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a permutation since otherwise
L
(
a
)=0=
L
(0) for some nonzero
a
and so we
get the inequality
L
◦
L
−
1
(
c
)
=
a
for any
c
and any nonzero
a
whichinitsturn
means
L
◦
L
−
1
=0,thatis,
L
=0.
By the definition of CCZ-equivalence and by the data obtained above we get
for CCZ-equivalent functions
F
and
F
that
F
=
F
2
◦
F
−
1
1
,
where
F
1
(
x
)=
L
1
(
x, F
(
x
)) =
L
(
x
)+
b,
F
2
(
x
)=
L
2
(
x, F
(
x
)) =
L
(
x
)+
L
(
F
(
x
)) +
b
with
b, b
∈
F
p
n
,
L, L
,L
linear and
L
a permutation. Note that
F
(
x
)=
L
F
−
1
1
(
x
)
+
L
F
F
−
1
1
(
x
)
+
b
=
A
(
x
)+
A
1
◦
F
◦
A
2
(
x
)
where
A
2
(
x
)=
F
−
1
F
−
1
is ane, and we
show below that the linear function
A
1
=
L
is a permutation. Indeed, the ane
function
(
x
) is an ane permutation,
A
=
L
◦
1
(
x, y
)=
L
(
x
)+
b, L
(
x
)+
L
(
y
)+
b
is a permutation, that is, the
system of two equations
L
(
x
)=0and
L
(
x
)+
L
(
y
) = 0 only has the solution
(0
,
0), and then
L
(
y
) = 0 should only have the solution 0 which implies that
L
is a permutation. Thus,
F
and
F
are EA-equivalent.
L
From the result above we get the following obvious corollaries.
Corollary 1.
If a PN function
F
is CCZ-equivalent to a DO polynomial
F
then
F
is also DO polynomial.
Corollary 2.
Perfect nonlinear DO polynomials
F
and
F
are CCZ-equivalent
if and only if they are linear equivalent.
Now it is obvious that CCZ-equivalence of two DO planar functions implies
strong isotopism of the corresponding commutative semifields.
It is also obvious that DO polynomials cannot be CCZ-equivalent to the PN
functions
x
(3
t
+1)
/
2
over
F
3
n
with gcd(
n, t
)=1,
t
odd. Indeed,
x
(3
t
+1)
/
2
is not
=2+
t−
1
3
t
+1
2
i
=1
3
t−i
.
DO polynomial because
5 On the Inequivalence of the Introduced PN Functions
with Known PN Mappings
Note that the functions of Theorems 1 and 2 are defined over
F
p
2
k
for any odd
prime
p
. Obviously, we can say the same only about PN functions of the cases
(i), (ii) and (iii), while the cases (v)-(viii) are defined only for
p
= 3 and cannot
cover all the functions of Theorems 1 and 2. So when proving CCZ-inequivalence
to the known PN functions we mainly concentrate our attention on the functions
(i), (ii), and (iii).
In the proposition below we show that any function which is CCZ-equivalent
to
x
2
should have some monomial of the form
x
2
p
t
for some
t
,0
≤
t<n
,inits
polynomial representation.
