New Families of ECM Curves for Cunningham Numbers

  • Éric Brier
  • Christophe Clavier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6197)


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.


Elliptic Curve Elliptic Curf Cyclic Subgroup Torsion Group Modular Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cohen, H.: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics, vol. 138. Springer, Heidelberg (1991)Google Scholar
  2. 2.
    Knapp, A.W.: Elliptic Curves. Princeton University Press, Princeton (1992)zbMATHGoogle Scholar
  3. 3.
    Koblitz, N.: Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics, vol. 97. Springer, Heidelberg (1993)zbMATHGoogle Scholar
  4. 4.
    Kubert, D.S.: Universal bounds on the torsion of elliptic curves. In: Proceedings of the London Mathematical Society, pp. 193–237 (1976)Google Scholar
  5. 5.
    Lenstra, A.K., Lenstra, H.W.: The Development of the Number Field Sieve. LNM, vol. 1554. Springer, Heidelberg (1993)zbMATHGoogle Scholar
  6. 6.
    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)Google Scholar
  7. 7.
    Lenstra, H.W.: Factoring integers with elliptic curves. Annals of Mathematics 126, 649–673 (1987)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Mazur, B.: Rational isogenies of prime degree. Invent. Math., 129–162 (1978)Google Scholar
  9. 9.
    Montgomery, P.L.: Speeding the pollard and elliptic curve methods of factorization. Mathematics of Computation 48, 243–264 (1987)zbMATHMathSciNetGoogle Scholar
  10. 10.
    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)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Éric Brier
    • 1
  • Christophe Clavier
    • 2
    • 3
  1. 1.Ingenico S.A.Guilherand-GrangesFrance
  2. 2.Institut d’Ingénierie Informatique de Limoges (3iL)Limoges
  3. 3.Département de Mathématiques et InformatiqueUniversité de Limoges – XLIMLimoges

Personalised recommendations