Journal of Cryptology

, Volume 6, Issue 3, pp 169–180 | Cite as

Modifications to the Number Field Sieve

  • Don Coppersmith


The Number Field Sieve, due to Lenstra et al. [LLMP] and Buhler et al. [BLP], is a new routine for factoring integers. We present here a modification of that sieve. We use the fact that certain smoothness computations can be reused, and thereby reduce the asymptotic running time of the Number Field Sieve. We also give a way to precompute tables which will be useful for factoring any integers in a large range.

Key words

Factoring Sieve methods 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A]
    L. M. Adleman, Factoring numbers using singular integers, Proc. 23rd Annual ACM Symposium on the Theory of Computing, 1991, pp. 64–71.Google Scholar
  2. [BLP]
    J. P. Buhler, H. W. Lenstra, Jr., and C. Pomerance, Factoring Integers with the Number Field Sieve, Springer-Verlag, Berlin, Lecture Notes in Mathematics, to appear.Google Scholar
  3. [CEP]
    E. R. Canfield, P. Erdös, and C. Pomerance, On a problem of Oppenheim concerning “Factorisatio Numerorum”, J. Number Theory 17 (1983), 1–28.Google Scholar
  4. [Co]
    D. Coppersmith, Solving Linear Equations over GF(2) II: Block Wiedemann Algorithm, Research Report RC 17293, IBM T. J. Watson Research Center, Yorktown Heights, NY, 17 October 1991. To appear in Math. Comput. Google Scholar
  5. [D]
    N. G. de Bruijn, On the number of positive integers ≤x and free of prime factors >y, II, Nederl. Akad. Wetersch. Indag. Math. 38 (1966), 239–247.Google Scholar
  6. [L]
    H. W. Lenstra, Jr., Factoring integers with elliptic curves, Ann. of Math. 126 (1987), 649–673.Google Scholar
  7. [LLMP]
    A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, and J. M. Pollard, The number field sieve, Proc 22nd Annual ACM Symposium on the Theory of Computing, 1990, pp. 564–572.Google Scholar
  8. [P]
    C. Pomerance, Fast, rigorous factorization and discrete logarithm algorithms, in: D. S. Johnson, T. Nishizeki, A Nozaki, and H. S. Wilf (eds), Discrete Algorithms and Complexity, Academic Press, Orlando, FL, 1987, pp. 119–143.Google Scholar
  9. [S]
    C. P. Schnorr, Refined analysis and improvements on some factoring algorithms, J. Algorithms 3 (1982), 101–127.Google Scholar
  10. [W]
    D. H. Wiedemann, Solving sparse linear equations over finite fields, IEEE Trans. Inform. Theory 32 (1986), 54–62.Google Scholar

Copyright information

© International Association for Cryptologic Research 1993

Authors and Affiliations

  • Don Coppersmith
    • 1
  1. 1.IBM Research DivisionT. J. Watson Research CenterYorktown HeightsUSA

Personalised recommendations