Cryptography Reference
In-Depth Information
b
7
b
6
b
5
b
4
b
3
b
2
b
1
b
0
0
1
1
0
0
0
1
1
s
7
s
6
s
5
s
4
s
3
s
2
s
1
s
0
11111000
01111100
00111110
00011111
10001111
11000111
11100011
11110001
+
=
.
Hence,
g
(
a
i,j
)=(
s
7
s
6
s
5
s
4
s
3
s
2
s
1
s
0
)
t
is the binary equivalent of the decimal digit appearing in the S-box at position
(
i,j
).
Observe that the column matrix, added on the left of the equality, is binary
for the decimal digit 99 (or equivalently, the hexadecimal digit 63).
We may consider the above in terms of polynomials. For instance,
f
◦
a
11
,
3
=8
·
11+3
−
9=82
has representation as the binary polynomial,
∈
F
2
8
=
x
6
+
x
4
+
x
F
2
[
x
]
/
(
m
(
x
))
,
where
m
(
x
)=
x
8
+
x
4
+
x
3
+
x
+1
is the irreducible
Rijndael polynomial
(see Example A.9 on page 489 in Appendix
A). The multiplicative inverse of 82 in
F
2
8
is given by
x
2
+1, so
(
b
7
,b
6
,b
5
,b
4
,b
3
,b
2
,b
1
,b
0
)=(0
,
0
,
0
,
0
,
0
,
1
,
0
,
1)
.
Thus
0
0
0
0
0
1
0
1
0
1
1
0
0
0
1
1
0
0
0
0
0
0
0
0
11111000
01111100
00111110
00011111
10001111
11000111
11100011
11110001
+
=
,
and 0 is the decimal entry in position (11
,
3) of the S-box:
In summary, all of the values values of
f
g
acting on the
a
i,j
are given by
the decimal representations in the following Rijndael
S
-box.
◦
Search WWH ::
Custom Search