Cryptography Reference
In-Depth Information
We can see that if we raise element
α
to successive powers 0, 1 and 2 we obtain
all the elements of field
F
4
with the exception of element 0. Indeed,
α
2
is still
equal to
(
α
+1)
modulo
ϕ
(
α
)
.Element
α
is called the
primitive element
of field
F
4
.
The elements of field
F
4
can also be represented in binary form:
F
4
:
{
00
,
01
,
10
,
11
}
The binary couples correspond to the four values taken by coecients
a
and
b
.
Primitive element of a Galois field
We call the primitive element of a Galois field
F
q
, an element of this field that,
when it is raised to successive powers
0
,
1
,
2
,
2)
;
q
=2
m
,makesit
possible to retrieve all the elements of the field except element 0. Every Galois
field has at least one primitive element. If
α
is a primitive element of field
F
q
then, the elements of this field are:
F
q
=
0
,α
0
,α
1
,
···
,
(
q
−
,α
q−
2
with
α
q−
1
=1
···
Note that in such a Galois field the "-" sign is equivalent to the "+" sign, that
is:
α
j
=
α
j
}
Observing that
2
α
j
=0
modulo 2, we can always add the zero quantity
2
α
j
−
∀
j
∈{
0
,
1
,
···
,
(
q
−
2)
to
α
j
and we thus obtain the above equality.
For example, for field
F
4
let us give the rules that govern the addition and
multiplication operations. All the operations are done modulo 2 and modulo
α
2
+
α
+1
.
−
α
2
+
0
1
α
α
2
0
0
1
α
1+
α
=
α
2
1+
α
2
=
α
1
1
0
1+
α
=
α
2
α
+
α
2
=1
α
α
0
α
2
α
2
1+
α
2
=
α
α
+
α
2
=1
0
Table 4.7 -
Addition in field
F
4
.
Minimal polynomial with coecients in
F
2
associ-
ated with an element of a Galois field
F
q
The minimal polynomial
m
β
(
x
)
with coecients in
F
2
associated with any
element
β
of a Galois field
F
q
, is a polynomial of degree at most equal to