<s>
In	O
number	O
theory	O
,	O
the	O
general	B-Algorithm
number	I-Algorithm
field	I-Algorithm
sieve	I-Algorithm
(	O
GNFS	B-Algorithm
)	O
is	O
the	O
most	O
efficient	B-General_Concept
classical	O
algorithm	O
known	O
for	O
factoring	O
integers	O
larger	O
than	O
.	O
</s>
<s>
It	O
is	O
a	O
generalization	O
of	O
the	O
special	B-Algorithm
number	I-Algorithm
field	I-Algorithm
sieve	I-Algorithm
:	O
while	O
the	O
latter	O
can	O
only	O
factor	O
numbers	O
of	O
a	O
certain	O
special	O
form	O
,	O
the	O
general	B-Algorithm
number	I-Algorithm
field	I-Algorithm
sieve	I-Algorithm
can	O
factor	O
any	O
number	O
apart	O
from	O
prime	O
powers	O
(	O
which	O
are	O
trivial	O
to	O
factor	O
by	O
taking	O
roots	O
)	O
.	O
</s>
<s>
The	O
principle	O
of	O
the	O
number	B-Algorithm
field	I-Algorithm
sieve	I-Algorithm
(	O
both	O
special	O
and	O
general	O
)	O
can	O
be	O
understood	O
as	O
an	O
improvement	O
to	O
the	O
simpler	O
rational	B-Algorithm
sieve	I-Algorithm
or	O
quadratic	B-Algorithm
sieve	I-Algorithm
.	O
</s>
<s>
The	O
general	B-Algorithm
number	I-Algorithm
field	I-Algorithm
sieve	I-Algorithm
,	O
on	O
the	O
other	O
hand	O
,	O
manages	O
to	O
search	O
for	O
smooth	O
numbers	O
that	O
are	O
subexponential	O
in	O
the	O
size	O
of	O
.	O
</s>
<s>
This	O
is	O
the	O
key	O
to	O
the	O
efficiency	O
of	O
the	O
number	B-Algorithm
field	I-Algorithm
sieve	I-Algorithm
.	O
</s>
<s>
In	O
order	O
to	O
achieve	O
this	O
speed-up	O
,	O
the	O
number	B-Algorithm
field	I-Algorithm
sieve	I-Algorithm
has	O
to	O
perform	O
computations	O
and	O
factorizations	O
in	O
number	O
fields	O
.	O
</s>
<s>
This	O
results	O
in	O
many	O
rather	O
complicated	O
aspects	O
of	O
the	O
algorithm	O
,	O
as	O
compared	O
to	O
the	O
simpler	O
rational	B-Algorithm
sieve	I-Algorithm
.	O
</s>
<s>
The	O
running	O
time	O
of	O
the	O
number	B-Algorithm
field	I-Algorithm
sieve	I-Algorithm
is	O
super-polynomial	O
but	O
sub-exponential	O
in	O
the	O
size	O
of	O
the	O
input	O
.	O
</s>
<s>
The	O
current	O
best-known	O
approach	O
for	O
this	O
search	O
is	O
lattice	B-Algorithm
sieving	I-Algorithm
;	O
to	O
get	O
acceptable	O
yields	O
,	O
it	O
is	O
necessary	O
to	O
use	O
a	O
large	O
factor	O
base	O
.	O
</s>
<s>
Having	O
enough	O
such	O
pairs	O
,	O
using	O
Gaussian	B-Algorithm
elimination	I-Algorithm
,	O
one	O
can	O
get	O
products	O
of	O
certain	O
r	O
and	O
of	O
the	O
corresponding	O
s	O
to	O
be	O
squares	O
at	O
the	O
same	O
time	O
.	O
</s>
<s>
It	O
should	O
be	O
remarked	O
that	O
the	O
use	O
of	O
Gaussian	B-Algorithm
elimination	I-Algorithm
does	O
not	O
give	O
the	O
optimal	O
run	O
time	O
of	O
the	O
algorithm	O
.	O
</s>
<s>
These	O
are	O
known	O
as	O
special	B-Algorithm
number	I-Algorithm
field	I-Algorithm
sieve	I-Algorithm
techniques	O
,	O
such	O
as	O
used	O
in	O
the	O
Cunningham	O
project	O
.	O
</s>
<s>
Polynomial	O
selection	O
is	O
normally	O
performed	O
by	O
GPL	B-License
software	O
written	O
by	O
Kleinjung	O
,	O
or	O
by	O
msieve	O
,	O
and	O
lattice	B-Algorithm
sieving	I-Algorithm
by	O
GPL	B-License
software	O
written	O
by	O
Franke	O
and	O
Kleinjung	O
;	O
these	O
are	O
distributed	O
in	O
GGNFS	O
.	O
</s>
