigates which orders in the endomorphism algebra of Drinfeld modules occur as endomorphism rings within isogeny classes, providing criteria and explicit computations for rank 3 modules to enhance understanding of their structure and cryptographic applications.

Contribution

It offers a necessary and sufficient condition for an order to be the endomorphism ring of a Drinfeld module with a field endomorphism algebra, specifically applied to rank 3 modules.

Findings

Provided criteria for endomorphism rings in isogeny classes

Explicitly computed orders for rank 3 Drinfeld modules

Enhanced understanding of isogeny graphs and cryptography applications

Abstract

The question we propose to answer throughout this paper is the following: Given an isogeny class of Drinfeld modules over a finite field, what are the orders of the corresponding endomorphism algebra (which is an isogeny invariant) that occur as endomorphism ring of a Drinfeld module in that isogeny class? It is worth mentioning that this question is different from the ones investigated by the authors Kuhn, Pink in [6] and Garai, Papikian in [3]. The former authors rather provided an answer to the question, given a Drinfeld module {\phi}, how does one efficiently compute the endomorphism ring of {\phi}? The importance of our question resides in the fact that it might be very helpful to better understand isogeny graphs of Drinfeld modules of higher rank (r > 2) and may be reopen the debate concerning the application to isogeny-based cryptography. We answer that question for the case whereby the endomorphism algebra is a field by providing a necessary and sufficient condition for a given order to be the endomorphism ring of a Drinfeld module. We apply our result to rank r = 3 Drinfeld modules and explicitly compute those orders occurring as endomorphism rings of rank 3 Drinfeld modules over a finite field.