Abstract
In this paper we study the difference between the 2-adic valuations of the cardinalities \( \# E( \mathbb {F}_{q^k} ) \) and \( \# E( \mathbb {F}_q ) \) of an elliptic curve E over \( \mathbb {F}_q \). We also deduce information about the structure of the 2-Sylow subgroup \( E[ 2^\infty ]( \mathbb {F}_{q^k} ) \) from the exponents of \( E[ 2^\infty ]( \mathbb {F}_q ) \).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blake, I.F., Seroussi, G., Smart, N.P.: Elliptic Curves in Cryptography, London Mathematical Society Lecture Note Series, vol. 265. Cambridge University Press, Cambridge (1999)
Cohen, H., Frey, G., Avanzi, R., Doche, C., Lange, T., Nguyen, K., Vercauteren, F.: Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman & Hall, London (2006)
Fouquet, M.: Anneau d’endomorphismes et cardinalité des courbes elliptiques : aspects algorithmiques. Ph.D. Thesis, École Polytechnique, Palaiseau (2001)
Freeman, D., Lauter, K.: Computing endomorphism rings of jacobians of genus 2 curves over finite fields. In: Proceedings of the First SAGA Conference, pp. 29-66. World Sci. Publ., Hackensack, NJ (2008)
Husemöller, D.: Elliptic Curves. Graduate Texts in Mathematics, 2nd edn. Springer, Berlin (2004)
Ionica, S., Joux, A.: Pairing the volcano. Math. Comput. 82(281), 581–603 (2013)
Miret, J.M., Moreno, R., Rio, A., Valls, M.: Determining the \( 2 \)-Sylow subgroup of an elliptic curve over a finite field. Math. Comput. 74(249), 411–427 (2005)
Miret, J.M., Moreno, R., Sadornil, D., Tena, J., Valls, M.: An algorithm to compute volcanoes of \( 2 \)-isogenies of elliptic curves over finite fields. Appl. Math. Comput. 176, 739–750 (2006)
Miret, J.M., Pujolàs, J., Valera, J.: On the \( \ell \)-adic valuation of the cardinality of elliptic curves over finite extensions of \(\mathbb{F} _q \). Arch. Math. 105, 261–269 (2015)
Rück, H.G.: A note on elliptic curves over finite fields. Math. Comput. 49(179), 301–304 (1987)
Schoof, R.: Nonsingular plane cubic curves over finite fields. J. Comb. Theory Ser. A 46, 183–211 (1987)
Voloch, J.F.: A note on elliptic curves over finite fields. Bull. Soc. Math. Fr. 116(4), 455–458 (1988)
Washington, L.C.: Introduction to Cyclotomic Fields. Graduate Texts in Mathematics, 2nd edn. Springer, Berlin (1997)
Waterhouse, W.C.: Abelian varieties over finite fields. Ann. Sci. Ècole Norm. Sup. 2, 521–560 (1969)
Acknowledgements
The authors would like to thank the anonymous referee, whose comments highly improved the quality of this publication. The research of the authors was supported in part by the Grants MTM2013-46949-P and MTM2017-83271-R (Spanish Ministerio de Economía y Competitividad) and 2017SGR-1158 (Generalitat de Catalunya).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Miret, J.M., Pujolàs, J. & Valera, J. On the 2-adic valuation of the cardinality of elliptic curves over finite extensions of \(\mathbb {F}_{\varvec{q}} \). Arch. Math. 111, 611–620 (2018). https://doi.org/10.1007/s00013-018-1245-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-018-1245-2