<s>
Badger	O
is	O
a	O
Message	B-Algorithm
Authentication	I-Algorithm
Code	I-Algorithm
(	O
MAC	O
)	O
based	O
on	O
the	O
idea	O
of	O
universal	B-Algorithm
hashing	I-Algorithm
and	O
was	O
developed	O
by	O
Boesgaard	O
,	O
Scavenius	O
,	O
Pedersen	O
,	O
Christensen	O
,	O
and	O
Zenner	O
.	O
</s>
<s>
It	O
is	O
constructed	O
by	O
strengthening	O
the	O
∆	O
-universal	O
hash	O
family	O
MMH	O
using	O
an	O
ϵ-almost	O
strongly	O
universal	O
(	O
ASU	O
)	O
hash	B-Error_Name
function	I-Error_Name
family	O
after	O
the	O
application	O
of	O
ENH	O
(	O
see	O
below	O
)	O
,	O
where	O
the	O
value	O
of	O
ϵ	O
is	O
.	O
</s>
<s>
Since	O
Badger	O
is	O
a	O
MAC	O
function	O
based	O
on	O
the	O
universal	B-Algorithm
hash	I-Algorithm
function	I-Algorithm
approach	O
,	O
the	O
conditions	O
needed	O
for	O
the	O
security	O
of	O
Badger	O
are	O
the	O
same	O
as	O
those	O
for	O
other	O
universal	B-Algorithm
hash	I-Algorithm
functions	I-Algorithm
such	O
as	O
UMAC	B-Algorithm
.	O
</s>
<s>
According	O
to	O
the	O
security	O
needs	O
,	O
user	O
can	O
choose	O
the	O
value	O
of	O
,	O
that	O
is	O
the	O
number	O
of	O
parallel	O
hash	B-Error_Name
trees	O
in	O
Badger	O
.	O
</s>
<s>
Hash	B-Error_Name
families	O
can	O
be	O
combined	O
in	O
order	O
to	O
obtain	O
new	O
hash	B-Error_Name
families	O
.	O
</s>
<s>
A	O
method	O
to	O
reduce	O
∆	O
-universal	O
hash	O
function	O
to	O
universal	B-Algorithm
hash	I-Algorithm
functions	I-Algorithm
will	O
be	O
described	O
in	O
the	O
following	O
.	O
</s>
<s>
Let	O
be	O
an	O
ϵ-AΔU	O
hash	B-Error_Name
family	O
from	O
a	O
set	O
A	O
to	O
a	O
set	O
B	O
.	O
</s>
<s>
The	O
ENH-family	O
is	O
constructed	O
based	O
on	O
the	O
universal	B-Algorithm
hash	I-Algorithm
family	O
NH	O
(	O
which	O
is	O
also	O
used	O
in	O
UMAC	B-Algorithm
)	O
:	O
</s>
<s>
It	O
is	O
a	O
-A	O
∆U	O
hash	B-Error_Name
family	O
.	O
</s>
<s>
where	O
an	O
-AU	O
universal	O
function	O
family	O
H*	O
is	O
used	O
to	O
hash	B-Error_Name
messages	O
of	O
any	O
size	O
onto	O
a	O
fixed	O
size	O
and	O
an	O
-ASU	O
function	O
family	O
F	O
is	O
used	O
to	O
guarantee	O
the	O
strong	O
universality	O
of	O
the	O
overall	O
construction	O
.	O
</s>
<s>
The	O
maximum	O
input	O
size	O
of	O
the	O
function	O
family	O
H*	O
is	O
and	O
the	O
output	O
size	O
is	O
128	O
bits	O
,	O
split	O
into	O
64	O
bits	O
each	O
for	O
the	O
message	O
and	O
the	O
hash	B-Error_Name
.	O
</s>
<s>
The	O
collision	B-Algorithm
probability	O
for	O
the	O
H*	O
-function	O
ranges	O
from	O
to	O
.	O
</s>
<s>
To	O
construct	O
the	O
strongly	O
universal	O
function	O
family	O
F	O
,	O
the	O
∆	O
-universal	O
hash	O
family	O
MMH*	O
is	O
transformed	O
into	O
a	O
strongly	O
universal	B-Algorithm
hash	I-Algorithm
family	O
by	O
adding	O
another	O
key	O
.	O
</s>
<s>
A	O
core	O
function	O
:	O
is	O
used	O
in	O
this	O
processing	O
phase	O
,	O
that	O
hashes	B-Error_Name
a	O
128-bit	O
string	O
to	O
a	O
64-bit	O
string	O
as	O
follows	O
:	O
</s>
<s>
This	O
finalization	O
phase	O
uses	O
the	O
Rabbit	B-Algorithm
stream	B-Algorithm
cipher	I-Algorithm
and	O
uses	O
both	O
key	O
setup	O
and	O
IV	O
setup	O
by	O
taking	O
the	O
finalization	O
key	O
final_key[j][i]	O
as	O
.	O
</s>
<s>
From	O
the	O
pseudocode	O
above	O
,	O
k	O
denotes	O
the	O
key	O
in	O
the	O
Rabbit	B-Algorithm
Key	O
Setup(K )	O
which	O
initializes	O
Rabbit	B-Algorithm
with	O
the	O
128-bit	O
key	O
k	O
.	O
M	O
denotes	O
the	O
message	O
to	O
be	O
hashed	O
and	O
|M|	O
denotes	O
the	O
length	O
of	O
the	O
message	O
in	O
bits	O
.	O
</s>
<s>
Boesgard	O
,	O
Christensen	O
and	O
Zenner	O
report	O
the	O
performance	O
of	O
Badger	O
measured	O
on	O
a	O
1.0GHz	O
Pentium	B-General_Concept
III	I-General_Concept
and	O
on	O
a	O
1.7GHz	O
Pentium	B-General_Concept
4	I-General_Concept
processor	O
.	O
</s>
<s>
The	O
speed-optimized	O
versions	O
were	O
programmed	O
in	O
assembly	O
language	O
inlined	O
in	O
C	O
and	O
compiled	O
using	O
the	O
Intel	O
C++	B-Language
7.1	O
compiler	O
.	O
</s>
<s>
denotes	O
the	O
amount	O
of	O
memory	O
required	O
to	O
store	O
the	O
internal	O
state	O
including	O
key	O
material	O
and	O
the	O
inner	O
state	O
of	O
the	O
Rabbit	B-Algorithm
stream	B-Algorithm
cipher	I-Algorithm
.	O
</s>
<s>
Setup	O
Pentium	B-General_Concept
III	I-General_Concept
Fin	O
.	O
</s>
<s>
Pentium	B-General_Concept
III	I-General_Concept
Setup	O
Pentium	B-General_Concept
III	I-General_Concept
Fin	O
.	O
</s>
<s>
MMH*	O
involves	O
a	O
construction	O
of	O
a	O
family	O
of	O
hash	B-Error_Name
functions	I-Error_Name
consisting	O
of	O
multilinear	O
functions	O
on	O
for	O
some	O
positive	O
integer	O
.	O
</s>
<s>
where	O
x	O
,	O
m	O
are	O
vectors	B-Application
,	O
and	O
the	O
functions	O
are	O
defined	O
as	O
follows	O
.	O
</s>
<s>
From	O
the	O
proof	O
above	O
,	O
is	O
the	O
collision	B-Algorithm
probability	O
of	O
the	O
attacker	O
in	O
1	O
round	O
,	O
so	O
on	O
average	O
verification	O
queries	O
will	O
suffice	O
to	O
get	O
one	O
message	O
accepted	O
.	O
</s>
<s>
To	O
reduce	O
the	O
collision	B-Algorithm
probability	O
,	O
it	O
is	O
necessary	O
to	O
choose	O
large	O
p	O
or	O
to	O
concatenate	O
such	O
MACs	O
using	O
independent	O
keys	O
so	O
that	O
the	O
collision	B-Algorithm
probability ]	O
becomes	O
.	O
</s>
<s>
By	O
theorem	O
1	O
,	O
the	O
collision	B-Algorithm
probability	O
is	O
about	O
ϵ	O
=	O
,	O
and	O
the	O
family	O
of	O
can	O
be	O
defined	O
as	O
ϵ-almost	O
∆	O
Universal	O
with	O
ϵ	O
=	O
.	O
</s>
<s>
The	O
value	O
of	O
k	O
that	O
describes	O
the	O
length	O
of	O
the	O
message	O
and	O
key	O
vectors	B-Application
has	O
several	O
effects	O
:	O
</s>
