<s>
based	O
on	O
a	O
trapdoor	O
function	O
whose	O
security	O
,	O
like	O
that	O
of	O
RSA	B-Architecture
,	O
is	O
related	O
to	O
the	O
difficulty	O
of	O
integer	O
factorization	O
.	O
</s>
<s>
The	O
Rabin	O
trapdoor	O
function	O
has	O
the	O
advantage	O
that	O
inverting	O
it	O
has	O
been	O
mathematically	O
proven	O
to	O
be	O
as	O
hard	O
as	O
factoring	O
integers	O
,	O
while	O
there	O
is	O
no	O
such	O
proof	O
known	O
for	O
the	O
RSA	B-Architecture
trapdoor	O
function	O
.	O
</s>
<s>
Public-key	B-Application
encryption	I-Application
schemes	O
based	O
on	O
the	O
Rabin	O
trapdoor	O
function	O
are	O
used	O
mainly	O
for	O
examples	O
in	O
textbooks	O
.	O
</s>
<s>
In	O
contrast	O
,	O
RSA	B-Architecture
is	O
the	O
basis	O
of	O
standard	O
public-key	B-Application
encryption	I-Application
schemes	O
such	O
as	O
RSAES-PKCS1-v1_5	O
and	O
RSAES-OAEP	B-Algorithm
that	O
are	O
used	O
widely	O
in	O
practice	O
.	O
</s>
<s>
The	O
Rabin	O
trapdoor	O
function	O
was	O
first	O
published	O
as	O
part	O
of	O
the	O
Rabin	B-Algorithm
signature	I-Algorithm
scheme	O
in	O
1978	O
by	O
Michael	O
O	O
.	O
Rabin	O
.	O
</s>
<s>
The	O
Rabin	B-Algorithm
signature	I-Algorithm
scheme	O
was	O
the	O
first	O
digital	O
signature	O
scheme	O
where	O
forging	O
a	O
signature	O
could	O
be	O
proven	O
to	O
be	O
as	O
hard	O
as	O
factoring	O
.	O
</s>
<s>
The	O
trapdoor	O
function	O
was	O
later	O
repurposed	O
in	O
textbooks	O
as	O
an	O
example	O
of	O
a	O
public-key	B-Application
encryption	I-Application
scheme	O
,	O
</s>
<s>
which	O
came	O
to	O
be	O
known	O
as	O
the	O
Rabin	B-Algorithm
cryptosystem	I-Algorithm
even	O
though	O
Rabin	O
never	O
published	O
it	O
as	O
an	O
encryption	O
scheme	O
.	O
</s>
<s>
Like	O
all	O
asymmetric	B-Application
cryptosystems	I-Application
,	O
the	O
Rabin	O
system	O
uses	O
a	O
key	B-Application
pair	I-Application
:	O
a	O
public	B-Application
key	I-Application
for	O
encryption	O
and	O
a	O
private	B-Application
key	I-Application
for	O
decryption	O
.	O
</s>
<s>
The	O
public	B-Application
key	I-Application
is	O
published	O
for	O
anyone	O
to	O
use	O
,	O
while	O
the	O
private	B-Application
key	I-Application
remains	O
known	O
only	O
to	O
the	O
recipient	O
of	O
the	O
message	O
.	O
</s>
<s>
The	O
keys	O
for	O
the	O
Rabin	B-Algorithm
cryptosystem	I-Algorithm
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>
The	O
Rabin	B-Algorithm
cryptosystem	I-Algorithm
can	O
be	O
used	O
to	O
create	O
and	O
verify	O
digital	O
signatures	O
.	O
</s>
<s>
Creating	O
a	O
signature	O
requires	O
the	O
private	B-Application
key	I-Application
.	O
</s>
<s>
Verifying	O
a	O
signature	O
requires	O
the	O
public	B-Application
key	I-Application
.	O
</s>
<s>
A	O
message	O
can	O
be	O
signed	O
with	O
a	O
private	B-Application
key	I-Application
as	O
follows	O
.	O
</s>
<s>
Use	O
a	O
cryptographic	B-Algorithm
hash	I-Algorithm
function	I-Algorithm
to	O
compute	O
,	O
where	O
the	O
double-bar	O
denotes	O
concatenation	O
.	O
</s>
<s>
Find	O
a	O
square	O
root	O
of	O
using	O
the	O
private	B-Application
key	I-Application
.	O
</s>
<s>
A	O
signature	O
for	O
a	O
message	O
can	O
be	O
verified	O
using	O
the	O
public	B-Application
key	I-Application
as	O
follows	O
.	O
</s>
<s>
This	O
is	O
the	O
major	O
disadvantage	O
of	O
the	O
Rabin	B-Algorithm
cryptosystem	I-Algorithm
and	O
one	O
of	O
the	O
factors	O
which	O
have	O
prevented	O
it	O
from	O
finding	O
widespread	O
practical	O
use	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
choose	O
plaintexts	O
with	O
special	O
structures	O
,	O
or	O
to	O
add	O
padding	B-Algorithm
,	O
to	O
eliminate	O
this	O
problem	O
.	O
</s>
<s>
These	O
restrictions	O
make	O
the	O
squaring	O
function	O
into	O
a	O
trapdoor	O
permutation	B-Algorithm
,	O
eliminating	O
the	O
ambiguity	O
.	O
</s>
<s>
This	O
is	O
more	O
efficient	O
than	O
RSA	B-Architecture
,	O
which	O
requires	O
the	O
calculation	O
of	O
at	O
least	O
a	O
cube	O
.	O
</s>
<s>
Here	O
the	O
efficiency	O
is	O
comparable	O
to	O
RSA	B-Architecture
.	O
</s>
<s>
Disambiguation	O
introduces	O
additional	O
computational	O
costs	O
,	O
and	O
is	O
what	O
has	O
prevented	O
the	O
Rabin	B-Algorithm
cryptosystem	I-Algorithm
from	O
finding	O
widespread	O
practical	O
use	O
.	O
</s>
<s>
Thus	O
,	O
Rabin	O
decryption	O
for	O
random	O
plaintext	O
is	O
at	O
least	O
as	O
hard	O
as	O
the	O
integer	O
factorization	O
problem	O
,	O
something	O
that	O
has	O
not	O
been	O
proven	O
for	O
RSA	B-Architecture
.	O
</s>
<s>
It	O
is	O
generally	O
believed	O
that	O
there	O
is	O
no	O
polynomial-time	O
algorithm	O
for	O
factoring	O
,	O
which	O
implies	O
that	O
there	O
is	O
no	O
efficient	O
algorithm	O
for	O
decrypting	O
a	O
random	O
Rabin-encrypted	O
value	O
without	O
the	O
private	B-Application
key	I-Application
.	O
</s>
<s>
The	O
Rabin	B-Algorithm
cryptosystem	I-Algorithm
does	O
not	O
provide	O
indistinguishability	O
against	O
chosen	O
plaintext	O
attacks	O
since	O
the	O
process	O
of	O
encryption	O
is	O
deterministic	O
.	O
</s>
<s>
The	O
Rabin	B-Algorithm
cryptosystem	I-Algorithm
is	O
insecure	O
against	O
a	O
chosen	O
ciphertext	O
attack	O
(	O
even	O
when	O
challenge	O
messages	O
are	O
chosen	O
uniformly	O
at	O
random	O
from	O
the	O
message	O
space	O
)	O
.	O
</s>
