<s>
The	O
Blum	O
–	O
Goldwasser	O
(	O
BG	O
)	O
cryptosystem	O
is	O
an	O
asymmetric	B-Application
key	I-Application
encryption	I-Application
algorithm	I-Application
proposed	O
by	O
Manuel	O
Blum	O
and	O
Shafi	O
Goldwasser	O
in	O
1984	O
.	O
</s>
<s>
Blum	O
–	O
Goldwasser	O
is	O
a	O
probabilistic	O
,	O
semantically	O
secure	O
cryptosystem	O
with	O
a	O
constant-size	O
ciphertext	B-General_Concept
expansion	I-General_Concept
.	O
</s>
<s>
The	O
encryption	O
algorithm	O
implements	O
an	O
XOR-based	O
stream	B-Algorithm
cipher	I-Algorithm
using	O
the	O
Blum-Blum-Shub	B-Algorithm
(	O
BBS	O
)	O
pseudo-random	O
number	O
generator	O
to	O
generate	O
the	O
keystream	B-Algorithm
.	O
</s>
<s>
Decryption	O
is	O
accomplished	O
by	O
manipulating	O
the	O
final	O
state	O
of	O
the	O
BBS	O
generator	O
using	O
the	O
private	B-Application
key	I-Application
,	O
in	O
order	O
to	O
find	O
the	O
initial	O
seed	O
and	O
reconstruct	O
the	O
keystream	B-Algorithm
.	O
</s>
<s>
BG	O
has	O
multiple	O
advantages	O
over	O
earlier	O
probabilistic	O
encryption	O
schemes	O
such	O
as	O
the	O
Goldwasser	B-Algorithm
–	I-Algorithm
Micali	I-Algorithm
cryptosystem	I-Algorithm
.	O
</s>
<s>
Secondly	O
,	O
BG	O
is	O
efficient	O
in	O
terms	O
of	O
storage	O
,	O
inducing	O
a	O
constant-size	O
ciphertext	B-General_Concept
expansion	I-General_Concept
regardless	O
of	O
message	O
length	O
.	O
</s>
<s>
The	O
Blum	B-Algorithm
–	I-Algorithm
Goldwasser	I-Algorithm
cryptosystem	I-Algorithm
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
public	O
and	O
private	B-Application
keys	I-Application
are	O
generated	O
as	O
follows	O
:	O
</s>
<s>
Then	O
is	O
the	O
public	B-Application
key	I-Application
and	O
the	O
pair	O
is	O
the	O
private	B-Application
key	I-Application
.	O
</s>
<s>
A	O
message	O
is	O
encrypted	O
with	O
the	O
public	B-Application
key	I-Application
as	O
follows	O
:	O
</s>
<s>
An	O
encrypted	O
message	O
can	O
be	O
decrypted	O
with	O
the	O
private	B-Application
key	I-Application
as	O
follows	O
:	O
</s>
<s>
The	O
Blum	O
–	O
Goldwasser	O
scheme	O
is	O
semantically-secure	O
based	O
on	O
the	O
hardness	O
of	O
predicting	O
the	O
keystream	B-Algorithm
bits	O
given	O
only	O
the	O
final	O
BBS	O
state	O
and	O
the	O
public	B-Application
key	I-Application
.	O
</s>
