<s>
The	O
Cayley	B-Algorithm
–	I-Algorithm
Purser	I-Algorithm
algorithm	I-Algorithm
was	O
a	O
public-key	B-Application
cryptography	I-Application
algorithm	O
published	O
in	O
early	O
1999	O
by	O
16-year-old	O
Irishwoman	O
Sarah	O
Flannery	O
,	O
based	O
on	O
an	O
unpublished	O
work	O
by	O
Michael	O
Purser	O
,	O
founder	O
of	O
Baltimore	O
Technologies	O
,	O
a	O
Dublin	O
data	O
security	O
company	O
.	O
</s>
<s>
It	O
has	O
since	O
been	O
found	O
to	O
be	O
flawed	O
as	O
a	O
public-key	B-Application
algorithm	O
,	O
but	O
was	O
the	O
subject	O
of	O
considerable	O
media	O
attention	O
.	O
</s>
<s>
During	O
a	O
work-experience	O
placement	O
with	O
Baltimore	O
Technologies	O
,	O
Flannery	O
was	O
shown	O
an	O
unpublished	O
paper	O
by	O
Michael	O
Purser	O
which	O
outlined	O
a	O
new	O
public-key	B-Application
cryptographic	O
scheme	O
using	O
non-commutative	O
multiplication	O
.	O
</s>
<s>
She	O
was	O
asked	O
to	O
write	O
an	O
implementation	O
of	O
this	O
scheme	O
in	O
Mathematica	B-Language
.	O
</s>
<s>
Before	O
this	O
placement	O
,	O
Flannery	O
had	O
attended	O
the	O
1998	O
ESAT	O
Young	O
Scientist	O
and	O
Technology	O
Exhibition	O
with	O
a	O
project	O
describing	O
already	O
existing	O
cryptographic	O
techniques	O
from	O
the	O
Caesar	O
cipher	O
to	O
RSA	B-Architecture
.	O
</s>
<s>
On	O
advice	O
from	O
her	O
mathematician	O
father	O
,	O
Flannery	O
decided	O
to	O
use	O
matrices	B-Architecture
to	O
implement	O
Purser	O
's	O
scheme	O
as	O
matrix	O
multiplication	O
has	O
the	O
necessary	O
property	O
of	O
being	O
non-commutative	O
.	O
</s>
<s>
As	O
the	O
resulting	O
algorithm	O
would	O
depend	O
on	O
multiplication	O
it	O
would	O
be	O
a	O
great	O
deal	O
faster	O
than	O
the	O
RSA	B-Architecture
algorithm	I-Architecture
which	O
uses	O
an	O
exponential	O
step	O
.	O
</s>
<s>
For	O
her	O
Intel	O
Science	O
Fair	O
project	O
Flannery	O
prepared	O
a	O
demonstration	O
where	O
the	O
same	O
plaintext	O
was	O
enciphered	O
using	O
both	O
RSA	B-Architecture
and	O
her	O
new	O
Cayley	B-Algorithm
–	I-Algorithm
Purser	I-Algorithm
algorithm	I-Algorithm
and	O
it	O
did	O
indeed	O
show	O
a	O
significant	O
time	O
improvement	O
.	O
</s>
<s>
Returning	O
to	O
the	O
ESAT	O
Young	O
Scientist	O
and	O
Technology	O
Exhibition	O
in	O
1999	O
,	O
Flannery	O
formalised	O
Cayley-Purser	B-Algorithm
'	O
s	O
runtime	O
and	O
analyzed	O
a	O
variety	O
of	O
known	O
attacks	O
,	O
none	O
of	O
which	O
were	O
determined	O
to	O
be	O
effective	O
.	O
</s>
<s>
Flannery	O
did	O
not	O
make	O
any	O
claims	O
that	O
the	O
Cayley	B-Algorithm
–	I-Algorithm
Purser	I-Algorithm
algorithm	I-Algorithm
would	O
replace	O
RSA	B-Architecture
,	O
knowing	O
that	O
any	O
new	O
cryptographic	O
system	O
would	O
need	O
to	O
stand	O
the	O
test	O
of	O
time	O
before	O
it	O
could	O
be	O
acknowledged	O
as	O
a	O
secure	O
system	O
.	O
</s>
<s>
Like	O
RSA	B-Architecture
,	O
Cayley-Purser	B-Algorithm
begins	O
by	O
generating	O
two	O
large	O
primes	O
p	O
and	O
q	O
and	O
their	O
product	O
n	O
,	O
a	O
semiprime	O
.	O
</s>
<s>
Next	O
,	O
consider	O
GL(2,n )	O
,	O
the	O
general	O
linear	O
group	O
of	O
2×2	O
matrices	B-Architecture
with	O
integer	O
elements	O
and	O
modular	O
arithmetic	O
mod	O
n	O
.	O
For	O
example	O
,	O
if	O
n	O
=	O
5	O
,	O
we	O
could	O
write	O
:	O
</s>
<s>
Let	O
and	O
be	O
two	O
such	O
matrices	B-Architecture
from	O
GL(2,n )	O
chosen	O
such	O
that	O
.	O
</s>
<s>
The	O
public	B-Application
key	I-Application
is	O
,	O
,	O
,	O
and	O
.	O
</s>
<s>
The	O
private	B-Application
key	I-Application
is	O
.	O
</s>
<s>
Then	O
,	O
to	O
encrypt	O
a	O
message	O
,	O
each	O
message	O
block	O
is	O
encoded	O
as	O
a	O
number	O
(	O
as	O
in	O
RSA	B-Architecture
)	O
and	O
they	O
are	O
placed	O
four	O
at	O
a	O
time	O
as	O
elements	O
of	O
a	O
plaintext	O
matrix	O
.	O
</s>
<s>
Recovering	O
the	O
private	B-Application
key	I-Application
from	O
is	O
computationally	O
infeasible	O
,	O
at	O
least	O
as	O
hard	O
as	O
finding	O
square	O
roots	O
mod	O
n	O
(	O
see	O
quadratic	O
residue	O
)	O
.	O
</s>
<s>
This	O
flaw	O
does	O
not	O
preclude	O
the	O
algorithm	O
's	O
use	O
as	O
a	O
mixed	O
private-key/public	O
-key	O
algorithm	O
,	O
if	O
the	O
sender	O
transmits	O
secretly	O
,	O
but	O
this	O
approach	O
presents	O
no	O
advantage	O
over	O
the	O
common	O
approach	O
of	O
transmitting	O
a	O
symmetric	B-Algorithm
encryption	I-Algorithm
key	O
using	O
a	O
public-key	B-Application
encryption	I-Application
scheme	O
and	O
then	O
switching	O
to	O
symmetric	B-Algorithm
encryption	I-Algorithm
,	O
which	O
is	O
faster	O
than	O
Cayley-Purser	B-Algorithm
.	O
</s>
