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f
W
of
f
with respect to
W
is
If
f
is bent with respect to
W
,the
partial dual
defined by the relation
1)
f
W
(
x,v
)
.
H
W
(
f
)(
x, v
)=(
−
f
W
is the usual dual of
f
,whichwe
Note that in the special case where
n
=
k
,
will denote by
f
.
In the remainder of this paper we shall make frequent use of the following
lemma.
Lemma 1.
For any
u
∈
V
n
we have
1)
u·x
i
wt(
x
)
=2
2
ω
n
i
−
wt(
u
)
,
(
−
x∈V
n
where
ω
=(1+
i
)
/
√
2
is a primitive
8
th root of unity.
Proof.
Write
u
=(
u
1
,u
2
,...,u
n
). By successively factoring out terms, we obtain
n
1)
u·x
i
wt(
x
)
=
1)
u
k
)
(
−
(1 +
i
(
−
x∈V
n
k
=1
n
=2
2
ω
(
−
1)
u
k
k
=1
=2
2
ω
n−
2wt(
u
)
=2
2
ω
n
i
−
wt(
u
)
.
Note that the preceding lemma shows that all ane functions
f
:
V
n
→
F
2
are
negabent (see also [3, Prop. 1]).
3 Transformations Preserving Bent-Negabentness
Several transformations that preserve the bent-negabent property have been
presented in [3]. Here we provide two new transformations.
It is known that, if
f
:
V
n
→
F
2
is a bent function, then the function given by
f
(
Ax
+
b
)+
c
·
x
+
d,
where
A
∈
GL
(2
,n
)
,b,c
∈
V
n
,d
∈
V
1
,
is also bent. Here,
GL
(2
,n
) is the general linear group of
n
F
2
.
These operations define a group whose action on
f
leaves the bent property of
f
invariant. Counterexamples show that these operations generally do not preserve
the negabent property of a Boolean function. It is therefore interesting to find
a subgroup of the bent-preserving operations that preserves also the negabent
property. The following theorem shows that, if we replace
GL
(2
,n
)by
O
(2
,n
),
the orthogonal group of
n
×
n
matrices over
×
n
matrices over
F
2
, we obtain such a subgroup.
