# Integer Factorization¶

Bill Hart’s quadratic sieve is included with Sage. The quadratic sieve is the best algorithm for factoring numbers of the form $$pq$$ up to around 100 digits. It involves searching for relations, solving a linear algebra problem modulo $$2$$, then factoring $$n$$ using a relation $$x^2 \equiv y^2 \mod n$$.

sage: qsieve(next_prime(2^90)*next_prime(2^91), time=True)   # not tested
([1237940039285380274899124357, 2475880078570760549798248507],
'14.94user 0.53system 0:15.72elapsed 98%CPU (0avgtext+0avgdata 0maxresident)k')


Using qsieve is twice as fast as Sage’s general factor command in this example. Note that Sage’s general factor command does nothing but call Pari’s factor C library function.

sage: time factor(next_prime(2^90)*next_prime(2^91))     # not tested
CPU times: user 28.71 s, sys: 0.28 s, total: 28.98 s
Wall time: 29.38 s
1237940039285380274899124357 * 2475880078570760549798248507


Obviously, Sage’s factor command should not just call Pari, but nobody has gotten around to rewriting it yet.

## GMP-ECM¶

Paul Zimmerman’s GMP-ECM is included in Sage. The elliptic curve factorization (ECM) algorithm is the best algorithm for factoring numbers of the form $$n=pm$$, where $$p$$ is not “too big”. ECM is an algorithm due to Hendrik Lenstra, which works by “pretending” that $$n$$ is prime, choosing a random elliptic curve over $$\ZZ/n\ZZ$$, and doing arithmetic on that curve–if something goes wrong when doing arithmetic, we factor $$n$$.

In the following example, GMP-ECM is over 10 times faster than Sage’s generic factor function. Again, this emphasizes that Sage’s generic factor command would benefit from a rewrite that uses GMP-ECM and qsieve.

sage: time ecm.factor(next_prime(2^40) * next_prime(2^300))    # not tested
CPU times: user 0.85 s, sys: 0.01 s, total: 0.86 s
Wall time: 1.73 s
[1099511627791,
2037035976334486086268445688409378161051468393665936250636140449354381299763336706183397533]
sage: time factor(next_prime(2^40) * next_prime(2^300))        # not tested
CPU times: user 23.82 s, sys: 0.04 s, total: 23.86 s
Wall time: 24.35 s
1099511627791 * 2037035976334486086268445688409378161051468393665936250636140449354381299763336706183397533