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We also enlarge the class of symmetry operations over which the bent-negabent
property of a Boolean function is preserved. In particular, we show that the bent-
negabent property is an invariant with respect to the action of the orthogonal group
on the input vector space. Finally we provide an upper bound on the algebraic
degree of any bent-negabent Boolean function from the Maiorana-McFarland class.
2No on
Let
V
n
be an
n
-dimensional vector space over
F
2
.Let
f
:
V
n
→
F
2
be a Boolean
function. The
Hadamard transform
of
f
is defined to be
2
x
1)
−
1)
f
(
x
)+
u·x
,
H
(
f
)(
u
):=(
−
−
u
∈
V
n
.
(
∈
V
n
The
nega-Hadamard transform
of
f
is defined to be
1)
−
2
x
1)
f
(
x
)+
u·x
i
wt(
x
)
,
N
(
f
)(
u
):=(
−
(
−
u
∈
V
n
,
∈
V
n
where
i
:=
√
−
1andwt(
.
) denotes the Hamming weight. The function
f
is called
bent
if
|H
(
f
)(
u
)
|
=1 forall
u
∈
V
n
.
Similarly,
f
is called
negabent
if
|N
(
f
)(
u
)
|
=1 forall
u
∈
V
n
.
If
f
is both bent and negabent, we say that
f
is
bent-negabent
.
Now let
f
:
V
n
⊕
V
n
→
F
2
be a Boolean function of the form
f
(
x, y
)=
σ
(
x
)
·
y
+
g
(
x
)
,
where
σ
:
V
n
→
V
n
and
g
:
V
n
→
F
2
. It is well known that this function is bent
if and only if
σ
is a permutation. The whole set of such bent functions forms the
Maiorana-McFarland class
.
In the remainder of this section we will introduce some further notation and
ausefullemma.Write
V
n
=
U
⊕
W
, where dim
W
=
k
and
k
≤
n
,sothat
dim
U
=
n
−
k
.Let
f
:
V
n
→
F
2
be a Boolean function. For each fixed
x
∈
U
we
may view
f
(
x,
) as a Boolean function on
W
. We define the
partial Hadamard
transform
of
f
with respect to
W
as
H
W
(
f
)(
x, v
):=2
−
2
y∈W
·
1)
f
(
x,y
)+
v·y
,
(
−
v
∈
W.
We say that
f
is
bent with respect to
W
if
|H
W
(
f
)(
x, v
)
|
=1 foreach
x
∈
U, v
∈
W.
