Pairing the Volcano
- 1k Downloads
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.
KeywordsElliptic Curve Elliptic Curf Stability Level Discrete Logarithm Endomorphism Ring
Unable to display preview. Download preview PDF.
- 2.Bisson, G., Sutherland, A.: Computing the endomorphism ring of an ordinary elliptic curve over a finite field. Journal of Number Theory (to appear 2010)Google Scholar
- 4.Broker, R., Lauter, K., Sutherland, A.: Computing modular polynomials with the chinese remainder theorem (2009), http://arxiv.org/abs/1001.0402
- 6.Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenkorper. Abh. Math. Sem. Hansischen Univ., vol. 14 (1941)Google Scholar
- 7.Fouquet, M.: Anneau d’endomorphismes et cardinalité des courbes elliptiques: aspects algorithmiques. PhD thesis, Ecole Polytechnique (2001)Google Scholar
- 9.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)Google Scholar
- 11.Ionica, S.: Algorithmique des couplages et cryptographie. PhD thesis, Université de Versailles St-Quentin-en-Yvelines (2010)Google Scholar
- 14.Kohel, D.: Endomorphism rings of elliptic curves over finite fields. PhD thesis, University of California, Berkeley (1996)Google Scholar
- 18.Montgomery, P.L.: A FFT extension of the elliptic curve method of factorization. PhD thesis, University of California (1992)Google Scholar
- 23.Sutherland, A.: Computing Hilbert Class Polynomials with the Chinese Remainder Theorem. Mathematics of Computation (2010)Google Scholar