The elliptic curve factorization method (ECM), also called elliptic curve factorization or 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 Pollard
rho factorization and Pollard p -1
factorization methods .
Zimmermann maintains a table of the largest factors found using the ECM. As of Oct. 2026, the largest prime
factor found using the ECM has 83 decimal digits .
This factor of
See also Atkin-Goldwasser-Kilian-Morain Certificate ,
Elliptic Curve Primality
Proving ,
Elliptic Pseudoprime ,
Prime
Factorization ,
Prime Factorization
Algorithms
Explore with Wolfram|Alpha
References 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 Brillhart, J.; Lehmer, D. H.;
Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorizations
of b n 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. https://maths-people.anu.edu.au/~brent/pub/pub156.html . Lenstra,
A. K. and Lenstra, H. W. Jr. "Algorithms in Number Theory."
In Handbook
of Theoretical Computer Science, Volume A: Algorithms and Complexity 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 . Referenced
on Wolfram|Alpha Elliptic Curve Factorization
Method
Cite this as:
Weisstein, Eric W. "Elliptic Curve Factorization
Method." From MathWorld https://mathworld.wolfram.com/EllipticCurveFactorizationMethod.html
Subject classifications
. It tends to be faster than the Pollard
rho factorization and Pollard p-1
factorization methods.
was found by R. Propper on Sep. 7, 2013
(Zimmermann).
