<s>
An	O
MDS	B-General_Concept
matrix	I-General_Concept
(	O
maximum	B-General_Concept
distance	I-General_Concept
separable	I-General_Concept
)	O
is	O
a	O
matrix	B-Architecture
representing	O
a	O
function	O
with	O
certain	O
diffusion	O
properties	O
that	O
have	O
useful	O
applications	O
in	O
cryptography	O
.	O
</s>
<s>
Technically	O
,	O
an	O
matrix	B-Architecture
over	O
a	O
finite	O
field	O
is	O
an	O
MDS	B-General_Concept
matrix	I-General_Concept
if	O
it	O
is	O
the	O
transformation	B-Algorithm
matrix	I-Algorithm
of	O
a	O
linear	B-Architecture
transformation	I-Architecture
from	O
to	O
such	O
that	O
no	O
two	O
different	O
-tuples	O
of	O
the	O
form	O
coincide	O
in	O
or	O
more	O
components	O
.	O
</s>
<s>
Let	O
be	O
the	O
matrix	B-Architecture
obtained	O
by	O
joining	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
to	O
.	O
</s>
<s>
Then	O
a	O
necessary	O
and	O
sufficient	O
condition	O
for	O
a	O
matrix	B-Architecture
to	O
be	O
MDS	O
is	O
that	O
every	O
possible	O
submatrix	O
obtained	O
by	O
removing	O
rows	O
from	O
is	O
non-singular	O
.	O
</s>
<s>
This	O
is	O
also	O
equivalent	O
to	O
the	O
following	O
:	O
all	O
the	O
sub-determinants	O
of	O
the	O
matrix	B-Architecture
are	O
non-zero	O
.	O
</s>
<s>
Then	O
a	O
binary	O
matrix	B-Architecture
(	O
namely	O
over	O
the	O
field	O
with	O
two	O
elements	O
)	O
is	O
never	O
MDS	O
unless	O
it	O
has	O
only	O
one	O
row	O
or	O
only	O
one	O
column	O
with	O
all	O
components	O
.	O
</s>
<s>
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
have	O
the	O
MDS	O
property	O
and	O
are	O
frequently	O
used	O
to	O
obtain	O
the	O
MDS	O
matrices	O
used	O
in	O
cryptographic	O
algorithms	O
.	O
</s>
<s>
MDS	O
matrices	O
are	O
used	O
for	O
diffusion	O
in	O
such	O
block	O
ciphers	O
as	O
AES	B-Algorithm
,	O
SHARK	B-Algorithm
,	O
Square	B-Algorithm
,	O
Twofish	B-Algorithm
,	O
Anubis	B-Algorithm
,	O
KHAZAD	B-Algorithm
,	O
Manta	O
,	O
Hierocrypt	B-Algorithm
,	O
Kalyna	O
and	O
Camellia	B-Algorithm
,	O
and	O
in	O
the	O
stream	B-Algorithm
cipher	I-Algorithm
MUGI	B-Algorithm
and	O
the	O
cryptographic	B-Algorithm
hash	I-Algorithm
function	I-Algorithm
Whirlpool	B-Algorithm
.	O
</s>
