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

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

2009-02-26

arxiv.org/abs/0902.4670v2

Journal of Number Theory 113 (2011), 815-831

Mathematics

Number Theory

16 pages (minor edits)

Scientific paper

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.

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