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defined by the functions of Theorems 1 and 2 cannot be isotopic to Albert's
commutative twisted fields. Besides, it is proven in [9] that, for any planar DO
function
F
, isotopism between the commutative semifield defined by
F
and a
commutative twisted field implies strong isotopism. Thus, by Corollary 4, when
2
k/
gcd(2
k, s
) is even, the PN functions (i
∗
)and(ii
∗
) define commutative semi-
fields nonisotopic to Albert's commutative twisted fields. Proposition 4 shows
that the condition on 2
k/
gcd(2
k, s
) is not necessary for these commutative semi-
fields to be nonisotopic. Indeed, cases (1) and (3) in Proposition 4 are defined
also for 2
k/
gcd(2
k, s
) odd, and it is proven in [2] that the Albert's commutative
twisted fields of order
p
n
and parameter
t
(where
n/
gcd(
n, t
) is odd) have the
middle and left nuclei of order
p
gcd(
n,t
)
.
Let
be a basis of
F
p
2
k
over
F
p
k
. The planar functions over
F
p
2
k
derived
from the Dickson semifields are
x
2
+
j
σ
x
q
{
1
,β
}
2
β
2
x
q
2
−
x
−
x
−
,
β
q
−
β
β
q
−
β
where
j
is a nonsquare in
F
p
k
,and1
Aut(
F
p
k
). Different choices of
β
and
σ
may give CCZ-inequivalent planar functions. However, as proven in
[13], all Dickson commutative semifields of order
p
2
k
have the middle nuclei of
order
p
k
. Since the orders of the respective nuclei of semifields are invariant
under isotopism then the commutative semifields defined by functions (1)-(3) of
Proposition 4 are nonisotopic to all Dickson semifields.
=
σ
∈
Corollary 5.
Functions
(1)
-
(3)
of Proposition
4
define commutative semifields
which are nonisotopic to all Dickson semifields.
Corollary 6.
The functions (i
∗
)
with
p
=5
and
n
=6
are CCZ-inequivalent to
any known PN functions and define commutative semifields nonisotopic to the
known ones.
Note that Proposition 4 also implies that in general the family of Cohen-Ganley
semifields is distinct from the families of semifields defined by (i
∗
)and(ii
∗
)with
p
= 3. Indeed, the Cohen-Ganley semifields of order 3
6
and 3
8
have the middle
nuclei of order 3
3
and 3
4
, respectively. Therefore, they are nonisotopic to the
commutative semifields defined by functions (1)-(2) of Proposition 4.
References
1. Albert, A.A.: On nonassociative division algebras. Trans. Amer. Math. Soc. 72,
296-309 (1952)
2. Albert, A.A.: Generalized twisted fields. Pacific J. Math. 11, 1-8 (1961)
3. Biham, E., Shamir, A.: Differential Cryptanalysis of DES-like Cryptosystems. Jour-
nal of Cryptology 4(1), 3-72 (1991)
4. Bracken, C., Byrne, E., Markin, N., McGuire, G.: New families of quadratic almost
perfect nonlinear trinomials and multinomials. Finite Fields and Applications (to
appear, 2008)
