Abstract
Recently, Kohel gave algorithms to compute the conductor of the endomorphism ring of an ordinary elliptic curve, given the cardinality of the curve. Using his work, we give a complete description of the structure of curves related via rational ℓ-degree isogenies, a structure we call a volcano. We explain how we can travel through this structure using modular polynomials. The computation of the structure is possible without knowing the cardinality of the curve, and that as a result, we deduce information on the cardinality.
The second author is on the leave from the French Department of Defense, Délégation Générale pour l’Armement. This research was partially supported by the French Ministry of Research — ACI Cryptologie.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Z. I. Borevitch and I. R. Chafarevitch. Théorie des nombres. Gauthiers-Villars, Paris, 1967.
J. Chao, O. Nakamura, K. Sobataka, and S. Tsujii. Construction of secure elliptic cryptosystems using CM tests and liftings. In K. Ohta and D. Pei, editors, Advances in Cryptology-ASIACRYPT’98, volume 1514 of Lecture Notes in Comput. Sci., pages 95–109. Springer-Verlag, 1998. Beijing, China.
J.-M. Couveignes, L. Dewaghe, and F. Morain. Isogeny cycles and the Schoof-Elkies-Atkin algorithm. Research Report LIX/RR/96/03, LIX, April 1996. Available at http://www.lix.polytechnique.fr/Labo/Francois.Morain/.
J.-M. Couveignes and F. Morain. Schoof’s algorithm and isogeny cycles. In ANTS-I, 1994.
D. H. Cox. Primes of the Form x 2 + ny 2. Wiley-Interscience, 1989.
M. Deuring. Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Sem. Hamburg, 14:197–272, 1941.
M. Fouquet. Anneau d’endomorphismes et cardinalité des courbes elliptiques: aspects algorithmiques. Thèse, École polytechnique, December 2001. Available at http://www.lix.polytechnique.fr/Labo/Mireille.Fouquet/.
M. Fouquet, P. Gaudry, and R. Harley. An extension of Satoh’s algorithm and its implementation. J. Ramanujan Math. Soc., December 2000.
S.D. Galbraith, F. Hess, and N.P. Smart. Extending the GHS weil descent attack. http://eprint.iacr.org/, 2001.
D. Kohel. Endomorphism rings of elliptic curves over finite fields. Phd thesis, University of California, Berkeley, 1996.
R. Lercier. Algorithmique des courbes elliptiques dans les corps finis. Thèse, École polytechnique, June 1997.
T. Satoh. The canonical lift of an ordinary elliptic curve over a finite field and its point counting. J. Ramanujan Math. Soc., 15:247–270, December 2000.
R. Schoof. Counting points on elliptic curves over finite fields. J. Théor. Nombres Bordeaux, 1995.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fouquet, M., Morain, F. (2002). Isogeny Volcanoes and the SEA Algorithm. In: Fieker, C., Kohel, D.R. (eds) Algorithmic Number Theory. ANTS 2002. Lecture Notes in Computer Science, vol 2369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45455-1_23
Download citation
DOI: https://doi.org/10.1007/3-540-45455-1_23
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43863-2
Online ISBN: 978-3-540-45455-7
eBook Packages: Springer Book Archive