Factoring integers with the number field sieve

  • J. P. Buhler
  • H. W. LenstraJr.
  • Carl Pomerance
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1554)


In 1990, the ninth Fermat number was factored into primes by means of a new algorithm, the “number field sieve”, which was proposed by John Pollard. The present paper is devoted to the description and analysis of a more general version of the number field sieve. It should be possible to use this algorithm to factor arbitrary integers into prime factors, not just integers of a special form like the ninth Fermat number. Under reasonable heuristic assumptions, the analysis predicts that the time needed by the general number field sieve to factor n is exp((c+o(1))(logn)1/3(loglogn)2/3) (for n → ∞), where c=(64/9)1/3=1.9223. This is asymptotically faster than all other known factoring algorithms, such as the quadratic sieve and the elliptic curve method.


Prime Number Number Field Ring Homomorphism Algebraic Integer Algebraic Number Theory 
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.


  1. 1.
    L. M. Adleman, Factoring numbers using singular integers, Proc. 23rd Annual ACM Symp. on Theory of Computing (STOC) (1991), 64–71.Google Scholar
  2. 2.
    E. Bach, Explicit bounds for primality testing and related problems, Math. Comp. 55 (1990), 355–380.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    N. Boston, W. Dabrowski, T. Foguel, P. Gies, D. Jackson, J. Leavitt, D. Ose, The proportion of fixed-point-free elements in a transitive permutation group, Comm. in Algebra, to appear.Google Scholar
  4. 4.
    J. Brillhart, M. Filaseta, A. Odlyzko, On an irreducibility theorem of A. Cohn, Can. J. Math. 33 (1981), 1055–1059.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, S. S. Wagstaff, Jr., Factorizations of b n ± 1, b=2, 3, 5, 6, 7, 10, 11, 12 up to high powers, second edition, Contemporary Mathematics 22, Amer. Math. Soc., Providence, 1988.zbMATHGoogle Scholar
  6. 6.
    J. A. Buchmann, H. W. Lenstra, Jr., Decomposing primes in number fields, in preparation.Google Scholar
  7. 7.
    P. J. Cameron, A. M. Cohen, On the number of fixed point free elements in a permutation group, Discrete Math. 106/107 (1992), 135–138.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    E. R. Canfield, P. Erdős, C. Pomerance, On a problem of Oppenheim concerning “factorisatio numerorum”, J. Number Theory 17 (1983), 1–28.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    J. W. S. Cassels, A. Fröhlich (eds), Algebraic number theory, Proceedings of an instructional conference, Academic Press, London, 1967.zbMATHGoogle Scholar
  10. 10.
    D. Coppersmith, Modifications to the number field sieve, J. Cryptology, to appear; IBM Research Report #RC 16264, Yorktown Heights, New York, 1990.Google Scholar
  11. 11.
    J.-M. Couveignes, Computing a square root for the number field sieve, this volume, pp. 95–102.Google Scholar
  12. 12.
    J. D. Dixon, Asymptotically fast factorization of integers, Math. Comp. 36 (1981), 255–260.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    W. Fulton, Intersection theory, Springer-Verlag, Berlin, 1984.CrossRefzbMATHGoogle Scholar
  14. 14.
    P. X. Gallagher, The large sieve and probabilistic Galois theory, in: H. G. Diamond (ed.), Analytic number theory, Proc. Symp. Pure Math. 24, Amer. Math. Soc., Providence, 1973, 91–101.CrossRefGoogle Scholar
  15. 15.
    D. Gordon, Discrete logarithms in GF(p) using the number field sieve, SIAM J. Discrete Math. 6 (1993), 124–138.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, 1967.CrossRefzbMATHGoogle Scholar
  17. 17.
    D. E. Knuth, The art of computer programming, volume 2, Seminumerical algorithms, second edition, Addison-Wesley, Reading, Mass., 1981.zbMATHGoogle Scholar
  18. 18.
    S. Landau, Factoring polynomials over algebraic number fields, SIAM J. Comput. 14 (1985), 184–195.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    S. Lang, Algebraic number theory, Addison-Wesley, Reading, Mass., 1970.zbMATHGoogle Scholar
  20. 20.
    A. K. Lenstra, Factorization of polynomials, in [29], 169–198.Google Scholar
  21. 21.
    A. K. Lenstra, Factoring polynomials over algebraic number fields, in: J. A. van Hulzen (ed.), Computer algebra, Lecture Notes in Comput. Sci. 162, Springer-Verlag, Berlin, 1983, 245–254.CrossRefGoogle Scholar
  22. 22.
    A. K. Lenstra, H. W. Lenstra, Jr., L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515–534.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, J. M. Pollard, The factorization of the ninth Fermat number, Math. Comp. 61 (1993), to appear.Google Scholar
  24. 24.
    A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, J. M. Pollard, The number field sieve, this volume, pp. 11–42. Extended abstract: Proc. 22nd Annual ACM Symp. on Theory of Computing (STOC) (1990), 564–572.Google Scholar
  25. 25.
    A. K. Lenstra, M. S. Manasse, Factoring with two large primes, Math. Comp., to appear.Google Scholar
  26. 26.
    H. W. Lenstra, Jr., Factoring integers with elliptic curves, Ann. of Math. 126 (1987), 649–673.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    H. W. Lenstra, Jr., Algorithms in algebraic number theory, Bull. Amer. Math. Soc. 26 (1992), 211–244.MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    H. W. Lenstra, Jr., C. Pomerance, A rigorous time bound for factoring integers, J. Amer. Math. Soc. 5 (1992), 483–516.MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    H. W. Lenstra, Jr., R. Tijdeman (eds), Computational methods in number theory, Mathematical Centre Tracts 154/155, Mathematisch Centrum, Amsterdam, 1982.zbMATHGoogle Scholar
  30. 30.
    M. A. Morrison, J. Brillhart, A method of factoring and the factorization of F 7, Math. Comp. 29 (1975), 183–205.MathSciNetzbMATHGoogle Scholar
  31. 31.
    J. M. Pollard, Factoring with cubic integers, this volume, pp. 4–10.Google Scholar
  32. 32.
    J. M. Pollard, The lattice sieve, this volume, pp. 43–49.Google Scholar
  33. 33.
    C. Pomerance, Analysis and comparison of some integer factoring algorithms, in [29], 89–139.Google Scholar
  34. 34.
    C. Pomerance, Fast, rigorous factorization and discrete logarithm algorithms, in: D. S. Johnson, T. Nishizeki, A. Nozaki, H. S. Wilf (eds), Discrete algorithms and complexity, Academic Press, Orlando, 1987, 119–143.Google Scholar
  35. 35.
    O. Schirokauer, On pro-finite groups and on discrete logarithms, Ph. D. thesis, University of California, Berkeley, May 1992.Google Scholar
  36. 36.
    B. Vallée, Generation of elements with small modular squares and provably fast integer factoring algorithms, Math. Comp. 56 (1991), 823–849.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    B. L. van der Waerden, Algebra, seventh edition, Springer-Verlag, Berlin, 1966.zbMATHGoogle Scholar
  38. 38.
    P. S. Wang, Factoring multivariate polynomials over algebraic number fields, Math. Comp. 30 (1976), 324–336.MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    P. J. Weinberger, L. P. Rothschild, Factoring polynomials over algebraic number fields, ACM Trans. Math. Software 2 (1976), 335–350.MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    E. Weiss, Algebraic number theory, McGraw-Hill, New York, 1963; reprinted, Chelsea, New York, 1976.zbMATHGoogle Scholar
  41. 41.
    D. Wiedemann, Solving sparse linear equations over finite fields, IEEE Trans. Inform. Theory 32 (1986), 54–62.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • J. P. Buhler
    • 1
  • H. W. LenstraJr.
    • 2
  • Carl Pomerance
    • 3
  1. 1.Department of MathematicsReed CollegePortlandUSA
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  3. 3.Department of MathematicsUniversity of GeorgiaAthensUSA

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