<s>
The	O
Karatsuba	B-Algorithm
algorithm	I-Algorithm
is	O
a	O
fast	B-Algorithm
multiplication	I-Algorithm
algorithm	O
.	O
</s>
<s>
It	O
is	O
a	O
divide-and-conquer	B-Algorithm
algorithm	I-Algorithm
that	O
reduces	O
the	O
multiplication	O
of	O
two	O
n-digit	O
numbers	O
to	O
three	O
multiplications	O
of	O
n/2	O
-digit	O
numbers	O
and	O
,	O
by	O
repeating	O
this	O
reduction	O
,	O
to	O
at	O
most	O
single-digit	O
multiplications	O
.	O
</s>
<s>
For	O
example	O
,	O
to	O
multiply	O
two	O
1024-digit	O
numbers	O
(	O
n	O
=	O
1024	O
=	O
210	O
)	O
,	O
the	O
traditional	O
algorithm	O
requires	O
(	O
210	O
)	O
2	O
=	O
1,048,576	O
single-digit	O
multiplications	O
,	O
whereas	O
the	O
Karatsuba	B-Algorithm
algorithm	I-Algorithm
requires	O
310	O
=	O
59,049	O
thus	O
being	O
~	O
17.758	O
times	O
faster	O
.	O
</s>
<s>
The	O
Karatsuba	B-Algorithm
algorithm	I-Algorithm
was	O
the	O
first	O
multiplication	B-Algorithm
algorithm	I-Algorithm
asymptotically	O
faster	O
than	O
the	O
quadratic	O
"	O
grade	O
school	O
"	O
algorithm	O
.	O
</s>
<s>
The	O
Toom	B-Algorithm
–	I-Algorithm
Cook	I-Algorithm
algorithm	I-Algorithm
(	O
1963	O
)	O
is	O
a	O
faster	O
generalization	O
of	O
Karatsuba	O
's	O
method	O
,	O
and	O
the	O
Schönhage	B-Algorithm
–	I-Algorithm
Strassen	I-Algorithm
algorithm	I-Algorithm
(	O
1971	O
)	O
is	O
even	O
faster	O
,	O
for	O
sufficiently	O
large	O
n	O
.	O
</s>
<s>
Andrey	O
Kolmogorov	O
conjectured	O
that	O
the	O
traditional	O
algorithm	O
was	O
asymptotically	B-General_Concept
optimal	I-General_Concept
,	O
meaning	O
that	O
any	O
algorithm	O
for	O
that	O
task	O
would	O
require	O
elementary	O
operations	O
.	O
</s>
<s>
The	O
article	O
had	O
been	O
written	O
by	O
Kolmogorov	O
and	O
contained	O
two	O
results	O
on	O
multiplication	O
,	O
Karatsuba	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
and	O
a	O
separate	O
result	O
by	O
Yuri	O
Ofman	O
;	O
it	O
listed	O
"	O
A	O
.	O
Karatsuba	O
and	O
Yu	O
.	O
</s>
<s>
The	O
basic	O
principle	O
of	O
Karatsuba	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
is	O
divide-and-conquer	B-Algorithm
,	O
using	O
a	O
formula	O
that	O
allows	O
one	O
to	O
compute	O
the	O
product	O
of	O
two	O
large	O
numbers	O
and	O
using	O
three	O
multiplications	O
of	O
smaller	O
numbers	O
,	O
each	O
with	O
about	O
half	O
as	O
many	O
digits	O
as	O
or	O
,	O
plus	O
some	O
additions	O
and	O
digit	O
shifts	O
.	O
</s>
<s>
This	O
basic	O
step	O
is	O
,	O
in	O
fact	O
,	O
a	O
generalization	O
of	O
a	O
similar	O
complex	O
multiplication	B-Algorithm
algorithm	I-Algorithm
,	O
where	O
the	O
imaginary	O
unit	O
is	O
replaced	O
by	O
a	O
power	O
of	O
the	O
base	O
.	O
</s>
<s>
Therefore	O
,	O
those	O
products	O
can	O
be	O
computed	O
by	O
recursive	O
calls	O
of	O
the	O
Karatsuba	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
for	O
some	O
constants	O
c	O
and	O
d	O
.	O
For	O
this	O
recurrence	O
relation	O
,	O
the	O
master	B-Algorithm
theorem	I-Algorithm
for	I-Algorithm
divide-and-conquer	I-Algorithm
recurrences	I-Algorithm
gives	O
the	O
asymptotic	O
bound	O
.	O
</s>
<s>
It	O
follows	O
that	O
,	O
for	O
sufficiently	O
large	O
n	O
,	O
Karatsuba	B-Algorithm
's	I-Algorithm
algorithm	I-Algorithm
will	O
perform	O
fewer	O
shifts	O
and	O
single-digit	O
additions	O
than	O
longhand	O
multiplication	O
,	O
even	O
though	O
its	O
basic	O
step	O
uses	O
more	O
additions	O
and	O
shifts	O
than	O
the	O
straightforward	O
formula	O
.	O
</s>
<s>
The	O
point	O
of	O
positive	O
return	O
depends	O
on	O
the	O
computer	B-Device
platform	I-Device
and	O
context	O
.	O
</s>
