<s>
The	O
Goldwasser	O
–	O
Micali	O
(	O
GM	O
)	O
cryptosystem	O
is	O
an	O
asymmetric	B-Application
key	I-Application
encryption	I-Application
algorithm	I-Application
developed	O
by	O
Shafi	O
Goldwasser	O
and	O
Silvio	O
Micali	O
in	O
1982	O
.	O
</s>
<s>
GM	O
has	O
the	O
distinction	O
of	O
being	O
the	O
first	O
probabilistic	O
public-key	B-Application
encryption	I-Application
scheme	O
which	O
is	O
provably	O
secure	O
under	O
standard	O
cryptographic	O
assumptions	O
.	O
</s>
<s>
Because	O
Goldwasser	O
–	O
Micali	O
produces	O
a	O
value	O
of	O
size	O
approximately	O
|N|	O
to	O
encrypt	O
every	O
single	O
bit	O
of	O
a	O
plaintext	O
,	O
GM	O
encryption	O
results	O
in	O
substantial	O
ciphertext	B-General_Concept
expansion	I-General_Concept
.	O
</s>
<s>
Thus	O
,	O
the	O
scheme	O
serves	O
mainly	O
as	O
a	O
proof	O
of	O
concept	O
,	O
and	O
more	O
efficient	O
provably-secure	O
schemes	O
such	O
as	O
ElGamal	B-Algorithm
have	O
been	O
developed	O
since	O
.	O
</s>
<s>
Goldwasser	O
–	O
Micali	O
consists	O
of	O
three	O
algorithms	O
:	O
a	O
probabilistic	O
key	O
generation	O
algorithm	O
which	O
produces	O
a	O
public	O
and	O
a	O
private	B-Application
key	I-Application
,	O
a	O
probabilistic	O
encryption	O
algorithm	O
,	O
and	O
a	O
deterministic	O
decryption	O
algorithm	O
.	O
</s>
<s>
The	O
modulus	O
used	O
in	O
GM	O
encryption	O
is	O
generated	O
in	O
the	O
same	O
manner	O
as	O
in	O
the	O
RSA	B-Architecture
cryptosystem	I-Architecture
.	O
</s>
<s>
(	O
See	O
RSA	B-Architecture
,	O
key	O
generation	O
for	O
details	O
.	O
)	O
</s>
<s>
The	O
public	B-Application
key	I-Application
consists	O
of	O
(	O
x	O
,	O
N	O
)	O
.	O
</s>
<s>
There	O
is	O
a	O
simple	O
reduction	B-Algorithm
from	O
breaking	O
this	O
cryptosystem	O
to	O
the	O
problem	O
of	O
determining	O
whether	O
a	O
random	O
value	O
modulo	O
N	O
with	O
Jacobi	O
symbol	O
+1	O
is	O
a	O
quadratic	O
residue	O
.	O
</s>
<s>
then	O
to	O
determine	O
if	O
a	O
given	O
value	O
x	O
is	O
a	O
quadratic	O
residue	O
modulo	O
N	O
,	O
we	O
test	O
A	O
to	O
see	O
if	O
it	O
can	O
break	O
the	O
cryptosystem	O
using	O
(	O
x	O
,	O
N	O
)	O
as	O
a	O
public	B-Application
key	I-Application
.	O
</s>
<s>
Furthermore	O
,	O
this	O
problem	O
is	O
random	O
self-reducible	O
,	O
which	O
ensures	O
that	O
for	O
a	O
given	O
N	O
,	O
every	O
public	B-Application
key	I-Application
is	O
just	O
as	O
secure	O
as	O
every	O
other	O
public	B-Application
key	I-Application
.	O
</s>
