Abstract
We study the cycle structure of permutations $$F(x)=x+\gamma f(x)$$
F
(
x
)
=
x
+
γ
f
(
x
)
on $$\mathbb {F}_{q^n}$$
F
q
n
, where $$f :\mathbb {F}_{q^n} \rightarrow \mathbb {F}_q$$
f
:
F
q
n
→
F
q
. We show that for a 1-homogeneous function f the cycle structure of F can be determined by calculating the cycle structure of certain induced mappings on parallel lines of $$\gamma \mathbb {F}_q$$
γ
F
q
. Using this observation we describe explicitly the cycle structure of two families of permutations over $$\mathbb {F}_{q^2}$$
F
q
2
: $$x+\gamma {{\,\mathrm{Tr}\,}}(x^{2q-1})$$
x
+
γ
Tr
(
x
2
q
-
1
)
, where $$q\equiv -1 \pmod 3$$
q
≡
-
1
(
mod
3
)
and $$\gamma \in \mathbb {F}_{q^2}$$
γ
∈
F
q
2
, with $$\gamma ^3=-\frac{1}{27}$$
γ
3
=
-
1
27
and $$x+\gamma {{\,\mathrm{Tr}\,}}\left( x^{\frac{2^{2s-1}+3\cdot 2^{s-1}+1}{3}}\right) $$
x
+
γ
Tr
x
2
2
s
-
1
+
3
·
2
s
-
1
+
1
3
, where $$q=2^s$$
q
=
2
s
, s odd and $$\gamma \in \mathbb {F}_{q^2}$$
γ
∈
F
q
2
, with $$\gamma ^{(q+1)/3}=1$$
γ
(
q
+
1
)
/
3
=
1
.
The cycle structure of a Markoff automorphism over finite fields
