The Elliptic Curve Method for Integer Factorization (ECM)¶
Sage includes GMPECM, which is a highly optimized implementation of Lenstra’s elliptic curve factorization method. See http://ecm.gforge.inria.fr/ for more about GMPECM. This file provides a Cython interface to the GMPECM library.
AUTHORS:
 Robert L Miller (20080121): library interface (clone of ecmfactor.c)
 Jeroen Demeyer (20120329): signal handling, documentation
 Paul Zimmermann (20110522) – added input/output of sigma
EXAMPLES:
sage: from sage.libs.libecm import ecmfactor
sage: result = ecmfactor(999, 0.00)
sage: result[0]
True
sage: result[1] in [3, 9, 27, 37, 111, 333, 999] or result[1]
True
sage: result = ecmfactor(999, 0.00, verbose=True)
Performing one curve with B1=0
Found factor in step 1: ...
sage: result[0]
True
sage: result[1] in [3, 9, 27, 37, 111, 333, 999] or result[1]
True
sage: ecmfactor(2^128+1,1000,sigma=227140902)
(True, 5704689200685129054721, 227140902)

sage.libs.libecm.
ecmfactor
(number, B1, verbose=False, sigma=0)¶ Try to find a factor of a positive integer using ECM (Elliptic Curve Method). This function tries one elliptic curve.
INPUT:
number
– positive integer to be factoredB1
– bound for step 1 of ECMverbose
(default: False) – print some debugging information
OUTPUT:
Either
(False, None)
if no factor was found, or(True, f)
if the factorf
was found.EXAMPLES:
sage: from sage.libs.libecm import ecmfactor
This number has a small factor which is easy to find for ECM:
sage: N = 2^167  1 sage: factor(N) 2349023 * 79638304766856507377778616296087448490695649 sage: ecmfactor(N, 2e5) (True, 2349023, ...)
If a factor was found, we can reproduce the factorization with the same sigma value:
sage: N = 2^167  1 sage: ecmfactor(N, 2e5, sigma=1473308225) (True, 2349023, 1473308225)
With a smaller B1 bound, we may or may not succeed:
sage: ecmfactor(N, 1e2) # random (False, None)
The following number is a Mersenne prime, so we don’t expect to find any factors (there is an extremely small chance that we get the input number back as factorization):
sage: N = 2^127  1 sage: N.is_prime() True sage: ecmfactor(N, 1e3) (False, None)
If we have several small prime factors, it is possible to find a product of primes as factor:
sage: N = 2^179  1 sage: factor(N) 359 * 1433 * 1489459109360039866456940197095433721664951999121 sage: ecmfactor(N, 1e3) # random (True, 514447, 3475102204)
We can ask for verbose output:
sage: N = 12^97  1 sage: factor(N) 11 * 43570062353753446053455610056679740005056966111842089407838902783209959981593077811330507328327968191581 sage: ecmfactor(N, 100, verbose=True) Performing one curve with B1=100 Found factor in step 1: 11 (True, 11, ...) sage: ecmfactor(N/11, 100, verbose=True) Performing one curve with B1=100 Found no factor. (False, None)