<s>
In	O
cryptography	O
,	O
the	O
QUAD	B-Algorithm
,	O
cipher	O
is	O
a	O
relatively	O
new	O
stream	B-Algorithm
cipher	I-Algorithm
,	O
which	O
was	O
designed	O
with	O
provable	O
security	O
arguments	O
in	O
mind	O
.	O
</s>
<s>
QUAD	B-Algorithm
relies	O
on	O
the	O
iteration	O
of	O
a	O
randomly	O
chosen	O
multivariate	O
quadratic	O
system	O
S=	O
( Q1	O
,	O
...	O
,	O
Qm	O
)	O
of	O
m	O
=	O
kn	O
equations	O
in	O
n	O
unknowns	O
over	O
a	O
finite	O
field	O
GF(q )	O
.	O
</s>
<s>
steps	O
in	O
order	O
to	O
produce	O
(	O
k	O
-1	O
)	O
n	O
GF(q )	O
keystream	B-Algorithm
values	O
at	O
each	O
iteration	O
.	O
</s>
<s>
QUAD	B-Algorithm
is	O
a	O
modern	O
stream	B-Algorithm
cipher	I-Algorithm
,	O
i.e.	O
</s>
<s>
it	O
uses	O
a	O
key	O
and	O
an	O
initialisation	O
value	O
(	O
IV	O
)	O
to	O
produce	O
a	O
keystream	B-Algorithm
sequence	O
.	O
</s>
<s>
The	O
security	O
of	O
the	O
keystream	B-Algorithm
generation	O
of	O
QUAD	B-Algorithm
is	O
provably	O
reducible	O
to	O
the	O
conjectured	O
intractability	O
of	O
the	O
MQ	O
problem	O
,	O
namely	O
solving	O
a	O
multivariate	O
system	O
of	O
quadratic	O
equations	O
.	O
</s>
<s>
The	O
first	O
proof	O
was	O
done	O
over	O
field	O
GF(2 )	O
for	O
an	O
old-fashioned	O
stream	B-Algorithm
cipher	I-Algorithm
(	O
where	O
the	O
key	O
is	O
the	O
initial	O
state	O
)	O
.	O
</s>
<s>
The	O
authors	O
recommend	O
to	O
use	O
a	O
version	O
of	O
QUAD	B-Algorithm
with	O
an	O
80-bit	O
key	O
,	O
80-bit	O
IV	O
and	O
an	O
internal	O
state	O
of	O
n	O
=	O
160	O
bits	O
.	O
</s>
<s>
It	O
outputs	O
160	O
keystream	B-Algorithm
bits	O
(	O
m	O
=	O
320	O
)	O
at	O
each	O
iteration	O
until	O
240	O
bits	O
of	O
keystream	B-Algorithm
have	O
been	O
produced	O
.	O
</s>
<s>
At	O
Eurocrypt	O
2006	O
,	O
speed	O
reports	O
were	O
presented	O
for	O
QUAD	B-Algorithm
instances	O
with	O
160-bit	O
state	O
and	O
output	O
block	O
over	O
the	O
fields	O
GF(2 )	O
,	O
GF(16 )	O
,	O
and	O
GF(256 )	O
.	O
</s>
<s>
This	O
analysis	O
(	O
which	O
also	O
covers	O
several	O
multivariate	O
public-key	O
schemes	O
as	O
well	O
as	O
the	O
QUAD	B-Algorithm
stream	B-Algorithm
cipher	I-Algorithm
)	O
studied	O
in	O
part	O
the	O
impact	O
of	O
changing	O
the	O
size	O
of	O
the	O
field	O
on	O
the	O
performances	O
without	O
considering	O
the	O
security	O
aspect	O
.	O
</s>
<s>
The	O
initial	O
security	O
theorem	O
for	O
QUAD	B-Algorithm
is	O
valid	O
for	O
the	O
field	O
GF(2 )	O
only	O
,	O
and	O
recommended	O
parameters	O
does	O
not	O
achieve	O
to	O
get	O
a	O
contradiction	O
with	O
the	O
proof	O
of	O
security	O
.	O
</s>
<s>
The	O
authors	O
of	O
QUAD	B-Algorithm
who	O
gave	O
the	O
security	O
theorem	O
acknowledged	O
that	O
a	O
break	O
of	O
QUAD	B-Algorithm
at	O
their	O
suggested	O
parameters	O
does	O
not	O
contradict	O
the	O
proof-of-security	O
theorems	O
when	O
they	O
proposed	O
the	O
scheme	O
at	O
Eurocrypt	O
2006	O
.	O
</s>
<s>
Yang	O
,	O
Chen	O
,	O
Bernstein	O
and	O
Chen	O
studied	O
the	O
security	O
of	O
the	O
different	O
parameter	O
sets	O
in	O
the	O
document	O
"	O
Analysis	O
of	O
Quad	B-Algorithm
"	O
and	O
found	O
some	O
of	O
them	O
very	O
insecure	O
.	O
</s>
<s>
Their	O
paper	O
discusses	O
both	O
theoretical	O
and	O
practical	O
aspects	O
of	O
attacking	O
QUAD	B-Algorithm
and	O
of	O
attacking	O
the	O
underlying	O
hard	O
problem	O
.	O
</s>
<s>
For	O
example	O
,	O
this	O
paper	O
shows	O
how	O
to	O
use	O
XL-Wiedemann	O
to	O
break	O
the	O
GF(256 )	O
instance	O
QUAD	B-Algorithm
(	O
256	O
,	O
20	O
,	O
20	O
)	O
in	O
approximately	O
266	O
Opteron	O
cycles	O
,	O
and	O
to	O
break	O
the	O
underlying	O
hard	O
problem	O
in	O
approximately	O
245	O
cycles	O
,	O
which	O
was	O
carried	O
out	O
successfully	O
.	O
</s>
<s>
However	O
,	O
according	O
to	O
this	O
paper	O
,	O
it	O
would	O
take	O
about	O
2110	O
to	O
solve	O
an	O
instance	O
of	O
the	O
QUAD(2,160,160 )	O
version	O
recommended	O
by	O
the	O
authors	O
of	O
QUAD	B-Algorithm
using	O
XL-Wiedemann	O
.	O
</s>
<s>
An	O
instance	O
that	O
will	O
be	O
provably	O
secure	O
would	O
be	O
QUAD(2,320,320 )	O
,	O
that	O
is	O
,	O
twice	O
as	O
wide	O
as	O
originally	O
proposed	O
.	O
</s>
<s>
A	O
security	O
theorem	O
can	O
also	O
be	O
proved	O
for	O
GF(q )	O
,	O
albeit	O
with	O
a	O
larger	O
looseness	O
factor	O
;	O
this	O
and	O
extensions	O
of	O
QUAD	B-Algorithm
for	O
more	O
efficient	O
implementations	O
is	O
proposed	O
by	O
Liu	O
et	O
al	O
.	O
</s>
