Cryptography Reference
In-Depth Information
addition
additive inverse
+ 01234
0 01234
1 12340
2 23401
3 34012
4 40123
−
0 = 0
−
1 = 4
−
2 = 3
−
3 = 2
−
4 = 1
multiplication
multiplicative inverse
×
01234
0 00000
1 01234
2 02413
3 03142
4 04321
0
−
1
does not exist
1
−
1
= 1
2
−
1
= 3
3
−
1
= 2
4
−
1
= 4
multiplicative inverse of the field elements. Using these tables, we can perform all
calculations in this field without using modular reduction explicitly.
A very important prime field is
GF
(2), which is the smallest finite field that
exists. Let's have a look at the multiplication and addition tables for the field.
Example 4.4.
Let's consider the small finite field
GF
(2)=
{
0
,
1
}
. Arithmetic is sim-
ply done modulo 2, yielding the following arithmetic tables:
multiplication
×
addition
+ 01
0 01
1 10
01
0 00
1 01
As we saw in Chap. 2 on stream ciphers,
GF
(2) addition, i.e., modulo 2 addition,
is equivalent to an XOR gate. What we learn from the example above is that
GF
(2)
multiplication is equivalent to the logical AND gate. The field
GF
(2) is important
for AES.
4.3.3 Extension Fields
GF
(2
m
)
In AES the finite field contains 256 elements and is denoted as
GF
(2
8
). This field
was chosen because each of the field elements can be represented by one byte. For
the S-Box and MixColumn transforms, AES treats every byte of the internal data
