Abstract
In this paper we describe our distributed implementation of two factoring algorithms, the elliptic curve method (ecm) and the multiple polynomial quadratic sieve algorithm (mpqs).
Since the summer of 1987, our ecm-implementation on a network of MicroVAX processors at DEC’s Systems Research Center has factored several most and more wanted numbers from the Cunningham project. In the summer of 1988, we implemented the multiple polynomial quadratic sieve algorithm on the same network. On this network alone, we are now able to factor any 100 digit integer, or to find 35 digit factors of numbers up to 150 digits long within one month.
To allow an even wider distribution of our programs we made use of electronic mail networks for the distribution of the programs and for inter-processor communication. Even during the initial stage of this experiment, machines all over the United States and at various places in Europe and Australia contributed 15 percent of the total factorization effort.
At all the sites where our program is running we only use cycles that would otherwise have been idle. This shows that the enormous computational task of factoring 100 digit integers with the current algorithms can be completed almost for free. Since we use a negligible fraction of the idle cycles of all the machines on the worldwide electronic mail networks, we could factor 100 digit integers within a few days with a little more help.
Chapter PDF
Similar content being viewed by others
Keywords
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.
References
L. Adleman, “The theory of computer viruses,” Proceedings Crypto 88, 1988.
E. Bach. J. Shallit. “Factoring with cyclotomic polynomials,” Proceedings 26th FOCS, 1985, pp 443–450.
G. Brassard, Modern Cryptology, Lecture Notes in Computer Science, vol. 325, 1988, Springer Verlag.
R.P. Brent, “Some integer factorization algorithms using elliptic curves,” Australian Computer Science Communications v. 8, 1986, pp 149–163.
R.P. Brent, G.L. Cohen, “A new lower bound for odd perfect numbers,” Math. Comp., to appear.
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, vol. 22, Providence: A.M.S., 1988.
T.R. Caron, R.D. Silverman, “Parallel implementation of the quadratic sieve,” J. Supercomputing, v. 1, 1988, pp 273–290.
A.J.C. Cunningham, H.J. Woodall, Factorisation of (y n ∓1). y = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers (n), London: Hodgson (1925).
J.A. Davis, D.B. Holdridge. “Factorization using the quadratic sieve algorithm,” Sandia National Laboratories Tech Rpt. SAND 83-1346, December 1983.
P.J. Denning, “The Science of Computing: Computer Viruses,” American Scientist, v. 76, May–June 1988.
A.K. Lenstra, H.W. Lenstra, Jr, “Algorithms in number theory,” in: J. van Leeuwen, A. Meyer, M. Nivat, M. Paterson, D. Perrin (eds.), Handbook of theoretical computer science, to appear; report 87-8, The University of Chicago, Department of Computer Science, May 1987.
A.K. Lenstra, M.S. Manasse, “Compact incremental Gaussian elimination over Z/2Z,” report 88-16, The University of Chicago, Department of Computer Science, October 1988.
H.W. Lenstra, Jr., “Factoring integers with elliptic curves,” Ann. of Math., v. 126, 1987, pp. 649–673.
P.L. Montgomery, “Modular multiplication without trial division,” Math. Comp., v. 44, 1985, pp 519–521.
P.L. Montgomery, “Speeding the Pollard and elliptic curve methods of factorization,” Math. Comp., v. 48, 1987, pp 243–264.
P.L. Montgomery, R.D. Silverman, “An FFT extension to the p-1 factoring algorithm,” manuscript, 1988.
A.M. Odlyzko, “Discrete logarithms and their cryptographic significance,” pp. 224–314; in: T. Beth, N. Cot, I. Ingemarsson (eds), Advances in cryptology, Springer Lecture Notes in Computer Science, vol. 209, 1985.
J.M. Pollard, “A Monte Carlo method for factorization,” BIT, v. 15, 1975, pp 331–334.
C. Pomerance, “Analysis and comparison of some integer factoring algorithms,” pp. 89–139; in: H.W. Lenstra, Jr., R. Tijdeman (eds), Computational methods in number theory, Mathematical Centre Tracts 154, 155, Mathematisch Centrum, Amsterdam, 1982.
C. Pomerance, J.W. Smith, R. Tuler, “A pipeline architecture for factoring large integers with the quadratic sieve algorithm,” SIAM J. Comput., v. 17, 1988, pp. 387–403.
H.J.J. te Riele, W.M. Lioen, D.T. Winter, “Factoring with the quadratic sieve on large vector computers,” report NM-R8805, 1988, Centrum voor Wiskunde en Informatica, Amsterdam.
R.L. Rivest, A. Shamir, L. Adleman, “A method for obtaining digital signatures and public-key cryptosystems,” Commun. ACM., v. 21, 1978, pp. 120–126.
E. Roberts, J. Ellis, “parmake and dp: Experience with a distributed, parallel implementation of make,” Proceedings from the Second Workshop on Large-Grained Parallelism, Software Engineering Institute, Carnegie-Mellon University, Report CMU/SEI-87-SR-5, November 1987.
R.D. Silverman, “The multiple polynomial quadratic sieve,” Math. Comp., v. 48, 1987, pp. 329–339.
K. Thompson, “Reflections on Trusting Trust,” Commun. ACM, v. 27, 1984, pp. 172–80.
D.H. Wiedemann, “Solving sparse linear equations over finite fields,” IEEE Transactions on Information Theory, v. 32, 1986, pp. 54–62.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lenstra, A.K., Manasse, M.S. (1990). Factoring by electronic mail. In: Quisquater, JJ., Vandewalle, J. (eds) Advances in Cryptology — EUROCRYPT ’89. EUROCRYPT 1989. Lecture Notes in Computer Science, vol 434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46885-4_35
Download citation
DOI: https://doi.org/10.1007/3-540-46885-4_35
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-53433-4
Online ISBN: 978-3-540-46885-1
eBook Packages: Springer Book Archive