Abstract. We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field Fq. 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.
Bisson and Sutherland REF have developed an algorithm to compute the endomorphism ring of an elliptic curve in subexponential time, using relation-finding techniques which largely overlap with ours.
Computing the endomorphism ring of an ordinary elliptic curve over a finite field
Journal of Number Theory 113 (2011), 815-831
Computer Science Mathematics