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In-Depth Information
Let the Boolean function
f
:
V
⊕
V
→
F
2
be given by
f
(
x
1
,...,x
m
,y
1
,...,y
m
)=
σ
(
x
1
,...,x
m
)
·
(
y
1
,...,y
m
)+
g
(
x
1
,...,x
m
)
,
(1)
where
σ
:
V
→
V
is of the form
σ
(
x
1
,...,x
m
)=(
ψ
1
(
x
1
)
,φ
1
(
x
1
)+
ψ
2
(
x
2
)
,...,φ
m−
1
(
x
m−
1
)+
ψ
m
(
x
m
))
and
g
:
V
→
F
2
is defined by
g
(
x
1
,...,x
m
)=
h
1
(
x
1
)+
h
2
(
x
2
)+
···
+
h
m
(
x
m
)
.
Here,
ψ
1
,...,ψ
m
,φ
1
,...,φ
m−
1
are permutations on
V
n
and
h
1
,...,h
m
:
V
n
→
F
2
are arbitrary Boolean functions. Explicitly,
f
reads
f
(
x
1
,...,x
m
,y
1
,...,y
m
)
m
=
ψ
1
(
x
1
)
·
y
1
+
h
1
(
x
1
)+
(
y
j
·
[
φ
j−
1
(
x
j−
1
)+
ψ
j
(
x
j
)] +
h
j
(
x
j
))
.
j
=2
Since
σ
is a permutation,
f
belongs to the Maiorana-McFarland class, and is
therefore bent. In the next theorem, we will identify configurations of
σ
and
g
so that
f
is also negabent.
Theorem 4.
Let
m
be a positive integer satisfying
m
≡
1(mod3)
,andlet
k
be an integer satisfying
0
<k<m
and
k
≡
0(mod3)
or
(
m
−
k
)
≡
1(mod3)
.
Let
f
be as in (1), where
σ
(
x
1
,...,x
m
)=(
x
1
,x
1
+
x
2
,...,x
k−
1
+
ψ
(
x
k
)
,φ
(
x
k
)+
x
k
+1
,...,x
m−
1
+
x
m
)
g
(
x
1
,...,x
m
)=
h
(
x
k
)
,
ψ, φ
are permutations on
V
n
,and
h
:
V
n
→
F
2
is an arbitrary Boolean function.
(In other words,
ψ
1
,...,ψ
m
,φ
1
,...,φ
m−
1
are identity maps except for
ψ
:=
ψ
k
and
φ
:=
φ
k
,and
h
1
,...,h
m
are zero except for
h
:=
h
k
.) Then
f
is bent-
negabent.
A lemma is required to prove the theorem.
Lemma 5.
Let
s
be a nonnegative integer. For any
u
1
,...,u
s
,z
s
+1
∈
V
n
define
s
1)
(
z
j
+1
+
u
j
)
·z
j
i
wt(
z
j
)
,
E
s
(
z
s
+1
):=
(
−
j
=1
z
j
∈V
n
where an empty product is defined to be equal to
1
. Then we have
⎧
⎨
2
sn/
2
ω
c
(
−
1)
a·z
s
+1
s
≡
if
0(mod3)
E
s
(
z
s
+1
)=
2
sn/
2
ω
c
(
1)
a·z
s
+1
i
−
wt(
z
s
+1
)
−
if
s
≡
1(mod3)
⎩
2
(
s
+1)
n/
2
ω
c
δ
z
s
+1
+
a
if
s
≡
2(mod3)
for some
c
V
n
.Here
δ
a
denotes the Kronecker delta function, i.e.,
δ
a
equals
1
if
a
=0
andiszerootherwise.
∈
Z
8
and
a
∈
