<s>
The	O
Lenstra	B-Algorithm
elliptic-curve	I-Algorithm
factorization	I-Algorithm
or	O
the	O
elliptic-curve	O
factorization	O
method	O
(	O
ECM	O
)	O
is	O
a	O
fast	O
,	O
sub-exponential	O
running	O
time	O
,	O
algorithm	O
for	O
integer	O
factorization	O
,	O
which	O
employs	O
elliptic	O
curves	O
.	O
</s>
<s>
The	O
second-fastest	O
is	O
the	O
multiple	B-Algorithm
polynomial	I-Algorithm
quadratic	I-Algorithm
sieve	I-Algorithm
,	O
and	O
the	O
fastest	O
is	O
the	O
general	B-Algorithm
number	I-Algorithm
field	I-Algorithm
sieve	I-Algorithm
.	O
</s>
<s>
The	O
Lenstra	B-Algorithm
elliptic-curve	I-Algorithm
factorization	I-Algorithm
is	O
named	O
after	O
Hendrik	O
Lenstra	O
.	O
</s>
<s>
,	O
it	O
is	O
still	O
the	O
best	O
algorithm	O
for	O
divisors	O
not	O
exceeding	O
50	O
to	O
60	O
digits	B-Application
,	O
as	O
its	O
running	O
time	O
is	O
dominated	O
by	O
the	O
size	O
of	O
the	O
smallest	O
factor	O
p	O
rather	O
than	O
by	O
the	O
size	O
of	O
the	O
number	O
n	O
to	O
be	O
factored	O
.	O
</s>
<s>
The	O
largest	O
factor	O
found	O
using	O
ECM	O
so	O
far	O
has	O
83	O
decimal	B-Application
digits	B-Application
and	O
was	O
discovered	O
on	O
7	O
September	O
2013	O
by	O
R	O
.	O
Propper	O
.	O
</s>
<s>
Increasing	O
the	O
number	O
of	O
curves	O
tested	O
improves	O
the	O
chances	O
of	O
finding	O
a	O
factor	O
,	O
but	O
they	O
are	O
not	O
linear	O
with	O
the	O
increase	O
in	O
the	O
number	O
of	O
digits	B-Application
.	O
</s>
<s>
The	O
Lenstra	B-Algorithm
elliptic-curve	I-Algorithm
factorization	I-Algorithm
method	O
to	O
find	O
a	O
factor	O
of	O
a	O
given	O
natural	O
number	O
works	O
as	O
follows	O
:	O
</s>
<s>
Although	O
there	O
is	O
no	O
proof	O
that	O
a	O
smooth	O
group	O
order	O
will	O
be	O
found	O
in	O
the	O
Hasse-interval	O
,	O
by	O
using	O
heuristic	B-Algorithm
probabilistic	O
methods	O
,	O
the	O
Canfield	O
–	O
Erdős	O
–	O
Pomerance	O
theorem	O
with	O
suitably	O
optimized	O
parameter	O
choices	O
,	O
and	O
the	O
L-notation	O
,	O
we	O
can	O
expect	O
to	O
try	O
curves	O
before	O
getting	O
a	O
smooth	O
group	O
order	O
.	O
</s>
<s>
This	O
heuristic	B-Algorithm
estimate	O
is	O
very	O
reliable	O
in	O
practice	O
.	O
</s>
<s>
The	O
Elliptic	B-Algorithm
Curve	I-Algorithm
Method	I-Algorithm
makes	O
use	O
of	O
the	O
failure	O
cases	O
of	O
the	O
addition	O
law	O
.	O
</s>
<s>
The	O
above	O
text	O
is	O
about	O
the	O
first	O
stage	O
of	O
elliptic	B-Algorithm
curve	I-Algorithm
factorisation	I-Algorithm
.	O
</s>
<s>
It	O
uses	O
Grover	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
to	O
roughly	O
double	O
the	O
length	O
of	O
the	O
primes	O
found	O
compared	O
to	O
standard	O
EECM	O
,	O
assuming	O
a	O
quantum	O
computer	O
with	O
sufficiently	O
many	O
qubits	O
and	O
of	O
comparable	O
speed	O
to	O
the	O
classical	O
computer	O
running	O
EECM	O
.	O
</s>
