New Classes Of Bent Functions Research Articles

Constructions of (vectorial) bent functions outside the completed Maiorana–McFarland class

Two new classes of bent functions derived from the Maiorana–McFarland (M) class, so-called C and D, were introduced by Carlet (1993) almost three decades ago. In Zhang (2020) sufficient conditions for specifying bent functions in C and D which are outside the completed M class, denoted by M#, were given. Furthermore in Pasalic et al. (2021) the notion of vectorial bent functions which are weakly or strongly outsideM#, referring respectively to the case whether some or all nonzero linear combinations (called components) of its coordinate functions are in class C (or D) but provably outside M#, was introduced. In this article we continue the work of finding new instances of vectorial bent functions weakly/strongly outside M# using a different approach. Namely, a generic method for the construction of vectorial bent (n,t)-functions of the form F(x,y)=G(x,y)+H(x,y), n=2m,t|m, was recently proposed in Bapić (2021), where G is a given bent (n,t)-function satisfying certain properties and H is an arbitrary (t,t)-function having certain form. We introduce a new superclass of bent functions SC which contains the classes D0 and C and whose members are provably outside M#. Most notably, using indicators of the form 1L⊥(x,y)+δ0(x) to define members of this class leads for the first time to modifications of the M class performed on sets rather than on affine subspaces. We also show that for suitable choices of H, the function F is a vectorial bent function weakly/strongly outside the class M#. In this context, a new concept of being almost strongly outside M# is introduced and some families of vectorial bent functions with this property are given. Furthermore, we provide two new families of vectorial bent functions strongly outside M# (considered to be an intrinsically hard problem) whose output dimension is greater than 2, thus giving first examples of such functions in the literature.

Vectorial bent functions weakly/strongly outside the completed Maiorana–McFarland class

Two new classes of bent functions derived from the Maiorana–McFarland (M) class, so-called C and D, were introduced by Carlet (1994) two decades ago. The difficulty of satisfying their defining conditions was emphasized in Mandal et al. (2016). In a recent work Zhang et al. (2017) a set of efficient sufficient conditions for specifying bent functions in C and D which are outside the completed M class, denoted by M#, was given. A natural follow up question is whether there is a possibility of extending this approach to the vectorial case. We introduce the property of vectorial bent functions that we call weakly or strongly outsideM#, referring respectively to the case whether some or all nonzero linear combinations (called components) of its coordinate functions are in class C (or D) but provably outside M#. For the first time, quite different to a straightforward vectorial extension of the Maiorana–McFarland class and the class of Dillon PSap, we show the existence of several classes of vectorial bent functions whose component functions come from different classes of bent functions, mainly from M and D, and in many cases being weakly outside M#. We also address a difficult problem of specifying vectorial bent functions whose all components are in class C but provably outside M#, thus being strongly outside M#. Even though we could only specify a class of such functions whose dimension of bent vector space is only two, thus F:F2n→F22, this is the very first evidence of their existence.

Further constructions of bent functions and their duals

In 2012, Carlet et al. developed two secondary constructions of bent functions (Advances in Mathematics of Communications, 6: 305-314) and proposed some applications for their constructions. However, the duals of bent functions in their constructions were not presented. In order to find more general applications to these constructions and obtain new classes of bent functions, an open problem was proposed by Carlet in 2014. Hence, in this study, a class of PS vectorial bent functions for answering that open problem, which also addresses another open problem on vectorial bent functions proposed by Mesnager in 2014, is constructed. In addition, a new secondary construction of bent functions that generalises one of Carlet et al.'s constructions in 2012 is presented. Based on that, two new classes of bent functions were obtained and their duals were presented explicitly. In particular, some self-dual bent functions are constructed. Moreover, it can be proved that our bent functions can be EA-inequivalent to those constructed by Carlet et al. in 2012.

Further study on constructing bent functions outside the completed Maiorana–McFarland class

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 ( ) class. In this study, the authors propose two similar methods for constructing bent functions outside the completed 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 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.

Further analysis of bent functions from [formula omitted] and [formula omitted] which are provably outside or inside [formula omitted

In early nineties Carlet (1994) introduced two new classes of bent functions, both derived from the Maiorana–McFarland (M) class, and named them C and D class, respectively. Apart from a subclass of D, denoted by D0 by Carlet, which is provably outside two main (completed) primary classes of bent functions, little is known about their efficient constructions. More importantly, both classes may easily remain in the underlying M class which has already been remarked in Mandal et al. (2016). Assuming the possibility of specifying a bent function f that belongs to one of these two classes (apart from D0), the most important issue is then to determine whether f is still contained in the known primary classes or lies outside their completed versions. In this article, we further elaborate on the analysis of the set of sufficient conditions given in Zhang et al. (2017) concerning the specification of bent functions in C and D which are provably outside M. It is shown that these conditions, related to bent functions in class D, can be relaxed so that even those permutations whose component functions admit linear structures still can be used in the design. It is also shown that monomial permutations of the form x2r+1 have inverses which are never quadratic for n>4, which gives rise to an infinite class of bent functions in C but outside M. Similarly, using a relaxed set of sufficient conditions for bent functions in D and outside M, one explicit infinite class of such bent functions is identified. We also extend the inclusion property of certain subclasses of bent functions in C and D, as addressed initially in Carlet (1994), Mandal et al. (2016) that are ultimately within the completed M class. Most notably, we specify other generic and explicit subclasses of D, denoted by Dk⋆ for k∈{1,…,n−2}, whose members are bent functions provably outside the completed M class.

Designing Plateaued Boolean Functions in Spectral Domain and Their Classification

The design of plateaued functions over $GF(2)^n$, also known as 3-valued Walsh spectra functions (taking the values from the set $\{0, \pm 2^{\lceil \frac{n+s}{2} \rceil}\}$), has been commonly approached by specifying a suitable algebraic normal form which then induces this particular Walsh spectral characterization. In this article, we consider the reversed design method which specifies these functions in the spectral domain by specifying a suitable allocation of the nonzero spectral values and their signs. We analyze the properties of trivial and nontrivial plateaued functions (as affine inequivalent distinct subclasses), which are distinguished by their Walsh support $S_f$ (the subset of $GF(2)^n$ having the nonzero spectral values) in terms of whether it is an affine subspace or not. The former class exactly corresponds to partially bent functions and admits linear structures, whereas the latter class may contain functions without linear structures. A simple sufficient condition on $S_f$, which ensures the nonexistence of linear structures, is derived and some generic design methods of nontrivial plateaued functions without linear structures are given. The extended affine equivalence of plateaued functions is also addressed using the concept of dual of plateaued functions. Furthermore, we solve the problem of specifying disjoint spectra (non)trivial plateaued functions of maximal cardinality whose concatenation can be used to construct bent functions in a generic manner. This approach may lead to new classes of bent functions due to large variety of possibilities to select underlying duals that define these disjoint spectra plateaued functions. An additional method of specifying affine inequivalent plateaued functions, obtained by applying a nonlinear transform to their input domain, is also given.

Constructing new APN functions and bent functions over finite fields of odd characteristic via the switching method

The switching method is a very powerful method to construct new APN functions and differentially 4-uniform permutations over finite fields of even characteristic. In this paper, using this method, we present several new constructions of infinite classes of nonpower APN functions and two new classes of bent functions in finite fields of odd characteristic.

Several New Classes of Bent Functions From Dillon Exponents

Several new classes of binary and p-ary regular bent functions are obtained in this paper. The bentness of all these functions is determined by some exponential sums over finite fields, most of which have close relations with the well-known Kloosterman sums.

A New Class of Bent Functions

In this letter, using techniques from linear algebra and coding theory, we characterize the quadratic Boolean functions represented by trace. We show that a linear combination of trace-terms over finite field can be determined to be bent by a polynomial GCD computation. Then we derive some new families of bent functions.

New cyclic difference sets with Singer parameters

The main result in this paper is a general construction of φ( m)/2 pairwise inequivalent cyclic difference sets with Singer parameters ( v, k, λ)=(2 m −1,2 m−1 ,2 m−2 ) for any m⩾3. The construction was conjectured by the second author at Oberwolfach in 1998. We also give a complete proof of related conjectures made by No, Chung and Yun and by No, Golomb, Gong, Lee and Gaal which produce another difference set for each m⩾7 not a multiple of 3. Our proofs exploit Fourier analysis on the additive group of GF(2 m ) and draw heavily on the theory of quadratic forms in characteristic 2. By-products of our results are a new class of bent functions and a new short proof of the exceptionality of the Müller–Cohen–Matthews polynomials. Furthermore, following the results of this paper, there are today no sporadic examples of difference sets with these parameters; i.e. every known such difference set belongs to a series given by a constructive theorem.

Generalized partial spreads

We exhibit a simple condition under which the sum (modulo 2) of characteristic functions of (n/2)-dimensional vector subspaces of (GF(2))/sup n/ (n even) is a Bent function. The "Fourier" transform of such a Bent function is the sum of the characteristic functions of the duals of these spaces. The class of Bent functions that we obtain contains the whole partial spreads class. Any element of Maiorana-McFarland's class or of class D is equivalent to one of its elements. Thus this new class gives a unified insight of both general classes of Bent functions studied by Dillon (1974) in his thesis. We deduce a way to construct new classes of Bent functions and exhibit an example.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>