The elliptic curve factorization method, abbreviated ECM and sometimes also called the Lenstra elliptic curve method, is a factorization algorithm that computes a large
multiple of a point on a random elliptic curve
modulo the number to be factored . It tends to be faster than the Pollard
rho factorization and Pollard
p-1 factorization methods.
Zimmermann maintains a table of the largest factors found using the ECM. As of Jan. 2009, the largest prime factor found using the ECM had 67 decimal digits. This factor of
was found by B. Dodson on Aug. 24, 2006 (Zimmermann).
Alpern, D. "Factorization Using the Elliptic Curve Method." https://www.alpertron.com.ar/ECM.HTM.Atkin,
A. O. L. and Morain, F. "Finding Suitable Curves for the Elliptic
Curve Method of Factorization." Math. Comput.60, 399-405, 1993.Brent,
R. P. "Some Integer Factorization Algorithms Using Elliptic Curves."
Austral. Comp. Sci. Comm.8, 149-163, 1986.Brent, R. P.
"Parallel Algorithms for Integer Factorisation." In Number
Theory and Cryptography (Ed. J. H. Loxton). New York: Cambridge
University Press, pp. 26-37, 1990.Brillhart, J.; Lehmer, D. H.;
Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorizations
of b-n+/-1, b=2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers, rev.
ed. Providence, RI: Amer. Math. Soc., p. lxxxiii, 1988.Eldershaw,
C. and Brent, R. P. "Factorization of Large Integers on Some Vector and
Parallel Computers." Australian National University, Technical Report TR-CS-95-01.
January 1995. http://cs.anu.edu.au/techreports/1995/TR-CS-95-01.html.Lenstra,
A. K. and Lenstra, H. W. Jr. "Algorithms in Number Theory."
In Handbook
of Theoretical Computer Science, Volume A: Algorithms and Complexity (Ed.
J. van Leeuwen). Amsterdam: Netherlands, Elsevier, pp. 673-715, 1990.Lenstra,
H. W. Jr. "Factoring Integers with Elliptic Curves." Ann.
Math.126, 649-673, 1987.Montgomery, P. L. "Speeding
the Pollard and Elliptic Curve Methods of Factorization." Math. Comput.48,
243-264, 1987.Zimmermann, P. "The ECMNET Project." https://members.loria.fr/PZimmermann/records/ecmnet.html.Zimmermann,
P. "ECM Top 100 Table." https://members.loria.fr/PZimmermann/records/top50.html.
. It tends to be faster than the Pollard
rho factorization and Pollard
p-1 factorization methods.
was found by B. Dodson on Aug. 24, 2006 (Zimmermann).
