Systematic Constructions of Bent-Negabent Functions, 2-Rotation Symmetric Bent-Negabent Functions and Their Duals

Bent-negabent functions have many important properties for their application in cryptography since they have the flat absolute spectrum under the both Walsh-Hadamard transform and nega-Hadamard transform. In this paper, we present four new systematic constructions of bent-negabent functions on $4k, 8k, 4k+2$ and $8k+2$ variables, respectively, by modifying the truth tables of two classes of quadratic bent-negabent functions with simple form. The algebraic normal forms and duals of these constructed functions are also determined. We further identify necessary and sufficient conditions for those bent-negabent functions which have the maximum algebraic degree. At last, by modifying the truth tables of a class of quadratic 2-rotation symmetric bent-negabent functions, we present a construction of 2-rotation symmetric bent-negabent functions with any possible algebraic degrees. Considering that there are probably no bent-negabent functions in the rotation symmetric class, it is the first significant attempt to construct bent-negabent functions in the generalized rotation symmetric class

Design and analysis of bent functions using M\mathcal{M}-subspaces

In this article, we provide the first systematic analysis of bent functions $f$ on $\mathbb{F}_2^{n}$ in the Maiorana-McFarland class $\mathcal{MM}$ regarding the origin and cardinality of their $\mathcal{M}$-subspaces, i.e., vector subspaces on which the second-order derivatives of $f$ vanish. By imposing restrictions on permutations $\pi$ of $\mathbb{F}_2^{n/2}$, we specify the conditions, such that Maiorana-McFarland bent functions $f(x,y)=x\cdot \pi(y) + h(y)$ admit a unique $\mathcal{M}$-subspace of dimension $n/2$. On the other hand, we show that permutations $\pi$ with linear structures give rise to Maiorana-McFarland bent functions that do not have this property. In this way, we contribute to the classification of Maiorana-McFarland bent functions, since the number of $\mathcal{M}$-subspaces is invariant under equivalence. Additionally, we give several generic methods of specifying permutations $\pi$ so that $f\in\mathcal{MM}$ admits a unique $\mathcal{M}$-subspace. Most notably, using the knowledge about $\mathcal{M}$-subspaces, we show that using the bent 4-concatenation of four suitably chosen Maiorana-McFarland bent functions, one can in a generic manner generate bent functions on $\mathbb{F}_2^{n}$ outside the completed Maiorana-McFarland class $\mathcal{MM}^\#$ for any even $n\geq 8$. Remarkably, with our construction methods it is possible to obtain inequivalent bent functions on $\mathbb{F}_2^8$ not stemming from two primary classes, the partial spread class $\mathcal{PS}$ and $\mathcal{MM}$. In this way, we contribute to a better understanding of the origin of bent functions in eight variables, since only a small fraction, of which size is about $2^{76}$, stems from $\mathcal{PS}$ and $\mathcal{MM}$, whereas the total number of bent functions on $\mathbb{F}_2^8$ is approximately $2^{106}$

Bent functions stemming from Maiorana-McFarland class being provably outside its completed version

In early nineties Carlet [1] introduced two new classes of bent functions, both derived from the Maiorana-McFarland ($\mathcal{M}$) class, and named them \cC and \cD class, respectively. Apart from a subclass of \cD, denoted by \cD_0 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 $\mathcal{M}$ class which has already been remarked in [21]. Assuming the possibility of specifying a bent function $f$ that belongs to one of these two classes (apart from \cD_0), 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 \cite{OutsideMM} concerning the specification of bent functions in \cC and \cD which are provably outside \cM. It is shown that these conditions, related to bent functions in class \cD, 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 $x^{2^r+1}$ have inverses which are never quadratic for $n >4$, which gives rise to an infinite class of bent functions in \cC but outside \cM. Similarly, using a relaxed set of sufficient conditions for bent functions in \cD and outside \cM, one explicit infinite class of such bent functions is identified. We also extend the inclusion property of certain subclasses of bent functions in \cC and \cD, as addressed initially in [1,21], that are ultimately within the completed $\mathcal{M}$ class. Most notably, we specify {\em another generic and explicit subclass} of \cD, which we call \cD_2^\star, whose members are bent functions provably outside the completed $\mathcal{M}$ class

Constructing new superclasses of bent functions from known ones

Some recent research articles [23, 24] addressed an explicit specification of indicators that specify bent functions in the so-called $\mathcal{C}$ and $\mathcal{D}$ classes, derived from the Maiorana- McFarland ($\mathcal{M}$) class by C. Carlet in 1994 [5]. Many of these bent functions that belong to $\mathcal{C}$ or $\mathcal{D}$ are provably outside the completed $\mathcal{M}$ class. Nevertheless, these modifications are performed on affine subspaces, whereas modifying bent functions on suitable subsets may provide us with further classes of bent functions. In this article, we exactly specify new families of bent functions obtained by adding together indicators typical for the $\mathcal{C}$ and $\mathcal{D}$ class, thus essentially modifying bent functions in $\mathcal{M}$ on suitable subsets instead of subspaces. It is shown that the modification of certain bent functions in $\mathcal{M}$ gives rise to new bent functions which are provably outside the completed $\mathcal{M}$ class. Moreover, we consider the so-called 4-bent concatenation (using four different bent functions on the same variable space) of the (non)modified bent functions in $\mathcal{M}$ and show that we can generate new bent functions in this way which do not belong to the completed $\mathcal{M}$ class either. This result is obtained by specifying explicitly the duals of four constituent bent functions used in the concatenation. The question whether these bent functions are also excluded from the completed versions of $\mathcal{PS}$, $\mathcal{C}$ or $\mathcal{D}$ remains open and is considered difficult due to the lack of membership indicators for these classes

Composition construction of new bent functions from known dually isomorphic bent functions

Bent functions are optimal combinatorial objects and have been studied over the last four decades. Secondary construction plays a central role in constructing bent functions since it may generate bent functions outside the primary classes of bent functions. In this study, we improve a theoretical framework of the secondary construction of bent functions in terms of the composition of Boolean functions. Based on this framework, we propose several constructions of bent functions through the composition of a balanced Boolean function and dually isomorphic (DI) bent functions defined herein. In addition, we present a construction of self-dual bent functions

The connection between quadratic bent-negabent functions and the Kerdock code

In this paper we prove that all bent functions in the Kerdock code, except for the coset of the symmetric quadratic bent function, are bent–negabent. In this direction, we characterize the set of quadratic bent–negabent functions and show some results connecting quadratic bent–negabent functions and the Kerdock code. Further, we note that there are bent–negabent preserving nonsingular transformations outside the well known class of orthogonal ones that might provide additional functions in the bent– negabent set. This is the first time we could identify non-orthogonal (nonsingular) linear transformations that preserve bent–negabent property for a special subset