<s>
The	O
Paillier	B-Algorithm
cryptosystem	I-Algorithm
,	O
invented	O
by	O
and	O
named	O
after	O
Pascal	B-Algorithm
Paillier	I-Algorithm
in	O
1999	O
,	O
is	O
a	O
probabilistic	O
asymmetric	B-Application
algorithm	I-Application
for	O
public	B-Application
key	I-Application
cryptography	I-Application
.	O
</s>
<s>
A	O
notable	O
feature	O
of	O
the	O
Paillier	B-Algorithm
cryptosystem	I-Algorithm
is	O
its	O
homomorphic	O
properties	O
along	O
with	O
its	O
non-deterministic	O
encryption	O
(	O
see	O
Electronic	O
voting	O
in	O
Applications	O
for	O
usage	O
)	O
.	O
</s>
<s>
However	O
,	O
given	O
the	O
Paillier	B-Algorithm
encryptions	I-Algorithm
of	O
two	O
messages	O
there	O
is	O
no	O
known	O
way	O
to	O
compute	O
an	O
encryption	O
of	O
the	O
product	O
of	O
these	O
messages	O
without	O
knowing	O
the	O
private	B-Application
key	I-Application
.	O
</s>
<s>
Paillier	B-Algorithm
cryptosystem	I-Algorithm
exploits	O
the	O
fact	O
that	O
certain	O
discrete	O
logarithms	O
can	O
be	O
computed	O
easily	O
.	O
</s>
<s>
Paillier	B-Algorithm
and	O
Pointcheval	O
however	O
went	O
on	O
to	O
propose	O
an	O
improved	O
cryptosystem	O
that	O
incorporates	O
the	O
combined	O
hashing	O
of	O
message	O
m	O
with	O
random	O
r	O
.	O
Similar	O
in	O
intent	O
to	O
the	O
Cramer	B-Algorithm
–	I-Algorithm
Shoup	I-Algorithm
cryptosystem	I-Algorithm
,	O
the	O
hashing	O
prevents	O
an	O
attacker	O
,	O
given	O
only	O
c	O
,	O
from	O
being	O
able	O
to	O
change	O
m	O
in	O
a	O
meaningful	O
way	O
.	O
</s>
<s>
Through	O
this	O
adaptation	O
the	O
improved	O
scheme	O
can	O
be	O
shown	O
to	O
be	O
IND-CCA2	O
secure	O
in	O
the	O
random	B-Application
oracle	I-Application
model	I-Application
.	O
</s>
<s>
The	O
homomorphic	O
property	O
of	O
Paillier	B-Algorithm
cryptosystem	I-Algorithm
is	O
sometime	O
used	O
to	O
build	O
Threshold	O
ECDSA	O
signature	O
.	O
</s>
