Abstract
We report the factorization of a 135-digit integer by the triple-large-prime variation of the multiple polynomial quadratic sieve. Previous workers [6][10] had suggested that using more than two large primes would be counterproductive, because of the greatly increased number of false reports from the sievers. We provide evidence that, for this number and our implementation, using three large primes is approximately 1.7 times as fast as using only two. The gain in efficiency comes from a sudden growth in the number of cycles arising from relations which contain three large primes. This effect, which more than compensates for the false reports, was not anticipated by the authors of [6] [10] but has become quite familiar from factorizations obtained using the number field sieve. We characterize the various types of cycles present, and give a semi-quantitative description of their rather mysterious behaviour.
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D. Atkins, M. Graff, A.K. Lenstra, P.C. Leyland, THE MAGIC WORDS ARE SQUEAMISH OSSIFRAGE, Proceedings Asiacrypt’94, LNCS 917, Springer-Verlag (1995), 265–277.
B. Carrier, S.S. Wagstaff Jr., Implementing the hypercube quadratic sieve with two large primes, Technical Report 2001-45, Purdue University CERIAS (2001) URL http://www.cerias.purdue.edu/papers/archive/2001-45.ps,pdf.
S. Cavallar, Strategies in filtering in the number field sieve, ANTS-IV, LNCS 1838, Springer-Verlag (2000) 209–231.
S. Cavallar, B. Dodson, A.K. Lenstra, P.C. Leyland, W. Lioen, P.L. Montgomery, B. Murphy, H. te Riele, P. Zimmermann, Factorization of RSA-140 using the number field sieve, Proceedings Asiacrypt’99, LNCS 1716, Springer-Verlag (1999), 195–207.
J. Cowie, B. Dodson, R.-M. Elkenbracht-Huizing, A.K. Lenstra, P.L. Montgomery, J. Zayer, A world wide number field sieve factoring record: on to 512 bits, Proceedings Asiacrypt’96, LNCS 1163, Springer-Verlag (1996), 382–394.
R. Crandall, C. Pomerance, Prime Numbers, a computational perspective, Springer (2001) 237.
B. Dodson, A.K. Lenstra, NFS with four large primes: an explosive experiment, Proceedings Crypto’95, LNCS 963, Springer-Verlag (1995) 372–385.
J.P.K. Doye, R.P. Sear, D. Frenkel, The effect of chain stiffness on the phase behaviour of isolated homopolymers, J. Chemical Physics 108, (1998) 2134–2142 URL http://brian.ch.cam.ac.uk/~jon/papers/homop/homop.html.
A.K. Lenstra, M.S. Manasse, Factoring by electronic mail, Proceedings Eurocrypt’89, LNCS 434, Springer-Verlag (1990) 355–371.
A.K. Lenstra, M.S. Manasse, Factoring with two large primes, Math. Comp. 63 (1994) 785–798.
P.L. Montgomery, A block Lanczos algorithm for finding dependencies over GF(2), Proceedings Eurocrypt’95, LNCS 921, Springer-Verlag (1995), 106–120.
P.L. Montgomery, Distributed Linear Algebra, 4th Workshop on Elliptic Curve Cryptography, Essen (2000), URL http://www.cacr.math.uwaterloo.ca/con-ferences/2000/ecc2000/montgomery.ppt,ps.
C. Pomerance, Analysis and comparison of some integer factoring algorithms in H.W. Lenstra Jr, R. Tijdeman, editors, Computational methods in number theory, Part I, Math. Centre Tracts, Math. Centrum (1982) 89–139.
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Leyland, P., Lenstra, A., Dodson, B., Muffett, A., Wagstaff, S. (2002). MPQS with Three Large Primes. In: Fieker, C., Kohel, D.R. (eds) Algorithmic Number Theory. ANTS 2002. Lecture Notes in Computer Science, vol 2369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45455-1_35
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DOI: https://doi.org/10.1007/3-540-45455-1_35
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