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Computing the $ \ell$-power torsion of an elliptic curve over a finite field

Authors: J. Miret, R. Moreno, A. Rio and M. Valls
Journal: Math. Comp. 78 (2009), 1767-1786
MSC (2000): Primary 11G20
Published electronically: October 29, 2008
MathSciNet review: 2501074
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Abstract: The algorithm we develop outputs the order and the structure, including generators, of the $ \ell$-Sylow subgroup of the group of rational points of an elliptic curve defined over a finite field. To do this, we do not assume any knowledge of the group order. We are able to choose points in such a way that a linear number of successive $ \ell$-divisions leads to generators of the subgroup under consideration. After the computation of a couple of polynomials, each division step relies on finding rational roots of polynomials of degree $ \ell$. We specify in complete detail the case $ \ell=3$, when the complexity of each trisection is given by the computation of cubic roots in finite fields.

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Additional Information

J. Miret
Affiliation: Department de Matemàtica, Universitat de Lleida, Jaume II 69, 25001-Lleida, Spain

R. Moreno
Affiliation: Department de Matemàtica, Universitat de Lleida, Jaume II 69, 25001-Lleida, Spain

A. Rio
Affiliation: Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Jordi Girona 1-3. 08034-Barcelona, Spain

M. Valls
Affiliation: Department de Matemàtica, Universitat de Lleida, Jaume II 69, 25001-Lleida, Spain

Received by editor(s): March 30, 2005
Received by editor(s) in revised form: May 28, 2008
Published electronically: October 29, 2008
Additional Notes: The first, second and fourth authors were supported in part by grant MTM2007-66842-C02-02.
The third author was supported in part by grants MTM2006-15038-C02-01 and 2005SGR 00443.
Article copyright: © Copyright 2008 American Mathematical Society

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