Abstract
AbstractWe suggest a construction of the minimal polynomial $$m_{\beta ^k}$$
m
β
k
of $$\beta ^k\in {\mathbb {F}}_{q^n}$$
β
k
∈
F
q
n
over $${\mathbb {F}}_q$$
F
q
from the minimal polynomial $$f= m_\beta $$
f
=
m
β
for all positive integers k whose prime factors divide $$q-1$$
q
-
1
. The computations of our construction are carried out in $${\mathbb {F}}_q$$
F
q
. The key observation leading to our construction is that for $$k \mid q-1$$
k
∣
q
-
1
holds $$\begin{aligned} m_{\beta ^k}(X^k) = \prod _{j=1}^{\frac{k}{t}} \zeta _{k}^{-jn} f (\zeta _{k}^j X), \end{aligned}$$
m
β
k
(
X
k
)
=
∏
j
=
1
k
t
ζ
k
-
j
n
f
(
ζ
k
j
X
)
,
where $$t= \max \{m\in {\mathbb {N}}: m \mid \gcd (n,k), f(X) = g (X^m), g \in {\mathbb {F}}_q[X]\}$$
t
=
max
{
m
∈
N
:
m
∣
gcd
(
n
,
k
)
,
f
(
X
)
=
g
(
X
m
)
,
g
∈
F
q
[
X
]
}
and $$\zeta _{k}$$
ζ
k
is a primitive k-th root of unity in $${\mathbb {F}}_q$$
F
q
. The construction allows to construct a large number of irreducible polynomials over $${\mathbb {F}}_q$$
F
q
of the same degree. Since different applications require different properties, this large number allows the selection of the candidates with the desired properties.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computer Science Applications