<s>
In	O
cryptography	O
,	O
a	O
ring	B-Algorithm
signature	I-Algorithm
is	O
a	O
type	O
of	O
digital	O
signature	O
that	O
can	O
be	O
performed	O
by	O
any	O
member	O
of	O
a	O
set	O
of	O
users	O
that	O
each	O
have	O
keys	O
.	O
</s>
<s>
Therefore	O
,	O
a	O
message	O
signed	O
with	O
a	O
ring	B-Algorithm
signature	I-Algorithm
is	O
endorsed	O
by	O
someone	O
in	O
a	O
particular	O
set	O
of	O
people	O
.	O
</s>
<s>
One	O
of	O
the	O
security	O
properties	O
of	O
a	O
ring	B-Algorithm
signature	I-Algorithm
is	O
that	O
it	O
should	O
be	O
computationally	O
infeasible	O
to	O
determine	O
which	O
of	O
the	O
set	O
's	O
members	O
 '	O
keys	O
was	O
used	O
to	O
produce	O
the	O
signature	O
.	O
</s>
<s>
Ring	B-Algorithm
signatures	I-Algorithm
are	O
similar	O
to	O
group	O
signatures	O
but	O
differ	O
in	O
two	O
key	O
ways	O
:	O
first	O
,	O
there	O
is	O
no	O
way	O
to	O
revoke	O
the	O
anonymity	O
of	O
an	O
individual	O
signature	O
;	O
and	O
second	O
,	O
any	O
set	O
of	O
users	O
can	O
be	O
used	O
as	O
a	O
signing	O
set	O
without	O
additional	O
setup	O
.	O
</s>
<s>
Ring	B-Algorithm
signatures	I-Algorithm
were	O
invented	O
by	O
Ron	O
Rivest	O
,	O
Adi	O
Shamir	O
,	O
and	O
Yael	O
Tauman	O
Kalai	O
,	O
and	O
introduced	O
at	O
ASIACRYPT	O
in	O
2001	O
.	O
</s>
<s>
The	O
name	O
,	O
ring	B-Algorithm
signature	I-Algorithm
,	O
comes	O
from	O
the	O
ring-like	O
structure	O
of	O
the	O
signature	O
algorithm	O
.	O
</s>
<s>
Party	O
i	O
can	O
compute	O
a	O
ring	B-Algorithm
signature	I-Algorithm
σ	O
on	O
a	O
message	O
m	O
,	O
on	O
input	O
(	O
m	O
,	O
Si	O
,	O
P1	O
,	O
...	O
,	O
Pn	O
)	O
.	O
</s>
<s>
Anyone	O
can	O
check	O
the	O
validity	O
of	O
a	O
ring	B-Algorithm
signature	I-Algorithm
given	O
σ	O
,	O
m	O
,	O
and	O
the	O
public	O
keys	O
involved	O
,	O
P1	O
,	O
...	O
,	O
Pn	O
.	O
</s>
<s>
If	O
a	O
ring	B-Algorithm
signature	I-Algorithm
is	O
properly	O
computed	O
,	O
it	O
should	O
pass	O
the	O
check	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
it	O
should	O
be	O
hard	O
for	O
anyone	O
to	O
create	O
a	O
valid	O
ring	B-Algorithm
signature	I-Algorithm
on	O
any	O
message	O
for	O
any	O
set	O
without	O
knowing	O
any	O
of	O
the	O
private	O
keys	O
for	O
that	O
set	O
.	O
</s>
<s>
In	O
the	O
original	O
paper	O
,	O
Rivest	O
,	O
Shamir	O
,	O
and	O
Tauman	O
described	O
ring	B-Algorithm
signatures	I-Algorithm
as	O
a	O
way	O
to	O
leak	O
a	O
secret	O
.	O
</s>
<s>
For	O
instance	O
,	O
a	O
ring	B-Algorithm
signature	I-Algorithm
could	O
be	O
used	O
to	O
provide	O
an	O
anonymous	O
signature	O
from	O
"	O
a	O
high-ranking	O
White	B-Application
House	I-Application
official	O
"	O
,	O
without	O
revealing	O
which	O
official	O
signed	O
the	O
message	O
.	O
</s>
<s>
Ring	B-Algorithm
signatures	I-Algorithm
are	O
right	O
for	O
this	O
application	O
because	O
the	O
anonymity	O
of	O
a	O
ring	B-Algorithm
signature	I-Algorithm
cannot	O
be	O
revoked	O
,	O
and	O
because	O
the	O
group	O
for	O
a	O
ring	B-Algorithm
signature	I-Algorithm
can	O
be	O
improvised	O
.	O
</s>
<s>
Here	O
the	O
sender	O
and	O
the	O
recipient	O
of	O
a	O
message	O
form	O
a	O
group	O
for	O
the	O
ring	B-Algorithm
signature	I-Algorithm
,	O
then	O
the	O
signature	O
is	O
valid	O
to	O
the	O
recipient	O
,	O
but	O
anyone	O
else	O
will	O
be	O
unsure	O
whether	O
the	O
recipient	O
or	O
the	O
sender	O
was	O
the	O
actual	O
signer	O
.	O
</s>
<s>
Threshold	O
ring	B-Algorithm
signatures	I-Algorithm
Unlike	O
standard	O
"	O
t-out-of-n	O
"	O
threshold	O
signature	O
,	O
where	O
t	O
of	O
n	O
users	O
should	O
collaborate	O
to	O
decrypt	O
a	O
message	O
,	O
this	O
variant	O
of	O
a	O
ring	B-Algorithm
signature	I-Algorithm
requires	O
t	O
users	O
to	O
cooperate	O
in	O
the	O
signing	O
protocol	B-Protocol
.	O
</s>
<s>
Linkable	O
ring	B-Algorithm
signatures	I-Algorithm
The	O
property	O
of	O
linkability	O
allows	O
one	O
to	O
determine	O
whether	O
any	O
two	O
signatures	O
have	O
been	O
produced	O
by	O
the	O
same	O
member	O
(	O
under	O
the	O
same	O
private	O
key	O
)	O
.	O
</s>
<s>
Traceable	O
ring	B-Algorithm
signature	I-Algorithm
In	O
addition	O
to	O
the	O
previous	O
scheme	O
the	O
public	O
key	O
of	O
the	O
signer	O
is	O
revealed	O
(	O
if	O
they	O
issue	O
more	O
than	O
one	O
signatures	O
under	O
the	O
same	O
private	O
key	O
)	O
.	O
</s>
<s>
An	O
e-voting	O
system	O
can	O
be	O
implemented	O
using	O
this	O
protocol	B-Protocol
.	O
</s>
<s>
CryptoNote	B-Protocol
implements	O
ring	B-Algorithm
signature	I-Algorithm
scheme	O
by	O
Fujisaki	O
and	O
Suzuki	O
in	O
p2p	O
payments	O
to	O
achieve	O
sender	O
's	O
untraceability	O
.	O
</s>
<s>
The	O
original	O
paper	O
describes	O
an	O
RSA	B-Architecture
based	O
ring	B-Algorithm
signature	I-Algorithm
scheme	O
,	O
as	O
well	O
as	O
one	O
based	O
on	O
Rabin	B-Algorithm
signatures	I-Algorithm
.	O
</s>
<s>
an	O
RSA	B-Architecture
public	O
key	O
in	O
the	O
case	O
of	O
RSA	B-Architecture
based	O
ring	B-Algorithm
signatures	I-Algorithm
)	O
.	O
</s>
<s>
The	O
equation	O
is	O
based	O
on	O
a	O
symmetric	B-Algorithm
encryption	I-Algorithm
function	I-Algorithm
:	O
</s>
<s>
Under	O
the	O
assumptions	O
of	O
RSA	B-Architecture
,	O
this	O
implies	O
knowledge	O
of	O
at	O
least	O
one	O
of	O
the	O
inverses	O
of	O
the	O
trap	O
door	O
functions	O
(	O
i.e.	O
</s>
<s>
Generating	O
a	O
ring	B-Algorithm
signature	I-Algorithm
involves	O
six	O
steps	O
.	O
</s>
<s>
Calculate	O
the	O
key	O
,	O
using	O
a	O
cryptographic	B-Algorithm
hash	I-Algorithm
function	I-Algorithm
.	O
</s>
<s>
This	O
step	O
assumes	O
a	O
random	B-Application
oracle	I-Application
for	O
,	O
since	O
will	O
be	O
used	O
as	O
key	O
for	O
.	O
</s>
<s>
Calculate	O
the	O
symmetric	B-Algorithm
key	I-Algorithm
.	O
</s>
<s>
Here	O
is	O
a	O
Python	B-Language
implementation	O
of	O
the	O
original	O
paper	O
using	O
RSA	B-Architecture
.	O
</s>
<s>
The	O
CryptoNote	B-Protocol
technology	O
uses	O
ring	B-Algorithm
signatures	I-Algorithm
.	O
</s>
<s>
The	O
cryptocurrency	O
ShadowCash	O
uses	O
traceable	O
ring	B-Algorithm
signature	I-Algorithm
to	O
anonymize	O
the	O
sender	O
of	O
a	O
transaction	O
.	O
</s>
<s>
However	O
,	O
these	O
were	O
originally	O
implemented	O
incorrectly	O
,	O
resulting	O
in	O
a	O
partial	O
de-anonymization	O
of	O
ShadowCash	O
from	O
their	O
first	O
implementation	O
until	O
February	O
2016	O
by	O
Monero	B-Protocol
Research	O
Labs	O
researcher	O
,	O
Shen	O
Noether	O
.	O
</s>
<s>
Monero	B-Protocol
uses	O
ring	B-Algorithm
signatures	I-Algorithm
to	O
obfuscate	O
the	O
true	O
spend	O
in	O
a	O
transaction	O
.	O
</s>
<s>
A	O
ring	B-Algorithm
signature	I-Algorithm
makes	O
use	O
of	O
a	O
user	O
's	O
account	O
keys	O
and	O
a	O
number	O
of	O
public	O
keys	O
(	O
also	O
known	O
as	O
outputs	O
)	O
pulled	O
from	O
the	O
blockchain	O
using	O
a	O
triangular	O
distribution	O
method	O
.	O
</s>
<s>
So	O
,	O
ring	B-Algorithm
signatures	I-Algorithm
ensure	O
that	O
transaction	O
outputs	O
are	O
untraceable	O
.	O
</s>
<s>
Moreover	O
,	O
there	O
are	O
no	O
fungibility	O
issues	O
with	O
Monero	B-Protocol
given	O
that	O
every	O
transaction	O
output	O
has	O
plausible	O
deniability	O
(	O
e.g.	O
</s>
