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 0 1 ,
···
, ( q
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
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