<s>
In	O
cryptography	O
,	O
a	O
random	B-Application
oracle	I-Application
is	O
an	O
oracle	O
(	O
a	O
theoretical	O
black	O
box	O
)	O
that	O
responds	O
to	O
every	O
unique	O
query	O
with	O
a	O
(	O
truly	O
)	O
random	O
response	O
chosen	O
uniformly	O
from	O
its	O
output	O
domain	O
.	O
</s>
<s>
Stated	O
differently	O
,	O
a	O
random	B-Application
oracle	I-Application
is	O
a	O
mathematical	O
function	O
chosen	O
uniformly	O
at	O
random	O
,	O
that	O
is	O
,	O
a	O
function	O
mapping	O
each	O
possible	O
query	O
to	O
a	O
(	O
fixed	O
)	O
random	O
response	O
from	O
its	O
output	O
domain	O
.	O
</s>
<s>
Random	B-Application
oracles	I-Application
as	O
a	O
mathematical	O
abstraction	O
were	O
first	O
used	O
in	O
rigorous	O
cryptographic	O
proofs	O
in	O
the	O
1993	O
publication	O
by	O
Mihir	O
Bellare	O
and	O
Phillip	O
Rogaway	O
(	O
1993	O
)	O
.	O
</s>
<s>
They	O
are	O
typically	O
used	O
when	O
the	O
proof	O
cannot	O
be	O
carried	O
out	O
using	O
weaker	O
assumptions	O
on	O
the	O
cryptographic	B-Algorithm
hash	I-Algorithm
function	I-Algorithm
.	O
</s>
<s>
A	O
system	O
that	O
is	O
proven	O
secure	O
when	O
every	O
hash	B-Error_Name
function	I-Error_Name
is	O
replaced	O
by	O
a	O
random	B-Application
oracle	I-Application
is	O
described	O
as	O
being	O
secure	O
in	O
the	O
random	B-Application
oracle	I-Application
model	I-Application
,	O
as	O
opposed	O
to	O
secure	O
in	O
the	O
standard	B-General_Concept
model	I-General_Concept
of	I-General_Concept
cryptography	I-General_Concept
.	O
</s>
<s>
Random	B-Application
oracles	I-Application
are	O
typically	O
used	O
as	O
an	O
idealised	O
replacement	O
for	O
cryptographic	B-Algorithm
hash	I-Algorithm
functions	I-Algorithm
in	O
schemes	O
where	O
strong	O
randomness	O
assumptions	O
are	O
needed	O
of	O
the	O
hash	B-Error_Name
function	I-Error_Name
's	O
output	O
.	O
</s>
<s>
However	O
,	O
it	O
only	O
proves	O
such	O
properties	O
in	O
the	O
random	B-Application
oracle	I-Application
model	I-Application
,	O
making	O
sure	O
no	O
major	O
design	O
flaws	O
are	O
present	O
.	O
</s>
<s>
It	O
is	O
in	O
general	O
not	O
true	O
that	O
such	O
a	O
proof	O
implies	O
the	O
same	O
properties	O
in	O
the	O
standard	B-General_Concept
model	I-General_Concept
.	O
</s>
<s>
Still	O
,	O
a	O
proof	O
in	O
the	O
random	B-Application
oracle	I-Application
model	I-Application
is	O
considered	O
better	O
than	O
no	O
formal	O
security	O
proof	O
at	O
all	O
.	O
</s>
<s>
Not	O
all	O
uses	O
of	O
cryptographic	B-Algorithm
hash	I-Algorithm
functions	I-Algorithm
require	O
random	B-Application
oracles	I-Application
:	O
schemes	O
that	O
require	O
only	O
one	O
or	O
more	O
properties	O
having	O
a	O
definition	O
in	O
the	O
standard	B-General_Concept
model	I-General_Concept
(	O
such	O
as	O
collision	O
resistance	O
,	O
preimage	O
resistance	O
,	O
second	O
preimage	O
resistance	O
,	O
etc	O
.	O
)	O
</s>
<s>
can	O
often	O
be	O
proven	O
secure	O
in	O
the	O
standard	B-General_Concept
model	I-General_Concept
(	O
e.g.	O
,	O
the	O
Cramer	B-Algorithm
–	I-Algorithm
Shoup	I-Algorithm
cryptosystem	I-Algorithm
)	O
.	O
</s>
<s>
Random	B-Application
oracles	I-Application
have	O
long	O
been	O
considered	O
in	O
computational	O
complexity	O
theory	O
,	O
and	O
many	O
schemes	O
have	O
been	O
proven	O
secure	O
in	O
the	O
random	B-Application
oracle	I-Application
model	I-Application
,	O
for	O
example	O
Optimal	B-Algorithm
Asymmetric	I-Algorithm
Encryption	I-Algorithm
Padding	I-Algorithm
,	O
RSA-FDH	B-Algorithm
and	O
Probabilistic	B-Algorithm
Signature	I-Algorithm
Scheme	I-Algorithm
.	O
</s>
<s>
In	O
1986	O
,	O
Amos	O
Fiat	O
and	O
Adi	O
Shamir	O
showed	O
a	O
major	O
application	O
of	O
random	B-Application
oracles	I-Application
–	O
the	O
removal	O
of	O
interaction	O
from	O
protocols	O
for	O
the	O
creation	O
of	O
signatures	O
.	O
</s>
<s>
In	O
1989	O
,	O
Russell	O
Impagliazzo	O
and	O
Steven	O
Rudich	O
showed	O
the	O
limitation	O
of	O
random	B-Application
oracles	I-Application
–	O
namely	O
that	O
their	O
existence	O
alone	O
is	O
not	O
sufficient	O
for	O
secret-key	O
exchange	O
.	O
</s>
<s>
In	O
their	O
definition	O
,	O
the	O
random	B-Application
oracle	I-Application
produces	O
a	O
bit-string	O
of	O
infinite	B-Application
length	O
which	O
can	O
be	O
truncated	O
to	O
the	O
length	O
desired	O
.	O
</s>
<s>
When	O
a	O
random	B-Application
oracle	I-Application
is	O
used	O
within	O
a	O
security	O
proof	O
,	O
it	O
is	O
made	O
available	O
to	O
all	O
players	O
,	O
including	O
the	O
adversary	O
or	O
adversaries	O
.	O
</s>
<s>
A	O
single	O
oracle	O
may	O
be	O
treated	O
as	O
multiple	O
oracles	O
by	O
pre-pending	O
a	O
fixed	O
bit-string	O
to	O
the	O
beginning	O
of	O
each	O
query	O
(	O
e.g.	O
,	O
queries	O
formatted	O
as	O
"	O
1|x	O
"	O
or	O
"	O
0|x	O
"	O
can	O
be	O
considered	O
as	O
calls	O
to	O
two	O
separate	O
random	B-Application
oracles	I-Application
,	O
similarly	O
"	O
00|x	O
"	O
,	O
"	O
01|x	O
"	O
,	O
"	O
10|x	O
"	O
and	O
"	O
11|x	O
"	O
can	O
be	O
used	O
to	O
represent	O
calls	O
to	O
four	O
separate	O
random	B-Application
oracles	I-Application
)	O
.	O
</s>
<s>
According	O
to	O
the	O
Church	O
–	O
Turing	O
thesis	O
,	O
no	O
function	O
computable	O
by	O
a	O
finite	O
algorithm	O
can	O
implement	O
a	O
true	O
random	B-Application
oracle	I-Application
(	O
which	O
by	O
definition	O
requires	O
an	O
infinite	B-Application
description	O
because	O
it	O
has	O
infinitely	O
many	O
possible	O
inputs	O
,	O
and	O
its	O
outputs	O
are	O
all	O
independent	O
from	O
each	O
other	O
and	O
need	O
to	O
be	O
individually	O
specified	O
by	O
any	O
description	O
)	O
.	O
</s>
<s>
In	O
fact	O
,	O
certain	O
artificial	O
signature	O
and	O
encryption	O
schemes	O
are	O
known	O
which	O
are	O
proven	O
secure	O
in	O
the	O
random	B-Application
oracle	I-Application
model	I-Application
,	O
but	O
which	O
are	O
trivially	O
insecure	O
when	O
any	O
real	O
function	O
is	O
substituted	O
for	O
the	O
random	B-Application
oracle	I-Application
.	O
</s>
<s>
Nonetheless	O
,	O
for	O
any	O
more	O
natural	O
protocol	O
a	O
proof	O
of	O
security	O
in	O
the	O
random	B-Application
oracle	I-Application
model	I-Application
gives	O
very	O
strong	O
evidence	O
of	O
the	O
practical	O
security	O
of	O
the	O
protocol	O
.	O
</s>
<s>
Instead	O
,	O
to	O
break	O
the	O
random	B-Application
oracle	I-Application
assumption	O
,	O
one	O
must	O
discover	O
some	O
unknown	O
and	O
undesirable	O
property	O
of	O
the	O
actual	O
hash	B-Error_Name
function	I-Error_Name
;	O
for	O
good	O
hash	B-Error_Name
functions	I-Error_Name
where	O
such	O
properties	O
are	O
believed	O
unlikely	O
,	O
the	O
considered	O
protocol	O
can	O
be	O
considered	O
secure	O
.	O
</s>
<s>
Although	O
the	O
Baker	O
–	O
Gill	O
–	O
Solovay	O
theorem	O
showed	O
that	O
there	O
exists	O
an	O
oracle	O
A	O
such	O
that	O
PA	O
=	O
NPA	O
,	O
subsequent	O
work	O
by	O
Bennett	O
and	O
Gill	O
,	O
showed	O
that	O
for	O
a	O
random	B-Application
oracle	I-Application
B	O
(	O
a	O
function	O
from	O
{0,1}n	O
to	O
 { 0 , 1 } 	O
such	O
that	O
each	O
input	O
element	O
maps	O
to	O
each	O
of	O
0	O
or	O
1	O
with	O
probability	O
1/2	O
,	O
independently	O
of	O
the	O
mapping	O
of	O
all	O
other	O
inputs	O
)	O
,	O
PB	O
⊊	O
NPB	O
with	O
probability	O
1	O
.	O
</s>
<s>
Similar	O
separations	O
,	O
as	O
well	O
as	O
the	O
fact	O
that	O
random	B-Application
oracles	I-Application
separate	O
classes	O
with	O
probability	O
0	O
or	O
1	O
(	O
as	O
a	O
consequence	O
of	O
the	O
Kolmogorov	O
's	O
zero	O
–	O
one	O
law	O
)	O
,	O
led	O
to	O
the	O
creation	O
of	O
the	O
Random	B-Application
Oracle	I-Application
Hypothesis	O
,	O
that	O
two	O
"	O
acceptable	O
"	O
complexity	O
classes	O
C1	O
and	O
C2	O
are	O
equal	O
if	O
and	O
only	O
if	O
they	O
are	O
equal	O
(	O
with	O
probability	O
1	O
)	O
under	O
a	O
random	B-Application
oracle	I-Application
(	O
the	O
acceptability	O
of	O
a	O
complexity	O
class	O
is	O
defined	O
in	O
BG81	O
)	O
.	O
</s>
<s>
This	O
hypothesis	O
was	O
later	O
shown	O
to	O
be	O
false	O
,	O
as	O
the	O
two	O
acceptable	O
complexity	O
classes	O
IP	O
and	O
PSPACE	O
were	O
shown	O
to	O
be	O
equal	O
despite	O
IPA	O
⊊	O
PSPACEA	O
for	O
a	O
random	B-Application
oracle	I-Application
A	O
with	O
probability	O
1	O
.	O
</s>
<s>
An	O
ideal	O
cipher	O
is	O
a	O
random	B-Algorithm
permutation	I-Algorithm
oracle	O
that	O
is	O
used	O
to	O
model	O
an	O
idealized	O
block	O
cipher	O
.	O
</s>
<s>
A	O
random	B-Algorithm
permutation	I-Algorithm
decrypts	O
each	O
ciphertext	O
block	O
into	O
one	O
and	O
only	O
one	O
plaintext	O
block	O
and	O
vice	O
versa	O
,	O
so	O
there	O
is	O
a	O
one-to-one	B-Algorithm
correspondence	I-Algorithm
.	O
</s>
<s>
Recent	O
works	O
showed	O
that	O
an	O
ideal	O
cipher	O
can	O
be	O
constructed	O
from	O
a	O
random	B-Application
oracle	I-Application
using	O
10-round	O
or	O
even	O
8-round	O
Feistel	B-Algorithm
networks	I-Algorithm
.	O
</s>
<s>
An	O
ideal	O
permutation	O
is	O
an	O
idealized	O
object	O
sometimes	O
used	O
in	O
cryptography	O
to	O
model	O
the	O
behaviour	O
of	O
a	O
permutation	O
whose	O
outputs	O
are	O
indistinguishable	O
from	O
those	O
of	O
a	O
random	B-Algorithm
permutation	I-Algorithm
.	O
</s>
<s>
As	O
a	O
random	B-Application
oracle	I-Application
is	O
an	O
abstraction	O
of	O
a	O
hash	B-Error_Name
function	I-Error_Name
,	O
it	O
makes	O
sense	O
to	O
assume	O
that	O
a	O
quantum	O
attacker	O
can	O
access	O
the	O
random	B-Application
oracle	I-Application
in	O
quantum	O
superposition	O
.	O
</s>
<s>
Many	O
of	O
the	O
classical	O
security	O
proofs	O
break	O
down	O
in	O
that	O
quantum	O
random	B-Application
oracle	I-Application
model	I-Application
and	O
need	O
to	O
be	O
revised	O
.	O
</s>
