Abstract
The general number field sieve is the asymptotically fastest—and by far most complex—factoring algorithm known. We have implemented this algorithm, including five practical improvements: projective polynomials, the lattice sieve, the large prime variation, character columns, and the positive square root method. In this paper we describe our implementation and list some factorizations we obtained, including the record factorization of 2523 − 1.
Thanks to Joe Buhler, Hendrik Lenstra, John Pollard, and Carl Pomerance for their helpful suggestions, and to Andrew Odlyzko for his help with the factorization of 2523 − 1. The first author was supported in part by a National Science Foundation Graduate Fellowship and by Bellcore.
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References
L.M. Adleman, Factoring numbers using singular integers, Proc. 23rd Annual ACM Symp. on Theory of Computing (STOC), New Orleans, May 6–8, 1991, 64–71.
W. Bosma, M.-P. van der Hulst, Primality proving with cyclotomy, Universiteit van Amsterdam, 1990.
J.P. Buhler, H.W. Lenstra, Jr., C. Pomerance, Factoring integers with the number field sieve, this volume, pp. 50–94.
J.-M. Couveignes, Computing a square root for the number field sieve, this volume, pp. 95–102.
J.A. Davis, D.B. Holdridge, Factorization using the quadratic sieve algorithm, Tech. Report SAND 83-1346, Sandia National Laboratories, Albuquerque, New Mexico, 1983.
B. Dixon, A.K. Lenstra, Factoring integers using SIMD sieves, Advances in Cryptology, Eurocrypt ′93, to appear.
D.E. Knuth, The art of computer programming, volume 2, Seminumerical algorithms, second edition, Addison-Wesley, Reading, Massachusetts, 1981.
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.
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.
F. Morain, Implementation of the Goldwasser-Kilian-Atkin primality testing algorithm, INRIA report 911, INRIA-Rocquencourt, 1988.
J.M. Pollard, The lattice sieve, this volume, pp. 43–49.
C. Pomerance, The quadratic sieve factoring algorithm, Lecture Notes in Comput. Sci. 209 (1985), 169–182.
D. Wiedemann, Solving sparse linear equations over finite fields, IEEE Trans. Inform. Theory 32 (1986), 54–62.
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© 1993 Springer-Verlag
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Bernstein, D.J., Lenstra, A.K. (1993). A general number field sieve implementation. 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/BFb0091541
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DOI: https://doi.org/10.1007/BFb0091541
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