In this paper we prove that in opposite to the cases of 6 and 8 variables, the Maiorana-McFarland construction does not describe the whole class of cubic bent functions in
n
variables for all
n
≥
10
.... Moreover, we show that for almost all values of
n
, these functions can simultaneously be homogeneous and have no affine derivatives.
In this paper, we put forward a new framework concerning the construction of (complete) permutations by setting certain restrictions on the coordinate functions. It is shown that our framework can ...construct much more (complete) permutations than the one in Pasalic et al. (2016). Moreover, the designed (complete) permutations can have no linear structure and better algebraic degree.
In this letter, we present a construction of bent functions which generalizes a work of Zhang et al. in 2016. Based on that, we obtain a cubic bent function in 10 variables and prove that, it has no ...affine derivative and does not belong to the completed Maiorana-McFarland class, which is opposite to all 6/8-variable cubic bent functions as they are inside the completed Maiorana-McFarland class. This is the first time a theoretical proof is given to show that the cubic bent functions in 10 variables can be outside the completed Maiorana-McFarland class. Before that, only a sporadic example with such properties was known by computer search. We also show that our function is EA-inequivalent to that sporadic one.
We use the well-known Maiorana–McFarland class to construct several important combinatorial structures. In the first place, we easily identify infinite classes of vectorial plateaued functions ...{F}:F2n→F2n such that all non-zero linear combinations of its component functions are also plateaued. More importantly, by setting certain restrictions on the component functions, the same approach also yields many infinite classes of permutations for any n≥6. Finally, we deduce some infinite classes of complete permutations, as a subclass of these permutations. Most notably, all these classes are of variable and controllable degree, the property being intrinsic to the construction method. The construction method is highly tweakable giving rise to many variations that again provide us with infinite classes of these structures.
In this article, we propose two secondary constructions of bent functions without any conditions on initial bent functions employed by these methods. It is shown that both methods generate bent ...functions that belong to the generalized Maiorana–McFarland (
GMM
n
/
2
+
k
) class of
n
-variable Boolean functions, with
n
even. The class
GMM
n
/
2
+
k
contains functions that can be viewed as a concatenation of
(
n
/
2
-
k
)
-variable (not necessarily distinct) affine functions, which was previously (mainly) used in the design of resilient Boolean functions. Most notably, we show that a subclass of bent functions generated by our first method is provably outside the completed Maiorana–McFarland class
MM
#
. This extremely large class of Boolean functions
GMM
n
/
2
+
k
, which is shown to properly include the standard Maiorana–McFarland class
MM
, may contain a significant subset of bent functions that are not the members of
MM
#
. In general, the inclusion of these bent functions, that are provably outside
MM
#
, into the completed partial spread class remains unknown.
The objective of this article is to broaden the understanding of the connections between bent functions and partial difference sets. Recently, the first two authors showed that the elements which a ...vectorial dual-bent function with certain additional properties maps to 0, form a partial difference set, which generalizes the connection between Boolean bent functions and Hadamard difference sets, and some later established connections between
p
-ary bent functions and partial difference sets to vectorial bent functions. We discuss the effects of coordinate transformations. As all currently known vectorial dual-bent functions
F
:
F
p
n
→
F
p
s
are linear equivalent to
l
-forms, i.e., to functions satisfying
F
(
β
x
)
=
β
l
F
(
x
)
for all
β
∈
F
p
s
, we investigate properties of partial difference sets obtained from
l
-forms. We show that they are unions of cosets of
F
p
s
∗
, which also can be seen as certain cyclotomic classes. We draw connections to known results on partial difference sets from cyclotomy. Motivated by experimental results, for a class of vectorial dual-bent functions from
F
p
n
to
F
p
s
, we show that the preimage set of the squares of
F
p
s
forms a partial difference set. This extends earlier results on
p
-ary bent functions.
In this paper, we consider constructions of rotation symmetric bent functions, which are of the forms: f c (x) = Σ i=1 m-1 c i (Σ j=0 n-1 x j x i+j ) + c m (Σ j=0 m-1 x j x m+j ) and f t (x) = Σ i=0 ...n-1 (x i x t+i x m+i + x i x t+i ) + Σ i=0 m-1 x i x m+i , where n = 2m, c i ϵ {0,1} (the subscript u of x u in the previous expressions is taken as u modulo n). For each case, a necessary and sufficient condition is obtained. To the best of our knowledge, this class of cubic rotation symmetric bent functions is the first example of an infinite class of nonquadratic rotation symmetric bent functions.
Generalized bent (gbent) functions is a class of functions f:Z2n→Zq, where q≥2 is a positive integer, that generalizes a concept of classical bent functions through their co-domain extension. A lot ...of research has recently been devoted towards derivation of the necessary and sufficient conditions when f is represented as a collection of Boolean functions. Nevertheless, apart from the necessary conditions that these component functions are bent when n is even (respectively semi-bent when n is odd), no general construction method has been proposed yet for n odd case. In this article, based on the use of the well-known Maiorana–McFarland (MM) class of functions, we give an explicit construction method of gbent functions, for any even q>2 when n is even and for any q of the form q=2r (for r>1) when n is odd. Thus, a long-term open problem of providing a general construction method of gbent functions, for odd n, has been solved. The method for odd n employs a large class of disjoint spectra semi-bent functions with certain additional properties which may be useful in other cryptographic applications.
Recently, the construction of bent functions that belong to the so-called
C
class and are provably outside the completed Maiorana-McFarland (
M
) class, introduced by Carlet almost three decades ago, ...has been addressed in several works. The main method for proving the class membership is based on a sufficient (but not necessary) condition that component functions of the permutation
π
that defines a bent function of the form
f
(
x
,
y
)
=
π
(
y
)
⋅
x
+
1
L
⊥
(
x
)
, where
x
,
y
∈
F
2
n
, (for a suitably chosen subspace
L
), do not admit non-trivial linear structures. The problem of finding such permutations and corresponding subspaces such that the pair additionally satisfies the so-called (
C
) property (
π
− 1
(
a
+
L
) is a flat for any
a
∈
F
2
n
) appears to be a difficult task. In this article, we provide a generic method for specifying such permutations which is based on a suitable space decomposition introduced by Baum and Neuwirth in the 1970’s. In contrast to this result, which gives many families of bent functions outside the completed
M
class, we also show that one cannot have the (
C
) property satisfied for permutations whose component functions are without linear structures, when the dimension of corresponding subspace
L
is relatively large. Furthermore, a class of vectorial bent functions
F
:
F
2
2
n
→
F
2
m
such that every component function of
F
is outside the completed
M
class (i.e.
F
is strongly outside
M
#
) is specified. The problem of increasing the output dimension
m
and especially specifying such functions with
m
=
n
seems to be difficult.
In the mid-sixties, Rothaus introduced the notion of bent function and later presented a secondary construction of bent functions (building new bent functions from already defined ones), called ...Rothaus’ construction. In Zhang et al. 2017 (‘Constructing bent functions outside the Maiorana–Mcfarland class using a general form of Rothaus,’ IEEE Transactions on Information Theory, 2017, vol. 63, no. 8, pp. 5336–5349.’) provided two constructions of bent functions using a general form of Rothaus and showed that the obtained classes lie outside the completed Maiorana–McFarland (${\rm {\cal M}{\cal M}}$MM) class. In this study, the authors propose two similar methods for constructing bent functions outside the completed ${\rm {\cal M}{\cal M}}$MM class but with significantly simplified sufficient conditions compared to those in Zhang et al. 2017. These simplified conditions do not induce any serious restrictions on the choice of permutations used in the construction apart from a simple requirement on their algebraic degree and the request that the component functions of one permutation do not admit linear structures. This enables us to generate a huge class of bent functions lying outside the completed ${\rm {\cal M}{\cal M}}$MM class. Even more importantly, they prove that the new classes of bent functions are affine inequivalent to the bent functions in Zhang et al. 2017.
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