It is shown that the generalized Rothaus construction of p -ary bent functions can be extended to a construction of a vectorial bent function with non-weakly regular components, for which in general ...the duals are not a bent function, i.e., they belong to the class of non-dual bent functions. This complements results on other two constructions of non-weakly regular bent functions, the generalized Maiorana-McFarland construction and the semi-direct sum, for which vectorial versions are presented and the properties of their duals are investigated in the literature. The distribution of the values of the Walsh transform of vectorial bent functions (with non-weakly regular components) is then analysed in detail. Among others, a condition on the values of the Walsh transform of a vectorial bent function from F n p to F m p is presented, which implies that m ≤ ⌈ n /2⌉. This refines a classical result by Nyberg, which states that for an ( n , m ) bent function, n even, with only regular components, m can be at most n /2. Some results on the weight distribution of codes obtained from vectorial bent functions with non-weakly regular components complement the article.
Bent partitions of V n (p) are quite powerful in constructing bent functions, vectorial bent functions and generalized bent functions, where V n (p) is an n -dimensional vector space over F p , n is ...an even positive integer and p is a prime. The classical examples of bent partitions are obtained from (partial) spreads. In 5, 19, two classes of bent partitions which are not obtained from (partial) spreads were presented. In 3, more bent partitions Γ 1 , Γ 2 , Γ * 1 , Γ * 2 , Θ 1 , Θ 2 were presented from (pre)semifields, including the bent partitions given in 5, 19. In this paper, we investigate the relations between bent partitions and vectorial dual-bent functions. For any prime p, we show that one can generate certain bent partitions (called bent partitions satisfying Condition C ) from certain vectorial dualbent functions (called vectorial dual-bent functions satisfying Condition A). In particular, when p is an odd prime, we show that bent partitions satisfying Condition C one-to-one correspond to vectorial dual-bent functions satisfying Condition A. We give an alternative proof that Γ 1 , Γ 2 , Γ * 1 , Γ * 2 , Θ 1 are bent partitions in terms of vectorial dual-bent functions. We present a secondary construction of vectorial dual-bent functions, which can be used to generate more bent partitions. We show that any weakly regular ternary bent function f : V n (3) → F 3 ( n is even) of 2-form can generate a bent partition. When such f is weakly regular but not regular, the generated bent partition from f is not coming from a normal bent partition, which answers an open problem proposed in 5. We give a sufficient condition on constructing partial difference sets from bent partitions, and when p is an odd prime, we provide a characterization of bent partitions satisfying Condition C in terms of partial difference sets.
In this survey, we revisit the Rothaus paper and the chapter of Dillon’s thesis dedicated to bent functions, and we describe the main results obtained on these functions during these last 40 years. ...We also cover more briefly super-classes of Boolean functions, vectorial bent functions and bent functions in odd characteristic.
Quadratic almost bent (AB) functions are characterized by the property that the duals of their component functions are bent functions. We prove that these duals are also quadratic and illustrate that ...these bent duals may give rise to vectorial bent functions (in certain cases having a maximal output dimension). A necessary and sufficient condition for ensuring bentness of the linear combinations of quadratic bent duals is provided. Moreover, we provide a rather detailed analysis related to the structure of quadratic AB functions in the spectral domain, more precisely with respect to their Walsh supports, their intersection and restrictions of these bent duals to suitable subspaces. In particular, we completely determine the intersection of Walsh supports of the coordinate (semi-bent) functions for Gold AB mappings. We also provide the design of quadratic AB functions in the spectral domain by identifying (using computer simulations) suitable sets of bent dual functions. For instance, when n=7, this approach provides several AB functions which are not CCZ-equivalent to Gold functions.
In this paper, we show that the maximum number of bent component functions of a vectorial function F : GF(2) n → GF(2) n is 2 n - 2 n/2 . We also show that it is very easy to construct such ...functions. However, it is a much more challenging task to find such functions in polynomial form F ∈ GF(2 n )x, where F has only a few terms. The only known power functions having such a large number of bent components are x d , where d = 2 n/2 + 1. In this paper, we show that the binomials F i (x) = x 2i (x + x(2 n/2 )) also have such a large number of bent components, and these binomials are inequivalent to the monomials x(2 n/2 +1) if 0 <; i <; n/2. In addition, the functions Fi have differential properties much better than x(2 n/2 +1). We also determine the complete Walsh spectrum of our functions when n/2 is odd and gcd(i, n/2) = 1.
Vectorial bent functions and their duals Çeşmelioğlu, Ayça; Meidl, Wilfried; Pott, Alexander
Linear algebra and its applications,
07/2018, Volume:
548
Journal Article
Peer reviewed
Motivated by the observation that for two (weakly regular) bent functions f,g for which also f+g is bent, the sum f⁎+g⁎ of their duals f⁎ and g⁎ is sometimes but not always bent, we initiate the ...study of duality for vectorial bent functions. We propose and investigate two concepts of self-duality for vectorial bent functions, self-duality and weak self-duality.
Bent functions
f
:
V
n
→
F
p
play an important role in constructing partial difference sets, where
V
n
denotes an
n
-dimensional vector space over
F
p
,
p
is an odd prime. In
2
,
3
, the so-called ...vectorial dual-bent functions are considered to construct partial difference sets. In
2
, Çeşmelioğlu
et al.
showed that for certain vectorial dual-bent functions
F
:
V
n
→
V
s
, the preimage set of 0 for
F
forms a partial difference set. In
3
, Çeşmelioğlu
et al.
showed that for a class of Maiorana-McFarland vectorial dual-bent functions
F
:
V
n
→
F
p
s
, the preimage set of the squares (non-squares) in
F
p
s
∗
for
F
forms a partial difference set. In this paper, we further study vectorial dual-bent functions and partial difference sets. We prove that for certain vectorial dual-bent functions
F
:
V
n
→
F
p
s
, the preimage set of the squares (non-squares) in
F
p
s
∗
for
F
and the preimage set of any coset of some subgroup of
F
p
s
∗
for
F
form partial difference sets. Furthermore, explicit constructions of partial difference sets are yielded from some (non-)quadratic vectorial dual-bent functions. In this paper, we illustrate that many results of using weakly regular
p
-ary bent functions to construct partial difference sets are special cases of our results. In
2
, the authors considered weakly regular
p
-ary bent functions
f
with
f
(
0
)
=
0
. They showed that if such a function
f
is an
l
-form with
g
c
d
(
l
-
1
,
p
-
1
)
=
1
for some integer
1
≤
l
≤
p
-
1
, then
f
is vectorial dual-bent. We prove that the converse also holds, which answers one open problem proposed in
3
.
In 2017, Tang et al. have introduced a generic construction for bent functions of the form
f
(
x
)
=
g
(
x
)
+
h
(
x
)
, where
g
is a bent function satisfying some conditions and
h
is a Boolean ...function. Recently, Zheng et al. (Discret Math 344:112473, 2021) generalized this result to construct large classes of bent vectorial Boolean functions from known ones in the form
F
(
x
)
=
G
(
x
)
+
h
(
X
)
, where
G
is a vectorial bent and
h
is a Boolean function. In this paper, we further generalize this construction to obtain vectorial bent functions of the form
F
(
x
)
=
G
(
x
)
+
H
(
X
)
, where
H
is also a vectorial Boolean function. This allows us to construct new infinite families of vectorial bent functions, EA-inequivalent to
G
, which was used in the construction. Most notably, specifying
H
(
x
)
=
h
(
T
r
1
n
(
u
1
x
)
,
…
,
T
r
1
n
(
u
t
x
)
)
, the function
h
:
F
2
t
→
F
2
t
can be chosen arbitrarily, which gives a relatively large class of different functions for a fixed function
G
. We also propose a method of constructing vectorial (
n
,
n
)-functions having maximal number of bent components.
In this paper, we study the vectorial bentness of an arbitrary quadratic form and construct two classes of linear codes of few weights from the quadratic forms. Let
q
be a prime power,
m
be a ...positive integer and
Q
:
F
q
m
→
F
q
be a quadratic form. We first show that
Q
is a vectorial bent function if and only if
Q
is non-degenerate and
(
q
+
1
)
m
is even (i.e. either
q
is odd or
m
is even). Furthermore, if
2
∣
(
q
+
1
)
m
and
Q
(
x
)
=
∑
i
=
0
m
-
1
Tr
q
m
/
q
(
a
i
x
q
i
+
1
)
(
a
i
≠
0
)
, we show that
Q
is vectorial bent if and only if the associated additive polynomial
L
Q
(
x
)
=
∑
i
(
a
i
+
a
m
-
i
q
i
)
x
q
i
is a permutation polynomial over
F
q
m
. If there is only one
a
i
≠
0
, we recover the constructions of Sidelnikov, Dembowski-Ostrom and Kasami of quadratic vectorial bent functions. We then construct two classes of linear codes
C
Q
′
and
C
Q
over
F
q
from
Q
and completely determine the weight distributions of our codes, showing that they are two-, three- or four-weight codes and contain optimal codes satisfying the Griesmer and Singleton bounds.
In this paper, we provide necessary and sufficient conditions for a function of the form F(x)=Trk 2k (Σi=1 t aix ri(2k -1)) to be bent. Three equivalent statements, all of them providing both the ...necessary and sufficient conditions, are derived. In particular, one characterization provides an interesting link between the bentness and the evaluation of F on the cyclic group of the (2 k +1)th primitive roots of unity in GF(2 2k ). More precisely, for this group of cardinality 2 k +1 given by U={u ∈ GF(2 2k ):u 2k +1=1}, it is shown that the property of being vectorial bent implies that Im(F)=GF(2 k )∪{0}, if F is evaluated on U, that is, F(u) takes all possible values of GF(2 k )* exactly once and the zero value is taken twice when u ranges over U. This condition is then reformulated in terms of the evaluation of certain elementary symmetric polynomials related to F, which in turn gives some necessary conditions on the coefficients ai (for binomial trace functions) that can be stated explicitly. Finally, we show that a bent trace monomial of Dillon's type Trk 2k (λx r(2k -1)) is never a vectorial bent function.
