<s>
Mental	B-Algorithm
poker	I-Algorithm
is	O
the	O
common	O
name	O
for	O
a	O
set	O
of	O
cryptographic	O
problems	O
that	O
concerns	O
playing	O
a	O
fair	O
game	O
over	O
distance	O
without	O
the	O
need	O
for	O
a	O
trusted	O
third	O
party	O
.	O
</s>
<s>
In	O
poker	O
,	O
this	O
could	O
translate	O
to	O
:	O
"	O
How	O
can	O
we	O
make	O
sure	O
no	O
player	O
is	O
stacking	O
the	O
deck	O
or	O
peeking	O
at	O
other	O
players	O
 '	O
cards	O
when	O
we	O
are	O
shuffling	B-Application
the	O
deck	O
ourselves	O
?	O
"	O
.	O
</s>
<s>
Several	O
protocols	O
for	O
doing	O
this	O
have	O
been	O
suggested	O
,	O
the	O
first	O
by	O
Adi	O
Shamir	O
,	O
Ron	O
Rivest	O
and	O
Len	O
Adleman	O
(	O
the	O
creators	O
of	O
the	O
RSA-encryption	O
protocol	O
)	O
.	O
</s>
<s>
This	O
protocol	O
was	O
the	O
first	O
example	O
of	O
two	O
parties	O
conducting	O
secure	B-Protocol
computation	I-Protocol
rather	O
than	O
secure	O
message	O
transmission	O
,	O
employing	O
cryptography	O
;	O
later	O
on	O
due	O
to	O
leaking	O
partial	O
information	O
in	O
the	O
original	O
protocol	O
,	O
this	O
led	O
to	O
the	O
definition	O
of	O
semantic	O
security	O
by	O
Shafi	O
Goldwasser	O
and	O
Silvio	O
Micali	O
.	O
</s>
<s>
The	O
concept	O
of	O
multi-player	O
mental	B-Algorithm
poker	I-Algorithm
was	O
introduced	O
in	O
Moti	O
Yung	O
's	O
1984	O
book	O
Cryptoprotocols	O
.	O
</s>
<s>
The	O
area	O
has	O
later	O
evolved	O
into	O
what	O
is	O
known	O
as	O
secure	B-Protocol
multi-party	I-Protocol
computation	I-Protocol
protocols	O
(	O
for	O
two	O
parties	O
,	O
and	O
multi	O
parties	O
as	O
well	O
)	O
.	O
</s>
<s>
One	O
possible	O
algorithm	O
for	O
shuffling	B-Application
cards	O
without	O
the	O
use	O
of	O
a	O
trusted	O
third	O
party	O
is	O
to	O
use	O
a	O
commutative	O
encryption	O
scheme	O
.	O
</s>
<s>
An	O
algorithm	O
for	O
shuffling	B-Application
cards	O
using	O
commutative	O
encryption	O
would	O
be	O
as	O
follows	O
:	O
</s>
<s>
Alice	O
shuffles	B-Application
the	O
cards	O
.	O
</s>
<s>
Bob	O
shuffles	B-Application
the	O
deck	O
.	O
</s>
<s>
(	O
Using	O
a	O
separate	O
key	O
for	O
each	O
card	O
even	O
in	O
the	O
initial	O
exchange	O
,	O
which	O
would	O
otherwise	O
make	B-Algorithm
this	I-Algorithm
scheme	I-Algorithm
secure	I-Algorithm
,	O
does	O
n't	O
work	O
since	O
the	O
cards	O
are	O
shuffled	O
before	O
they	O
're	O
returned	O
.	O
)	O
</s>
<s>
The	O
work	O
is	O
concerned	O
with	O
general-purpose	O
operations	O
(	O
masking	O
and	O
unmasking	O
cards	O
,	O
shuffling	B-Application
and	O
re-shuffling	O
,	O
inserting	O
a	O
card	O
into	O
a	O
stack	O
,	O
etc	O
.	O
)	O
</s>
<s>
The	O
correctness	O
of	O
operations	O
can	O
be	O
checked	O
by	O
using	O
zero-knowledge	B-Protocol
proofs	I-Protocol
,	O
so	O
that	O
players	O
do	O
not	O
need	O
to	O
reveal	O
their	O
strategy	O
to	O
verify	O
the	O
game	O
's	O
correctness	O
.	O
</s>
<s>
To	O
date	O
,	O
mental	B-Algorithm
poker	I-Algorithm
approaches	O
based	O
on	O
the	O
standard	O
Alice-Bob	O
protocol	O
(	O
above	O
)	O
do	O
not	O
offer	O
high	O
enough	O
performance	O
for	O
real-time	O
online	O
play	O
.	O
</s>
<s>
A	O
recent	O
paper	O
by	O
Golle	O
 [ GOL05 ] 	O
describes	O
a	O
mental	B-Algorithm
poker	I-Algorithm
protocol	O
that	O
achieves	O
significantly	O
higher	O
performance	O
by	O
exploiting	O
the	O
properties	O
of	O
the	O
poker	O
game	O
to	O
move	O
away	O
from	O
the	O
encrypt-shuffle	O
model	O
.	O
</s>
<s>
Rather	O
than	O
shuffle	B-Application
the	O
cards	O
and	O
then	O
deal	O
as	O
needed	O
,	O
with	O
the	O
new	O
approach	O
,	O
the	O
players	O
generate	O
(	O
encrypted	O
)	O
random	O
numbers	O
on	O
the	O
fly	O
,	O
which	O
are	O
used	O
to	O
select	O
the	O
next	O
card	O
.	O
</s>
<s>
The	O
Elgamal	B-Algorithm
encryption	I-Algorithm
scheme	O
is	O
just	O
one	O
example	O
of	O
a	O
well-known	O
system	O
with	O
these	O
properties	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
this	O
scheme	O
turns	O
out	O
to	O
be	O
2-4	O
times	O
faster	O
(	O
as	O
measured	O
by	O
the	O
total	O
number	O
of	O
modular	O
exponentiations	O
)	O
than	O
the	O
best-known	O
protocol	O
 [ JAK99 ] 	O
that	O
does	O
full	O
shuffling	B-Application
using	O
mix-networks	B-Protocol
.	O
</s>
<s>
Note	O
that	O
the	O
random	B-Error_Name
number	I-Error_Name
generation	I-Error_Name
is	O
secure	O
as	O
long	O
as	O
any	O
one	O
player	O
is	O
generating	O
valid	O
random	O
numbers	O
.	O
</s>
<s>
Any	O
mental	B-Algorithm
poker	I-Algorithm
protocol	O
that	O
relies	O
on	O
the	O
players	O
to	O
perform	O
the	O
encryption	O
is	O
bound	O
by	O
the	O
requirement	O
that	O
every	O
player	O
encrypt	O
every	O
card	O
that	O
is	O
dealt	O
.	O
</s>
<s>
The	O
protocol	O
for	O
choosing	O
cards	O
without	O
shuffling	B-Application
may	O
be	O
adapted	O
so	O
that	O
the	O
encryption	O
is	O
handled	O
by	O
two	O
or	O
more	O
servers	O
.	O
</s>
<s>
Servers	O
S1	O
and	O
S2	O
encrypt	O
and	O
shuffle	B-Application
a	O
deck	O
of	O
cards	O
,	O
and	O
publish	O
a	O
non-malleable	O
commitment	O
to	O
some	O
permutation	B-Algorithm
of	O
encrypted	O
cards	O
to	O
the	O
players	O
.	O
</s>
<s>
The	O
random	O
number	O
is	O
used	O
as	O
an	O
index	O
into	O
the	O
random	O
permutation	B-Algorithm
,	O
the	O
appropriate	O
player	O
gains	O
"	O
ownership	O
"	O
of	O
the	O
specified	O
card	O
,	O
and	O
the	O
servers	O
send	O
that	O
player	O
the	O
keys	O
to	O
read	O
the	O
card	O
's	O
value	O
.	O
</s>
