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

Gaetan Bisson 1, 2 Andrew V. Sutherland 3
1 EIPSI - Eindhoven Institute for the Protection of Systems and Information
2 CARAMEL - Cryptology, Arithmetic: Hardware and Software
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
3 Department of Mathematics [MIT]
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.
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Submitted on : Tuesday, May 12, 2009 - 11:14:03 AM
Last modification on : Tuesday, December 18, 2018 - 4:18:25 PM

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Gaetan Bisson, Andrew V. Sutherland. Computing the endomorphism ring of an ordinary elliptic curve over a finite field. Journal of Number Theory, Elsevier, 2011, Elliptic Curve Cryptography, 131 (5), pp.815--831. ⟨10.1016/j.jnt.2009.11.003⟩. ⟨inria-00383155⟩

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