<s>
In	O
mathematics	O
,	O
elliptic	B-Algorithm
curve	I-Algorithm
primality	I-Algorithm
testing	I-Algorithm
techniques	O
,	O
or	O
elliptic	B-Algorithm
curve	I-Algorithm
primality	I-Algorithm
proving	I-Algorithm
(	O
ECPP	O
)	O
,	O
are	O
among	O
the	O
quickest	O
and	O
most	O
widely	O
used	O
methods	O
in	O
primality	B-Algorithm
proving	I-Algorithm
.	O
</s>
<s>
The	O
concept	O
of	O
using	O
elliptic	B-Algorithm
curves	I-Algorithm
in	I-Algorithm
factorization	I-Algorithm
had	O
been	O
developed	O
by	O
H	O
.	O
W	O
.	O
Lenstra	O
in	O
1985	O
,	O
and	O
the	O
implications	O
for	O
its	O
use	O
in	O
primality	B-Algorithm
testing	I-Algorithm
(	O
and	O
proving	O
)	O
followed	O
quickly	O
.	O
</s>
<s>
Primality	B-Algorithm
testing	I-Algorithm
is	O
a	O
field	O
that	O
has	O
been	O
around	O
since	O
the	O
time	O
of	O
Fermat	B-Language
,	O
in	O
whose	O
time	O
most	O
algorithms	O
were	O
based	O
on	O
factoring	O
,	O
which	O
become	O
unwieldy	O
with	O
large	O
input	O
;	O
modern	O
algorithms	O
treat	O
the	O
problems	O
of	O
determining	O
whether	O
a	O
number	O
is	O
prime	O
and	O
what	O
its	O
factors	O
are	O
separately	O
.	O
</s>
<s>
Although	O
many	O
current	O
tests	O
result	O
in	O
a	O
probabilistic	O
output	O
(	O
N	O
is	O
either	O
shown	O
composite	O
,	O
or	O
probably	O
prime	O
,	O
such	O
as	O
with	O
the	O
Baillie	B-Algorithm
–	I-Algorithm
PSW	I-Algorithm
primality	I-Algorithm
test	I-Algorithm
or	O
the	O
Miller	B-Algorithm
–	I-Algorithm
Rabin	I-Algorithm
test	I-Algorithm
)	O
,	O
the	O
elliptic	O
curve	O
test	O
proves	O
primality	O
(	O
or	O
compositeness	O
)	O
with	O
a	O
quickly	O
verifiable	O
certificate	B-Algorithm
.	O
</s>
<s>
Previously-known	O
prime-proving	O
methods	O
such	O
as	O
the	O
Pocklington	B-Algorithm
primality	I-Algorithm
test	I-Algorithm
required	O
at	O
least	O
partial	O
factorization	O
of	O
in	O
order	O
to	O
prove	O
that	O
is	O
prime	O
.	O
</s>
<s>
ECPP	O
is	O
currently	O
in	O
practice	O
the	O
fastest	O
known	O
algorithm	O
for	O
testing	O
the	O
primality	O
of	O
general	O
numbers	O
,	O
but	O
the	O
worst-case	B-General_Concept
execution	I-General_Concept
time	I-General_Concept
is	O
not	O
known	O
.	O
</s>
<s>
ECPP	O
works	O
the	O
same	O
way	O
as	O
most	O
other	O
primality	B-Algorithm
tests	I-Algorithm
do	O
,	O
finding	O
a	O
group	O
and	O
showing	O
its	O
size	O
is	O
such	O
that	O
is	O
prime	O
.	O
</s>
<s>
attempts	O
to	O
verify	O
the	O
certificate	B-Algorithm
.	O
</s>
<s>
The	O
step	O
that	O
takes	O
the	O
most	O
CPU	B-Device
time	O
is	O
the	O
certificate	B-Algorithm
generation	O
,	O
because	O
factoring	O
over	O
a	O
class	O
field	O
must	O
be	O
performed	O
.	O
</s>
<s>
The	O
certificate	B-Algorithm
can	O
be	O
verified	O
quickly	O
,	O
allowing	O
a	O
check	O
of	O
operation	O
to	O
take	O
very	O
little	O
time	O
.	O
</s>
<s>
We	O
will	O
now	O
state	O
a	O
proposition	O
on	O
which	O
to	O
base	O
our	O
test	O
,	O
which	O
is	O
analogous	O
to	O
the	O
Pocklington	O
criterion	O
,	O
and	O
gives	O
rise	O
to	O
the	O
Goldwasser	O
–	O
Kilian	O
–	O
Atkin	O
form	O
of	O
the	O
elliptic	B-Algorithm
curve	I-Algorithm
primality	I-Algorithm
test	I-Algorithm
.	O
</s>
<s>
If	O
we	O
can	O
write	O
m	O
in	O
the	O
form	O
where	O
is	O
a	O
small	O
integer	O
and	O
q	O
a	O
large	O
probable	O
prime	O
(	O
a	O
number	O
that	O
passes	O
a	O
probabilistic	O
primality	B-Algorithm
test	I-Algorithm
,	O
for	O
example	O
)	O
,	O
then	O
we	O
do	O
not	O
discard	O
E	O
.	O
Otherwise	O
,	O
we	O
discard	O
our	O
curve	O
and	O
randomly	O
select	O
another	O
triple	O
(	O
a	O
,	O
x	O
,	O
y	O
)	O
to	O
start	O
over	O
.	O
</s>
<s>
This	O
leads	O
to	O
a	O
nested	O
certificate	B-Algorithm
where	O
at	O
each	O
level	O
we	O
have	O
an	O
elliptic	O
curve	O
E	O
,	O
an	O
m	O
and	O
the	O
prime	O
in	O
doubt	O
,	O
q	O
.	O
</s>
<s>
Also	O
,	O
if	O
our	O
N	O
is	O
of	O
length	O
k	O
,	O
then	O
the	O
algorithm	O
creates	O
a	O
certificate	B-Algorithm
of	O
size	O
that	O
can	O
be	O
verified	O
in	O
.	O
</s>
<s>
For	O
some	O
forms	O
of	O
numbers	O
,	O
it	O
is	O
possible	O
to	O
find	O
'	O
short-cuts	O
'	O
to	O
a	O
primality	B-Algorithm
proof	I-Algorithm
.	O
</s>
<s>
There	O
has	O
been	O
a	O
method	O
in	O
use	O
for	O
some	O
time	O
to	O
verify	O
primality	O
of	O
Mersenne	O
numbers	O
,	O
known	O
as	O
the	O
Lucas	B-Algorithm
–	I-Algorithm
Lehmer	I-Algorithm
test	I-Algorithm
.	O
</s>
<s>
The	O
following	O
method	O
is	O
drawn	O
from	O
the	O
paper	O
Primality	B-Algorithm
Test	I-Algorithm
for	O
using	O
Elliptic	O
Curves	O
,	O
by	O
Yu	O
Tsumura	O
.	O
</s>
