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F
=
A
1
◦
A
2
, where the mappings
A
1
,A
2
are ane (resp. linear) permuta-
tions. Functions
F
and
F
are called
extended ane equivalent
(EA-equivalent)
if
F
=
A
1
◦
F
◦
A
2
+
A
, where the mappings
A, A
1
,A
2
are ane, and where
A
1
,A
2
are permutations.
Two mappings
F
and
F
from
F
p
n
to itself are called
Carlet-Charpin-Zinoviev
equivalent
(CCZ-equivalent) if for some ane permutation
F
◦
L
of
F
p
n
the im-
age of the graph of
F
is the graph of
F
,thatis,
L
(
G
F
)=
G
F
where
G
F
=
(
x, F
(
x
))
{
. Differential uniformity
is invariant under CCZ-equivalence. EA-equivalence is a particular case of CCZ-
equivalence and any permutation is CCZ-equivalent to its inverse. In [5], it is
proven that CCZ-equivalence is even more general. In the present paper we prove
that for PN functions CCZ-equivalence coincides with EA-equivalence.
Almost all known planar functions are DO polynomials. Recall that a function
F
is called
Dembowski-Ostrom polynomial
(DO polynomial) if
(
x, F
(
x
))
|
x
∈
F
p
n
}
and
G
F
=
{
|
x
∈
F
p
n
}
a
kj
x
p
k
+
p
j
.
F
(
x
)=
0
≤k,j<n
When
p
is odd the notion of planar DO polynomial is closely connected to the
notion of
commutative semifield
. A ring with left and right distributivity and
with no zero divisors is called a
presemifield
. A presemifield with a multiplicative
identity is called a
semifield
. Any finite presemifield can be represented by
S
=
(
F
p
n
,
+
,
), where (
F
p
n
,
+) is the additive group of
F
p
n
and
xy
=
φ
(
x, y
)with
φ
a function from
F
p
n
onto
F
p
n
,see[9].
Let
S
1
=(
F
p
n
,
+
,
)and
S
2
=(
F
p
n
,
+
,
) be two presemifields. They are
called
isotopic
if there exist three linear permutations
L, M, N
over
F
p
n
such
that
∗
L
(
x
∗
y
)=
M
(
x
)
N
(
y
)
,
for any
x, y
∈
F
p
n
. The triple (
M, N, L
) is called the
isotopism
between
S
1
and
S
2
.If
M
=
N
then
S
1
and
S
2
are called
strongly isotopic
.
Let
S
be a finite semifield. The subsets
N
l
(
S
)=
{α ∈
S
:(
αx
)
y
=
α
(
xy
) for all
x, y ∈
S
},
N
m
(
S
)=
{α ∈
S
:(
xα
)
y
=
x
(
αy
) for all
x, y ∈
S
},
N
r
(
S
)=
{α ∈
S
:(
xy
)
α
=
x
(
yα
) for all
x, y ∈
S
},
are called the
left, middle
and
right nucleus
of
S
, respectively, and the set
N
(
S
)=
N
l
(
S
)
N
r
(
S
) is called the
nucleus
. These sets are finite
fields and, if
S
is commutative then
N
l
(
S
)=
N
r
(
S
). The nuclei measure how
far
S
is from being associative.
The orders of the respective nuclei are invariant
under isotopism
[9].
Let
S
=(
F
p
n
,
+
,
) be a commutative presemifield which does not contain
an identity. To create a semifield from
S
choose any
a
∩
N
m
(
S
)
∩
∈
F
p
n
and define a new
multiplication
∗
by
(
xa
)
∗
(
ay
)=
xy
