<s>
In	O
computational	O
number	O
theory	O
,	O
a	O
factor	B-Algorithm
base	I-Algorithm
is	O
a	O
small	O
set	O
of	O
prime	O
numbers	O
commonly	O
used	O
as	O
a	O
mathematical	O
tool	O
in	O
algorithms	O
involving	O
extensive	O
sieving	O
for	O
potential	O
factors	O
of	O
a	O
given	O
integer	O
.	O
</s>
<s>
A	O
factor	B-Algorithm
base	I-Algorithm
is	O
a	O
relatively	O
small	O
set	O
of	O
distinct	O
prime	O
numbers	O
P	O
,	O
sometimes	O
together	O
with	O
-1	O
.	O
</s>
<s>
Say	O
we	O
want	O
to	O
factorize	O
an	O
integer	O
n	O
.	O
We	O
generate	O
,	O
in	O
some	O
way	O
,	O
a	O
large	O
number	O
of	O
integer	O
pairs	O
(	O
x	O
,	O
y	O
)	O
for	O
which	O
,	O
,	O
and	O
can	O
be	O
completely	O
factorized	O
over	O
the	O
chosen	O
factor	B-Algorithm
base	I-Algorithm
—	O
that	O
is	O
,	O
all	O
their	O
prime	O
factors	O
are	O
in	O
P	O
.	O
</s>
<s>
In	O
practice	O
,	O
several	O
integers	O
x	O
are	O
found	O
such	O
that	O
has	O
all	O
of	O
its	O
prime	O
factors	O
in	O
the	O
pre-chosen	O
factor	B-Algorithm
base	I-Algorithm
.	O
</s>
<s>
We	O
represent	O
each	O
expression	O
as	O
a	O
vector	O
of	O
a	O
matrix	B-Architecture
with	O
integer	O
entries	O
being	O
the	O
exponents	O
of	O
factors	O
in	O
the	O
factor	B-Algorithm
base	I-Algorithm
.	O
</s>
<s>
This	O
essentially	O
reformulates	O
the	O
problem	O
into	O
a	O
system	O
of	O
linear	O
equations	O
,	O
which	O
can	O
be	O
solved	O
using	O
numerous	O
methods	O
such	O
as	O
Gaussian	B-Algorithm
elimination	I-Algorithm
;	O
in	O
practice	O
advanced	O
methods	O
like	O
the	O
block	O
Lanczos	O
algorithm	O
are	O
used	O
,	O
that	O
take	O
advantage	O
of	O
certain	O
properties	O
of	O
the	O
system	O
.	O
</s>
<s>
If	O
repeated	O
attempts	O
to	O
factor	O
fail	O
we	O
can	O
try	O
again	O
using	O
a	O
different	O
factor	B-Algorithm
base	I-Algorithm
.	O
</s>
<s>
Factor	B-Algorithm
bases	I-Algorithm
are	O
used	O
in	O
,	O
for	O
example	O
,	O
Dixon	B-Algorithm
's	I-Algorithm
factorization	I-Algorithm
,	O
the	O
quadratic	B-Algorithm
sieve	I-Algorithm
,	O
and	O
the	O
number	B-Algorithm
field	I-Algorithm
sieve	I-Algorithm
.	O
</s>
<s>
Factor	B-Algorithm
bases	I-Algorithm
are	O
also	O
used	O
in	O
the	O
Index	O
calculus	O
algorithm	O
for	O
computing	O
discrete	O
logarithms	O
.	O
</s>
