<s>
Algebraic-group	B-Algorithm
factorisation	I-Algorithm
algorithms	I-Algorithm
are	O
algorithms	O
for	O
factoring	O
an	O
integer	O
N	O
by	O
working	O
in	O
an	O
algebraic	O
group	O
defined	O
modulo	O
N	O
whose	O
group	O
structure	O
is	O
the	O
direct	O
sum	O
of	O
the	O
'	O
reduced	O
groups	O
 '	O
obtained	O
by	O
performing	O
the	O
equations	O
defining	O
the	O
group	O
arithmetic	O
modulo	O
the	O
unknown	O
prime	O
factors	O
p1	O
,	O
p2	O
,	O
...	O
By	O
the	O
Chinese	O
remainder	O
theorem	O
,	O
arithmetic	O
modulo	O
N	O
corresponds	O
to	O
arithmetic	O
in	O
all	O
the	O
reduced	O
groups	O
simultaneously	O
.	O
</s>
<s>
If	O
t	O
is	O
a	O
quadratic	O
residue	O
,	O
the	O
p+1	B-Algorithm
method	O
degenerates	O
to	O
a	O
slower	O
form	O
of	O
the	O
p1	O
method	O
.	O
</s>
<s>
If	O
the	O
algebraic	O
group	O
is	O
an	O
elliptic	O
curve	O
,	O
the	O
one-sided	O
identities	O
can	O
be	O
recognised	O
by	O
failure	O
of	O
inversion	O
in	O
the	O
elliptic-curve	O
point	O
addition	O
procedure	O
,	O
and	O
the	O
result	O
is	O
the	O
elliptic	B-Algorithm
curve	I-Algorithm
method	I-Algorithm
;	O
Hasse	O
's	O
theorem	O
states	O
that	O
the	O
number	O
of	O
points	O
on	O
an	O
elliptic	O
curve	O
modulo	O
p	O
is	O
always	O
within	O
of	O
p	O
.	O
</s>
<s>
All	O
three	O
of	O
the	O
above	O
algebraic	O
groups	O
are	O
used	O
by	O
the	O
package	O
,	O
which	O
includes	O
efficient	O
implementations	O
of	O
the	O
two-stage	O
procedure	O
,	O
and	O
an	O
implementation	O
of	O
the	O
PRAC	O
group-exponentiation	O
algorithm	O
which	O
is	O
rather	O
more	O
efficient	O
than	O
the	O
standard	O
binary	B-Algorithm
exponentiation	I-Algorithm
approach	O
.	O
</s>
