<s>
The	O
Damgård	B-Algorithm
–	I-Algorithm
Jurik	I-Algorithm
cryptosystem	I-Algorithm
is	O
a	O
generalization	O
of	O
the	O
Paillier	B-Algorithm
cryptosystem	I-Algorithm
.	O
</s>
<s>
It	O
uses	O
computations	O
modulo	O
where	O
is	O
an	O
RSA	B-Architecture
modulus	O
and	O
a	O
(	O
positive	O
)	O
natural	O
number	O
.	O
</s>
<s>
Paillier	B-Algorithm
's	O
scheme	O
is	O
the	O
special	O
case	O
with	O
.	O
</s>
<s>
Like	O
Paillier	B-Algorithm
,	O
the	O
security	O
of	O
Damgård	O
–	O
Jurik	O
can	O
be	O
proven	O
under	O
the	O
decisional	O
composite	O
residuosity	O
assumption	O
.	O
</s>
<s>
For	O
instance	O
could	O
be	O
as	O
in	O
Paillier	B-Algorithm
's	O
original	O
scheme	O
.	O
</s>
<s>
Apply	O
a	O
recursive	O
version	O
of	O
the	O
Paillier	B-Algorithm
decryption	O
mechanism	O
to	O
obtain	O
.	O
</s>
<s>
At	O
the	O
cost	O
of	O
no	O
longer	O
containing	O
the	O
classical	O
Paillier	B-Algorithm
cryptosystem	I-Algorithm
as	O
an	O
instance	O
,	O
Damgård	O
–	O
Jurik	O
can	O
be	O
simplified	O
in	O
the	O
following	O
way	O
:	O
</s>
<s>
Using	O
recursive	O
Paillier	B-Algorithm
decryption	O
this	O
gives	O
us	O
directly	O
the	O
plaintext	O
m	O
.	O
</s>
