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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
L. M. Adleman, Factoring numbers using singular integers, Proc. 23rd Annual ACM Symposium on the Theory of Computing, 1991, pp. 64–71.
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.
E. R. Canfield, P. Erdös, and C. Pomerance, On a problem of Oppenheim concerning “Factorisatio Numerorum”, J. Number Theory 17 (1983), 1–28.
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.
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.
H. W. Lenstra, Jr., Factoring integers with elliptic curves, Ann. of Math. 126 (1987), 649–673.
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.
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.
C. P. Schnorr, Refined analysis and improvements on some factoring algorithms, J. Algorithms 3 (1982), 101–127.
D. H. Wiedemann, Solving sparse linear equations over finite fields, IEEE Trans. Inform. Theory 32 (1986), 54–62.
Communicated by Andrew M. Odlyzko
About this article
Cite this article
Coppersmith, D. Modifications to the Number Field Sieve. J. Cryptology 6, 169–180 (1993). https://doi.org/10.1007/BF00198464
- Sieve methods