<s>
In	O
cryptography	O
,	O
a	O
Schnorr	B-Algorithm
signature	I-Algorithm
is	O
a	O
digital	O
signature	O
produced	O
by	O
the	O
Schnorr	B-Algorithm
signature	I-Algorithm
algorithm	I-Algorithm
that	O
was	O
described	O
by	O
Claus	O
Schnorr	O
.	O
</s>
<s>
Typically	O
a	O
Schnorr	B-Algorithm
group	I-Algorithm
is	O
used	O
.	O
</s>
<s>
All	O
users	O
agree	O
on	O
a	O
cryptographic	B-Algorithm
hash	I-Algorithm
function	I-Algorithm
.	O
</s>
<s>
Just	O
as	O
with	O
the	O
closely	O
related	O
signature	O
algorithms	O
DSA	O
,	O
ECDSA	O
,	O
and	O
ElGamal	B-Algorithm
,	O
reusing	O
the	O
secret	O
nonce	O
value	O
on	O
two	O
Schnorr	B-Algorithm
signatures	I-Algorithm
of	O
different	O
messages	O
will	O
allow	O
observers	O
to	O
recover	O
the	O
private	O
key	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
Schnorr	B-Algorithm
signatures	I-Algorithm
,	O
this	O
simply	O
requires	O
subtracting	O
values	O
:	O
</s>
<s>
Therefore	O
,	O
(	O
as	O
per	O
Fiat	O
and	O
Shamir	O
's	O
arguments	O
)	O
,	O
it	O
is	O
secure	O
if	O
is	O
modeled	O
as	O
a	O
random	B-Application
oracle	I-Application
.	O
</s>
<s>
Its	O
security	O
can	O
also	O
be	O
argued	O
in	O
the	O
generic	B-General_Concept
group	I-General_Concept
model	I-General_Concept
,	O
under	O
the	O
assumption	O
that	O
is	O
"	O
random-prefix	O
preimage	O
resistant	O
"	O
and	O
"	O
random-prefix	O
second-preimage	O
resistant	O
"	O
.	O
</s>
<s>
In	O
2012	O
,	O
Seurin	O
provided	O
an	O
exact	O
proof	O
of	O
the	O
Schnorr	B-Algorithm
signature	I-Algorithm
scheme	O
.	O
</s>
<s>
Namely	O
,	O
under	O
the	O
ROMDL	O
assumption	O
,	O
any	O
algebraic	O
reduction	O
must	O
lose	O
a	O
factor	O
in	O
its	O
time-to-success	O
ratio	O
,	O
where	O
is	O
a	O
function	O
that	O
remains	O
close	O
to	O
1	O
as	O
long	O
as	O
"	O
is	O
noticeably	O
smaller	O
than	O
1	O
"	O
,	O
where	O
is	O
the	O
probability	O
of	O
forging	O
an	O
error	O
making	O
at	O
most	O
queries	O
to	O
the	O
random	B-Application
oracle	I-Application
.	O
</s>
<s>
In	O
Schnorr	O
's	O
original	O
1991	O
paper	O
,	O
it	O
was	O
suggested	O
that	O
since	O
collision	O
resistance	O
in	O
the	O
hash	B-Algorithm
is	O
not	O
required	O
,	O
then	O
therefore	O
shorter	O
hash	B-Algorithm
functions	I-Algorithm
may	O
be	O
just	O
as	O
secure	O
,	O
and	O
indeed	O
recent	O
developments	O
suggest	O
that	O
a	O
t-bit	O
security	O
level	O
can	O
be	O
achieved	O
with	O
3t-bit	O
signatures	O
.	O
</s>
