<s>
The	O
Cramer	O
–	O
Shoup	O
system	O
is	O
an	O
asymmetric	B-Application
key	I-Application
encryption	I-Application
algorithm	I-Application
,	O
and	O
was	O
the	O
first	O
efficient	O
scheme	O
proven	O
to	O
be	O
secure	O
against	O
adaptive	O
chosen	O
ciphertext	O
attack	O
using	O
standard	O
cryptographic	O
assumptions	O
.	O
</s>
<s>
Developed	O
by	O
Ronald	O
Cramer	O
and	O
Victor	O
Shoup	O
in	O
1998	O
,	O
it	O
is	O
an	O
extension	O
of	O
the	O
ElGamal	B-Algorithm
cryptosystem	I-Algorithm
.	O
</s>
<s>
In	O
contrast	O
to	O
ElGamal	B-Algorithm
,	O
which	O
is	O
extremely	O
malleable	O
,	O
Cramer	O
–	O
Shoup	O
adds	O
other	O
elements	O
to	O
ensure	O
non-malleability	O
even	O
against	O
a	O
resourceful	O
attacker	O
.	O
</s>
<s>
This	O
non-malleability	O
is	O
achieved	O
through	O
the	O
use	O
of	O
a	O
universal	B-Algorithm
one-way	I-Algorithm
hash	I-Algorithm
function	I-Algorithm
and	O
additional	O
computations	O
,	O
resulting	O
in	O
a	O
ciphertext	O
which	O
is	O
twice	O
as	O
large	O
as	O
in	O
ElGamal	B-Algorithm
.	O
</s>
<s>
This	O
security	O
definition	O
is	O
currently	O
the	O
strongest	O
definition	O
known	O
for	O
a	O
public	B-Application
key	I-Application
cryptosystem	I-Application
:	O
it	O
assumes	O
that	O
the	O
attacker	O
has	O
access	O
to	O
a	O
decryption	O
oracle	O
which	O
will	O
decrypt	O
any	O
ciphertext	O
using	O
the	O
scheme	O
's	O
secret	O
decryption	O
key	O
.	O
</s>
<s>
This	O
began	O
to	O
change	O
during	O
the	O
late	O
1990s	O
,	O
particularly	O
when	O
Daniel	O
Bleichenbacher	O
demonstrated	O
a	O
practical	O
adaptive	O
chosen	O
ciphertext	O
attack	O
against	O
SSL	B-Protocol
servers	O
using	O
a	O
form	O
of	O
RSA	B-Architecture
encryption	I-Architecture
.	O
</s>
<s>
These	O
techniques	O
are	O
secure	O
under	O
a	O
standard	O
set	O
of	O
cryptographic	O
assumptions	O
(	O
without	O
random	B-Application
oracles	I-Application
)	O
,	O
however	O
they	O
rely	O
on	O
complex	O
zero-knowledge	B-Protocol
proof	I-Protocol
techniques	O
,	O
and	O
are	O
inefficient	O
in	O
terms	O
of	O
computational	O
cost	O
and	O
ciphertext	O
size	O
.	O
</s>
<s>
A	O
variety	O
of	O
other	O
approaches	O
,	O
including	O
Bellare/Rogaway	O
'	O
s	O
OAEP	B-Algorithm
and	O
Fujisaki	O
–	O
Okamoto	O
achieve	O
efficient	O
constructions	O
using	O
a	O
mathematical	O
abstraction	O
known	O
as	O
a	O
random	B-Application
oracle	I-Application
.	O
</s>
<s>
Unfortunately	O
,	O
to	O
implement	O
these	O
schemes	O
in	O
practice	O
requires	O
the	O
substitution	O
of	O
some	O
practical	O
function	O
(	O
e.g.	O
,	O
a	O
cryptographic	B-Algorithm
hash	I-Algorithm
function	I-Algorithm
)	O
in	O
place	O
of	O
the	O
random	B-Application
oracle	I-Application
.	O
</s>
<s>
Alice	O
publishes	O
,	O
along	O
with	O
the	O
description	O
of	O
,	O
as	O
her	O
public	B-Application
key	I-Application
.	O
</s>
<s>
To	O
encrypt	O
a	O
message	O
to	O
Alice	O
under	O
her	O
public	B-Application
key	I-Application
,	O
</s>
<s>
,	O
where	O
H( )	O
is	O
a	O
universal	B-Algorithm
one-way	I-Algorithm
hash	I-Algorithm
function	I-Algorithm
(	O
or	O
a	O
collision-resistant	O
cryptographic	B-Algorithm
hash	I-Algorithm
function	I-Algorithm
,	O
which	O
is	O
a	O
stronger	O
requirement	O
)	O
.	O
</s>
