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(vii) the function over
F
3
10
corresponding to the Penttila-Williams semifield
[22];
(viii) the function over
F
3
8
corresponding to the Coulter-Henderson-Kosick
semifield [10].
The representations of the PN functions corresponding to the cases (iii), (v)-
(vii), can be found in [18,19]. The only known PN functions which are not DO
polynomials are the power functions
x
3
t
+1
2
over
F
3
n
,where
t
is odd and gcd(
t, n
)=1[8,17].
Let
p
be an odd prime,
s
and
k
positive integers, and
n
=2
k
. In the present
paper we introduce the following new infinite classes of perfect nonlinear DO
polynomials over
F
p
n
:
−
(
bx
)
p
s
+1
p
k
+
k−
1
(bx)
p
s
+1
i
=0
c
i
x
p
i
(
p
k
+1)
,
(i
∗
)
where
k−
1
i
=0
c
i
x
p
i
is a permutation over
F
p
n
with coecients in
F
p
k
,
b
∈
=gcd
p
s
+
F
p
n
,andgcd(
k
+
s,
2
k
)=gcd(
k
+
s, k
), gcd(
p
s
+1
,p
k
+1)
1
,
(
p
k
+1)
/
2
.
b
x
p
s
+1
+(
bx
p
s
+1
)
p
k
+
cx
p
k
+1
+
k−
1
i
=1
r
i
x
p
k
+
i
+
p
i
,
(ii
∗
)
∈
F
p
n
is not a square,
c
where
b
∈
F
p
n
\
F
p
k
,and
r
i
∈
F
p
k
,0
≤
i<k
,and
gcd(
k
+
s, n
)=gcd(
k
+
s, k
).
We show that in general these functions are CCZ-inequivalent to previously
known PN functions and define new commutative semifields.
2 A New Family of PN Multinomials
In [20] Ness gives a list of planar DO trinomials over
F
p
n
for
p
8which
were found with a computer. Investigation of these functions has led us to the
following family of planar DO polynomials.
≤
7,
n
≤
Theorem 1.
Let
p
be an odd prime,
s
and
k
positive integers such that
gcd(
p
s
+1
,p
k
+1)
=gcd
p
s
+1
,
(
p
k
+1)
/
2
and
gcd(
k
+
s,
2
k
)=gcd(
k
+
s, k
)
.
Let also
n
=2
k
,
b
∈
F
p
n
,and
k−
1
i
=0
c
i
x
p
i
be a permutation over
F
p
n
with
coecients in
F
p
k
. Then the function
(
bx
)
p
s
+1
p
k
k−
1
F
(
x
)=(
bx
)
p
s
+1
c
i
x
p
i
(
p
k
+1)
−
+
i
=0
is PN over
F
p
n
.
