Abstract
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.
The authors wish to thank Dan Bernstein, Arjeh Cohen, Michael Filaseta, Andrew Granville, Arjen Lenstra, Victor Miller, Robert Rumely, and Robert Silverman for their helpful suggestions. The authors were supported by NSF under Grants No. DMS 90-12989, No. DMS 90-02939, and No. DMS 90-02538, respectively. The second and third authors are grateful to the Institute for Advanced Study (Princeton), where part of the work on which this paper is based was done.
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References
L. M. Adleman, Factoring numbers using singular integers, Proc. 23rd Annual ACM Symp. on Theory of Computing (STOC) (1991), 64–71.
E. Bach, Explicit bounds for primality testing and related problems, Math. Comp. 55 (1990), 355–380.
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.
J. Brillhart, M. Filaseta, A. Odlyzko, On an irreducibility theorem of A. Cohn, Can. J. Math. 33 (1981), 1055–1059.
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.
J. A. Buchmann, H. W. Lenstra, Jr., Decomposing primes in number fields, in preparation.
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.
E. R. Canfield, P. Erdős, C. Pomerance, On a problem of Oppenheim concerning “factorisatio numerorum”, J. Number Theory 17 (1983), 1–28.
J. W. S. Cassels, A. Fröhlich (eds), Algebraic number theory, Proceedings of an instructional conference, Academic Press, London, 1967.
D. Coppersmith, Modifications to the number field sieve, J. Cryptology, to appear; IBM Research Report #RC 16264, Yorktown Heights, New York, 1990.
J.-M. Couveignes, Computing a square root for the number field sieve, this volume, pp. 95–102.
J. D. Dixon, Asymptotically fast factorization of integers, Math. Comp. 36 (1981), 255–260.
W. Fulton, Intersection theory, Springer-Verlag, Berlin, 1984.
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.
D. Gordon, Discrete logarithms in GF(p) using the number field sieve, SIAM J. Discrete Math. 6 (1993), 124–138.
B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, 1967.
D. E. Knuth, The art of computer programming, volume 2, Seminumerical algorithms, second edition, Addison-Wesley, Reading, Mass., 1981.
S. Landau, Factoring polynomials over algebraic number fields, SIAM J. Comput. 14 (1985), 184–195.
S. Lang, Algebraic number theory, Addison-Wesley, Reading, Mass., 1970.
A. K. Lenstra, Factorization of polynomials, in [29], 169–198.
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.
A. K. Lenstra, H. W. Lenstra, Jr., L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 515–534.
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.
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.
A. K. Lenstra, M. S. Manasse, Factoring with two large primes, Math. Comp., to appear.
H. W. Lenstra, Jr., Factoring integers with elliptic curves, Ann. of Math. 126 (1987), 649–673.
H. W. Lenstra, Jr., Algorithms in algebraic number theory, Bull. Amer. Math. Soc. 26 (1992), 211–244.
H. W. Lenstra, Jr., C. Pomerance, A rigorous time bound for factoring integers, J. Amer. Math. Soc. 5 (1992), 483–516.
H. W. Lenstra, Jr., R. Tijdeman (eds), Computational methods in number theory, Mathematical Centre Tracts 154/155, Mathematisch Centrum, Amsterdam, 1982.
M. A. Morrison, J. Brillhart, A method of factoring and the factorization of F 7, Math. Comp. 29 (1975), 183–205.
J. M. Pollard, Factoring with cubic integers, this volume, pp. 4–10.
J. M. Pollard, The lattice sieve, this volume, pp. 43–49.
C. Pomerance, Analysis and comparison of some integer factoring algorithms, in [29], 89–139.
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.
O. Schirokauer, On pro-finite groups and on discrete logarithms, Ph. D. thesis, University of California, Berkeley, May 1992.
B. Vallée, Generation of elements with small modular squares and provably fast integer factoring algorithms, Math. Comp. 56 (1991), 823–849.
B. L. van der Waerden, Algebra, seventh edition, Springer-Verlag, Berlin, 1966.
P. S. Wang, Factoring multivariate polynomials over algebraic number fields, Math. Comp. 30 (1976), 324–336.
P. J. Weinberger, L. P. Rothschild, Factoring polynomials over algebraic number fields, ACM Trans. Math. Software 2 (1976), 335–350.
E. Weiss, Algebraic number theory, McGraw-Hill, New York, 1963; reprinted, Chelsea, New York, 1976.
D. Wiedemann, Solving sparse linear equations over finite fields, IEEE Trans. Inform. Theory 32 (1986), 54–62.
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Buhler, J.P., Lenstra, H.W., Pomerance, C. (1993). Factoring integers with the number field sieve. In: Lenstra, A.K., Lenstra, H.W. (eds) The development of the number field sieve. Lecture Notes in Mathematics, vol 1554. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0091539
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DOI: https://doi.org/10.1007/BFb0091539
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