Abstract
Isogeny volcanoes are graphs whose vertices are elliptic curves and whose edges are ℓ-isogenies. Algorithms allowing to travel on these graphs were developed by Kohel in his thesis (1996) and later on, by Fouquet and Morain (2001). However, up to now, no method was known, to predict, before taking a step on the volcano, the direction of this step. Hence, in Kohel’s and Fouquet-Morain algorithms, we take many steps before choosing the right direction. In particular, ascending or horizontal isogenies are usually found using a trial-and-error approach. In this paper, we propose an alternative method that efficiently finds all points P of order ℓ such that the subgroup generated by P is the kernel of an horizontal or an ascending isogeny. In many cases, our method is faster than previous methods.
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
Belding, J., Broker, R., Enge, A., Lauter, K.: Computing Hilbert Class Polynomials. In: van der Poorten, A.J., Stein, A. (eds.) ANTS-VIII 2008. LNCS, vol. 5011, pp. 282–295. Springer, Heidelberg (2008)
Bisson, G., Sutherland, A.: Computing the endomorphism ring of an ordinary elliptic curve over a finite field. Journal of Number Theory (to appear 2010)
Blake, I.F., Seroussi, G., Smart, N.P.: Advances in Elliptic Curve Cryptography. London Mathematical Society Lecture Note Series, vol. 317. Cambridge University Press, Cambridge (2005)
Broker, R., Lauter, K., Sutherland, A.: Computing modular polynomials with the chinese remainder theorem (2009), http://arxiv.org/abs/1001.0402
Cox, D.A.: Primes of the Form x 2 + ny 2: Fermat, class field theory, and complex multiplication. John Wiley & Sons, Inc., Chichester (1989)
Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenkorper. Abh. Math. Sem. Hansischen Univ., vol. 14 (1941)
Fouquet, M.: Anneau d’endomorphismes et cardinalité des courbes elliptiques: aspects algorithmiques. PhD thesis, Ecole Polytechnique (2001)
Fouquet, M., Morain, F.: Isogeny Volcanoes and the SEA Algorithm. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 276–291. Springer, Heidelberg (2002)
Frey, G.: Applications of arithmetical geometry to cryptographic constructions. In: Proceedings of the Fifth International Conference on Finite Fields and Applications, pp. 128–161. Springer, Heidelberg (2001)
Grabher, P., Großschädl, J., Page, D.: On software parallel implementation of cryptographic pairings. In: Avanzi, R.M., Keliher, L., Sica, F. (eds.) SAC 2008. LNCS, vol. 5381, pp. 35–50. Springer, Heidelberg (2009)
Ionica, S.: Algorithmique des couplages et cryptographie. PhD thesis, Université de Versailles St-Quentin-en-Yvelines (2010)
Joux, A., Nguyen, K.: Separating decision Diffie–Hellman from computational Diffie–Hellman in cryptographic groups. Journal of Cryptology 16(4), 239–247 (2003)
Lenstra Jr., H.W.: Complex multiplication structure of elliptic curves. Journal of Number Theory 56(2), 227–241 (1996)
Kohel, D.: Endomorphism rings of elliptic curves over finite fields. PhD thesis, University of California, Berkeley (1996)
Miller, V.S.: The Weil pairing, and its efficient calculation. Journal of Cryptology 17(4), 235–261 (2004)
Miret, J., Moreno, R., Sadornil, D., Tena, J., Valls, M.: An algorithm to compute volcanoes of 2-isogenies of elliptic curves over finite fields. Applied Mathematics and Computation 176(2), 739–750 (2006)
Miret, J., Moreno, R., Sadornil, D., Tena, J., Valls, M.: Computing the height of volcanoes of l-isogenies of elliptic curves over finite fields. Applied Mathematics and Computation 196(1), 67–76 (2008)
Montgomery, P.L.: A FFT extension of the elliptic curve method of factorization. PhD thesis, University of California (1992)
Ruck, H.-G.: A note on elliptic curves over finite fields. Mathematics of Computation 179, 301–304 (1987)
Schoof, R.: Counting points on elliptic curves over finite fields. Journal de Theorie des Nombres de Bordeaux 7, 219–254 (1995)
Silverman, J.H.: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 106. Springer, Heidelberg (1986)
Silverman, J.H.: Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 151. Springer, Heidelberg (1994)
Sutherland, A.: Computing Hilbert Class Polynomials with the Chinese Remainder Theorem. Mathematics of Computation (2010)
Vélu, J.: Isogenies entre courbes elliptiques. Comptes Rendus De L’Academie Des Sciences Paris, Serie I-Mathematique, Serie A. 273, 238–241 (1971)
von zur Gathen, J., Shoup, V.: Computing Frobenius maps and factoring polynomials. Computational Complexity 2, 187–224 (1992)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ionica, S., Joux, A. (2010). Pairing the Volcano. In: Hanrot, G., Morain, F., Thomé, E. (eds) Algorithmic Number Theory. ANTS 2010. Lecture Notes in Computer Science, vol 6197. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14518-6_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-14518-6_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14517-9
Online ISBN: 978-3-642-14518-6
eBook Packages: Computer ScienceComputer Science (R0)