Computing the endomorphism ring of an ordinary elliptic curve over a finite field

TL;DR

This paper introduces two subexponential algorithms for computing the endomorphism ring of an ordinary elliptic curve over a finite field, providing complexity bounds and a verification method.

Contribution

The paper presents novel algorithms with complexity bounds for determining the endomorphism ring of elliptic curves, including a verification certificate.

Findings

Algorithms operate under heuristic assumptions

Complexity depends on log q and discriminant D_E

Provides a method for endomorphism ring verification

Abstract

We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field F_q. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of log q, while our bound for the second algorithm depends primarily on log |D_E|, where D_E is the discriminant of the order isomorphic to End(E). As a byproduct, our method yields a short certificate that may be used to verify that the endomorphism ring is as claimed.