Abstract
In this paper we study structures related to torsion of elliptic curves defined over number fields. The aim is to build families of elliptic curves more efficient to help factoring numbers of special form, including numbers from the Cunningham Project. We exhibit a family of curves with rational ℤ/4ℤ×ℤ/4ℤ torsion and positive rank over the field ℚ(ζ 8) and a family of elliptic curves with rational ℤ/6ℤ×ℤ/3ℤ torsion and positive rank over the field ℚ(ζ 3). These families have been used in finding new prime factors for the numbers 2972 + 1 and 21048 + 1. Along the way, we classify and give a parameterization of modular curves for some torsion subgroups.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Cohen, H.: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics, vol. 138. Springer, Heidelberg (1991)
Knapp, A.W.: Elliptic Curves. Princeton University Press, Princeton (1992)
Koblitz, N.: Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics, vol. 97. Springer, Heidelberg (1993)
Kubert, D.S.: Universal bounds on the torsion of elliptic curves. In: Proceedings of the London Mathematical Society, pp. 193–237 (1976)
Lenstra, A.K., Lenstra, H.W.: The Development of the Number Field Sieve. LNM, vol. 1554. Springer, Heidelberg (1993)
Lenstra, A.K., Lenstra, H.W., Manasse, M.S., Pollard, J.M.: The Factorization of the Ninth Fermat Number. In: Mathematics of Computation, vol. 61. American Mathematical Society, Providence (1993)
Lenstra, H.W.: Factoring integers with elliptic curves. Annals of Mathematics 126, 649–673 (1987)
Mazur, B.: Rational isogenies of prime degree. Invent. Math., 129–162 (1978)
Montgomery, P.L.: Speeding the pollard and elliptic curve methods of factorization. Mathematics of Computation 48, 243–264 (1987)
Zimmermann, P., Dodson, B.: Twenty Years of ECM. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 525–542. Springer, Heidelberg (2006)
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
Brier, É., Clavier, C. (2010). New Families of ECM Curves for Cunningham Numbers. 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_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-14518-6_11
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)