(Short Paper) Parameter Trade-Offs for NFS and ECM

  • Kazumaro AokiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11049)


This paper analyzes two factoring algorithms, NFS (Number Field Sieve) and ECM (Elliptic Curve Method). The previous results only minimize their running times, however, we may need to minimize the storage size or running time with smaller success probability. We provide these trade-offs, L[s] (\(s\le 1/3\)) memory requires \(L[1-2s]\) running time for NFS, for example. This can be interpreted that NFS requires much more running time when reducing memory complexity.


Number Field Sieve NFS Elliptic Curve Method ECM Time complexity Memory complexity Success probability 


  1. 1.
    Shor, P.W.: Algorithms for quantum computation: discrete logarithms and factoring. In: 35th Annual Symposium on Foundations of Computer Science (FOCS 1994), pp. 124–134. IEEE Computer Society (1994)Google Scholar
  2. 2.
    Lenstra, A.K., Lenstra Jr., H.W. (eds.): The Development of the Number Field Sieve. LNM, vol. 1554. Springer, Heidelberg (1993). CrossRefzbMATHGoogle Scholar
  3. 3.
    Coppersmith, D.: Modifications to the number field sieve. J. Cryptology 6(3), 169–180 (1993)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Kleinjung, T.: On polynomial selection for the general number field sieve. Math. Comput. 75(256), 2037–2047 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Aoki, K., Ueda, H.: Sieving using bucket sort. In: Lee, P.J. (ed.) ASIACRYPT 2004. LNCS, vol. 3329, pp. 92–102. Springer, Heidelberg (2004). CrossRefGoogle Scholar
  6. 6.
    Papadopoulos, J.: A self-tuning filtering implementation for the number field sieve, CADO workshop on integer factorization.
  7. 7.
    Aoki, K., Franke, J., Kleinjung, T., Lenstra, A.K., Osvik, D.A.: A kilobit special number field sieve factorization. In: Kurosawa, K. (ed.) ASIACRYPT 2007. LNCS, vol. 4833, pp. 1–12. Springer, Heidelberg (2007). CrossRefGoogle Scholar
  8. 8.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. Applied Mathematics Ser. vol. 55. National Bureau of Standards (1964). Tenth Printing, December 1972, with correctionsGoogle Scholar
  9. 9.
    Crandall, R., Pomerance, C.: Prime Numbers. A Computational Perspective. Springer, New York (2001). CrossRefzbMATHGoogle Scholar
  10. 10.
    Coppersmith, D.: Solving homogeneous linear equations over GF(2) via block Wiedemann algorithm. Math. Comput. 62(205), 333–350 (1994)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Lenstra Jr., H.W.: Factoring integers with elliptic curves. Ann. Math. 126(3), 649–673 (1987)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.NTT Secure Platform LaboratoriesNippon Telegraph and Telephone CorporationTokyoJapan

Personalised recommendations