## Determining the $2$-Sylow subgroup of an elliptic curve over a finite field

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## Abstract:

In this paper we describe an algorithm that outputs the order and the structure, including generators, of the $2$-Sylow subgroup of an elliptic curve over a finite field. To do this, we do not assume any knowledge of the group order. The results that lead to the design of this algorithm are of inductive type. Then a right choice of points allows us to reach the end within a linear number of successive halvings. The algorithm works with abscissas, so that halving of rational points in the elliptic curve becomes computing of square roots in the finite field. Efficient methods for this computation determine the efficiency of our algorithm.## References

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## Additional Information

**J. Miret**- Affiliation: Department de Matemàtica, Universitat de Lleida, Jaume II 69, 25001-Lleida, Spain
- Email: miret@eup.udl.es
**R. Moreno**- Affiliation: Department de Matemàtica, Universitat de Lleida, Jaume II 69, 25001-Lleida, Spain
- Email: ramiro@eup.udl.es
**A. Rio**- Affiliation: Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Pau Gargallo 5, 08028-Barcelona, Spain
- MR Author ID: 364632
- Email: ana.rio@upc.es
**M. Valls**- Affiliation: Department de Matemàtica, Universitat de Lleida, Jaume II 69, 25001-Lleida, Spain
- Email: magda@eup.udl.es
- Received by editor(s): March 5, 2003
- Received by editor(s) in revised form: May 3, 2003
- Published electronically: March 4, 2004
- Additional Notes: The first, second and fourth authors were supported in part by grant BFM2000-1113-C02-02.

The third author was supported in part by grant BFM2000-0794-C02-02. - © Copyright 2004 American Mathematical Society
- Journal: Math. Comp.
**74**(2005), 411-427 - MSC (2000): Primary 11G20
- DOI: https://doi.org/10.1090/S0025-5718-04-01640-0
- MathSciNet review: 2085900