# The Elliptic Curve Method for Integer Factorization (ECM)¶

The Elliptic Curve Method for Integer Factorization (ECM)

Sage includes GMP-ECM, which is a highly optimized implementation of Lenstra’s elliptic curve factorization method. See http://ecm.gforge.inria.fr/ for more about GMP-ECM. This file provides a Cython interface to the GMP-ECM library.

AUTHORS:

• Robert L Miller (2008-01-21): library interface (clone of ecmfactor.c)
• Jeroen Demeyer (2012-03-29): signal handling, documentation

EXAMPLES:

sage: from sage.libs.libecm import ecmfactor
sage: result = ecmfactor(999, 0.00)
sage: result in [(True, 27), (True, 37), (True, 999)]
True
sage: result = ecmfactor(999, 0.00, verbose=True)
Performing one curve with B1=0
Found factor in step 1: ...
sage: result in [(True, 27), (True, 37), (True, 999)]
True

sage.libs.libecm.ecmfactor(number, B1, verbose=False)

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 factored
• B1 – bound for step 1 of ECM
• verbose (default: False) – print some debugging information

OUTPUT:

Either (False, None) if no factor was found, or (True, f) if the factor f 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)


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)


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)


TESTS:

Check that ecmfactor can be interrupted (factoring a large prime number):

sage: import sage.tests.interrupt
sage: try:
...     sage.tests.interrupt.interrupt_after_delay()
...     ecmfactor(2^521-1, 1e7)
... except KeyboardInterrupt:
...     print "Caught KeyboardInterrupt"
Caught KeyboardInterrupt


Some special cases:

sage: ecmfactor(1, 100)
(True, 1)
sage: ecmfactor(0, 100)
Traceback (most recent call last):
...
ValueError: Input number (0) must be positive


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