Bent Functions Research Articles

Quantum Advantage in Cryptography with a Low-Connectivity Quantum Annealer

The application in cryptography of quantum algorithms for prime factorization fostered the interest in quantum computing. However, quantum computers, and particularly quantum annealers, can also be helpful to construct secure cryptographic keys. Indeed, finding robust Boolean functions for cryptography is an important problem in sequence ciphers, block ciphers, and hash functions, among others. Due to the super-exponential size $\mathcal{O}(2^{2^n})$ of the associated space, finding $n$-variable Boolean functions with global cryptographic constraints is computationally hard. This problem has already been addressed employing generic low-connected incoherent D-Wave quantum annealers. However, the limited connectivity of the Chimera graph, together with the exponential growth in the complexity of the Boolean function design problem, limit the problem scalability. Here, we propose a special-purpose coherent quantum annealing architecture with three couplers per qubit, designed to optimally encode the bent function design problem. A coherent quantum annealer with this tree-type architecture has the potential to solve the $8$-variable bent function design problem, which is classically unsolved, with only $127$ physical qubits and $126$ couplers. This paves the way to reach useful quantum supremacy within the framework of quantum annealing for cryptographic purposes.

Several new infinite families of bent functions via second order derivatives

Inspired by a recent work of Tang et al. on constructing bent functions [14, IEEE TIT, 63(1): 6149-6157, 2017], we introduce a property (Pτ) of any Boolean function that its second order derivatives vanish at any direction (ui,uj) for some τ-subset {u1,…,uτ} of $\mathbb {F}_{2^{n}}$ , and then establish a link between this property and the construction of Tang et al. (IEEE Trans. Inf. Theory 63(10), 6149–6157 2017). It enables us to find more bent functions efficiently. We construct (at least) five new infinite families of bent functions from some known functions: the Gold’s bent functions and some quadratic non-monomial bent functions, Leander’s monomial bent functions, Canteaut-Charpin-Kyureghyan’s monomial bent functions, and the Maiorana-McFarland class of bent functions, respectively. Our result generalizes some recent works on bent functions. We also provide the corresponding dual functions in all our constructions except the quadratic non-monomial one. It also turns out that we can get new bent functions outside the Maiorana-McFarland completed class.

An answer to an open problem of Mesnager on bent functions

In 2014, Mesnager proposed two open problems in [4] on the construction of bent functions. One problem has been settled by Tang et al. in 2017. However, the other is still outstanding, which is solved in this letter by considering a class of PS− vectorial bent functions.

On q-ary bent and plateaued functions

We obtain the following results. For any prime p the minimal Hamming distance between distinct regular p-ary bent functions of 2n variables is equal to $$p^n$$ . The number of p-ary regular bent functions at the distance $$p^n$$ from the quadratic bent function $$Q_n=x_1x_2+\cdots +x_{2n-1}x_{2n}$$ is equal to $$p^n(p^{n-1}+1)\cdots (p+1)(p-1)$$ for $$p>2$$ . The Hamming distance between distinct binary s-plateaued functions of n variables is not less than $$2^{\frac{s+n-2}{2}}$$ and the Hamming distance between distinct ternary s-plateaued functions of n variables is not less than $$3^{\frac{s+n-1}{2}}$$ . These bounds are tight. For $$p=3$$ we prove an upper bound on nonlinearity of ternary functions in terms of their correlation immunity. Moreover, functions reaching this bound are plateaued. For $$p=2$$ analogous result are well known but for large p it seems impossible. Constructions and some properties of p-ary plateaued functions are discussed.

A general framework for secondary constructions of bent and plateaued functions

In this work, we employ the concept of {\em composite representation} of Boolean functions, which represents an arbitrary Boolean function as a composition of one Boolean function and one vectorial function, for the purpose of specifying new secondary constructions of bent/plateaued functions. This representation gives a better understanding of the existing secondary constructions and it also allows us to provide a general construction framework of these objects. This framework essentially gives rise to an {\em infinite number} of possibilities to specify such secondary construction methods (with some induced sufficient conditions imposed on initial functions) and in particular we solve several open problems in this context. We provide several explicit methods for specifying new classes of bent/plateaued functions and demonstrate through examples that the imposed initial conditions can be easily satisfied. Our approach is especially efficient when defining new bent/plateaued functions on larger variable spaces than initial functions. For instance, it is shown that the indirect sum methods and Rothaus' construction are just special cases of this general framework and some explicit extensions of these methods are given. In particular, similarly to the basic indirect sum method of Carlet, we show that it is possible to derive (many) secondary constructions of bent functions without any additional condition on initial functions apart from the requirement that these are bent functions. In another direction, a few construction methods that generalize the secondary constructions which do not extend the variable space of the employed initial functions are also proposed.

Systematic Methods of Constructing Bent Functions and 2-Rotation Symmetric Bent Functions

In this paper, we first present two systematic constructions of bent functions by modifying the truth tables of Rothaus’s bent function and Maiorana-McFarland’s bent function respectively. The number of the newly constructed bent functions by modifying the truth table of Rothaus’s bent function is also determined. The methods of constructing self-dual bent functions are then given after the dual functions of these bent functions being determined. Finally, as an application, two constructions of 2-rotation symmetric bent functions are presented in this paper.

Experimental study of multi-stable morphing structures actuated by pneumatic actuation

In this paper, we proposed a new pneumatic actuation method inspired from a soft robot, which could drive the multi-stable carbon fiber-reinforced polymer (CFRP) structure transition among five stable configurations under air pressure. Multi-stable structure was manufactured by the carbon fiber prepreg laid in asymmetrical and anti-symmetric layup methods and stacked at the joint during the preparation process according to the designed scheme. In experiments, multi-stable structures with two different fibers layups were prepared to study the snap-through behaviors. The snap-through process of two kinds of multi-stable structures was investigated by experiment. The parameters used to characterize efficiency of the pneumatic actuator actuating the multi-stable structures, including air pressure and snap time, were measured. The influence of the pneumatic actuator on the curvature of multi-stable structures was discussed by comparing the shape characteristic of multi-stable structures with or without pneumatic actuation, and it showed pneumatic actuator has little effect to the structure. The results demonstrated that the proposed pneumatic actuation could be used to drive multi-stable morphing structure flexibly. The combination of multi-stable laminates develops the function of multiple point bending and shape retention of pneumatic actuation.

Shifted plateaued functions and their differential properties

A bent4 function is a Boolean function with a flat spectrum with respect to a certain unitary transform $\mathcal {T}$ . It was shown previously that a Boolean function f in an even number of variables is bent4 if and only if f + σ is bent, where σ is a certain quadratic function depending on $\mathcal {T}$ . Hence bent4 functions are also called shifted bent functions. Similarly, a Boolean function f in an odd number of variables is bent4 if and only if f + σ is a semibent function satisfying some additional properties. In this article, for the first time, we analyse in detail the effect of the shifts on plateaued functions, on partially bent functions and on the linear structures of Boolean functions. We also discuss constructions of bent and bent4 functions from partially bent functions and study the differential properties of partially bent4 functions, unifying the previous work on partially bent functions.

Duals of non-weakly regular bent functions are not weakly regular and generalization to plateaued functions

It is known that the dual of a weakly regular bent function is again weakly regular. On the other hand, the dual of a non-weakly regular bent function may not even be a bent function. In 2013, Çesmelioğlu, Meidl and Pott pointed out that the existence of a non-weakly regular bent function having weakly regular bent dual is an open problem. In this paper, we prove that for an odd prime p and n∈Z+, if f:Fpn→Fp is a non-weakly regular bent function such that its dual f⁎ is bent, then f⁎⁎(−x)=f(x), and f⁎ is non-weakly regular, which solves the open problem. We also generalize our results to plateaued functions.

Correction to: Algebraic Construction of Near-Bent and APN Functions

Remark 4.1 and Remark 4.5 in Section 4 will be true only if a is a power of two. Hence, the proofs of Theorem 4.2 and Theorem 4.6 hold in the case of a being a power of 2. More generally, both Theorems 4.2 and 4.6 are true by observing the change of variable x → y + 1 in the Walsh-Hadamard Transform, as it is a permutation on the finite field, i.e., we get exactly opposite values of the function f(x) = Tr( x)d, where d is the Niho exponent. Since the function f(x) = Tr(x)d is near-bent function, the function f(x) = Tr(x+1)d is also a near bent function

Linear codes with few weights from cyclotomic classes and weakly regular bent functions

Linear codes with few weights constructed from defining sets have been extensively studied due to their applications in data storage systems, secret sharing schemes and authentication codes. In this paper, inspired by the works of Li et al. (Appl Algebra Eng Commun Comput 28(1):11–30, 2017) and Jian et al. (Finite Fields Appl 57:92–107, 2019), we propose a class of five-weight linear codes by choosing the defining set from cyclotomic classes, which includes a class of two-weight linear codes as a special case, and we also present two classes of two or three-weight linear codes by employing weakly regular bent functions. Besides, we obtain a class of two-weight optimal punctured codes with respect to the Griesmer bound.

Polynomial Criterion for Abelian Difference Sets

Difference sets are subsets of a group satisfying certain combinatorial property with respect to the group operation. They can be characterized using an equality in the group ring of the corresponding group. In this paper, we exploit the special structure of the group ring of an Abelian group to establish a one-to one correspondence of the class of difference sets with specific parameters in that group with the set of all complex solutions of a specified system of polynomial equations. The correspondence also develops some tests for a Boolean function to be a bent function.

Improved lower bound for algebraic immunity of a subclass of MM bent Boolean functions

It is known that algebraic immunity of an n-variable Boolean function is upper bounded by []. In literature, it was shown algebraic immunity of a subclass of Maiorana-McFarland (MM) bent Boolean functions is lower bounded by 2 and upper bounded by . In this paper, this lower bound is further improved. Given a Boolean function of -variables with algebraic immunity k ≥ 4, the improved lower bound helps a cryptographer to find a MM bent function of this class whose algebraic immunity is at least k − 2.

Cubic bent functions outside the completed Maiorana-McFarland class

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 nge 10. Moreover, we show that for almost all values of n, these functions can simultaneously be homogeneous and have no affine derivatives.

EA-inequivalence of bent functions

The question of establishing EA-inequivalence among the classes of bent functions remains in general an open problem. The EA-inequivalence is also relevant in classifying bent functions within the same class. This paper is an attempt to investigate these questions for the Maiorana–McFarland ($$\mathcal{M}$$) class and so-called class $$\mathcal{H}$$ of bent functions. For cubic bent functions of the form $$Tr_1^t(xy^{2^i+1})$$ in $$\mathcal{M}$$, the necessary and sufficient conditions related to EA-equivalence are derived. It is also shown that in most of the cases, at least over finite fields of relatively small order, bent functions within $$\mathcal{H}$$ are EA-inequivalent.

Generalized bent Boolean functions and strongly regular Cayley graphs

In this paper we define the (edge-weighted) Cayley graph associated to a generalized Boolean function, introduce a notion of strong regularity and give several of its properties. We show some connections between this concept and generalized bent functions (gbent), that is, functions with flat generalized Walsh–Hadamard spectrum. In particular, we find a complete characterization of quartic gbent functions in terms of the strong regularity of their associated Cayley graph.

On [formula omitted]-nearly bent Boolean functions

For each non-constant Boolean function q, Klapper introduced the notion of q-transforms of Boolean functions. The q-transform of a Boolean function f is related to the Hamming distances from f to the functions obtainable from q by nonsingular linear change of basis.In this work, we discuss the existence of q-nearly bent functions, a new family of Boolean functions characterized by the q-transform. We prove that any balanced Boolean functions (linear or non-linear) are q-nearly bent if q has weight one, which gives a positive answer to an open question (whether there exist non-affine q-nearly bent functions) proposed by Klapper. We also prove a necessary condition for checking when a function is not q-nearly bent.

New Classes of Traveling Waves in a Planar Kirchhoff Beam with Nonlinear Bending Stiffness

New traveling wave solutions are presented for motion of an inextensible, unshearable, planar Kirchhoff beam endowed with rotary inertia and a generalized strain energy function for bending which models nonlinear stiffening and softening. It is shown that although sonic waves (i.e., wave traveling at the bar speed in the beam) do not exist for constant bending stiffness, nonlinear bending stiffness supports new classes of solutions that admit nonlinear waves traveling with axial tension or compression. In addition, it is shown that waves traveling at the bar wave speed may be compactons (i.e., solitary waves of finite span). Surprisingly, for a large class of constant and nonlinear bending stiffness, initially circular rods do not support traveling waves.

Balanced crossover operators in Genetic Algorithms

Abstract In several combinatorial optimization problems arising in cryptography and design theory, the admissible solutions must often satisfy a balancedness constraint, such as being represented by bitstrings with a fixed number of ones. For this reason, several works in the literature tackling these optimization problems with Genetic Algorithms (GA) introduced new balanced crossover operators which ensure that the offspring has the same balancedness characteristics of the parents. However, the use of such operators has never been thoroughly motivated, except for some generic considerations about search space reduction. In this paper, we undertake a rigorous statistical investigation on the effect of balanced and unbalanced crossover operators against three optimization problems from the area of cryptography and coding theory: nonlinear balanced Boolean functions, binary Orthogonal Arrays (OA) and bent functions. In particular, we consider three different balanced crossover operators (each with two variants: “left-to-right” and “shuffled”), two of which have never been published before, and compare their performances with classic one-point crossover. We are able to confirm that the balanced crossover operators perform better than one-point crossover. Furthermore, in two out of three crossovers, the “left-to-right” version performs better than the “shuffled” version.

Gowers U_2 norm as a measure of nonlinearity for Boolean functions and their generalizations

<p style='text-indent:20px;'>In this paper, we investigate the Gowers <inline-formula><tex-math id="M2">\begin{document}$ U_2 $\end{document}</tex-math></inline-formula> norm for generalized Boolean functions, and <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-bent functions. The Gowers <inline-formula><tex-math id="M4">\begin{document}$ U_2 $\end{document}</tex-math></inline-formula> norm of a function is a measure of its resistance to affine approximation. Although nonlinearity serves the same purpose for the classical Boolean functions, it does not extend easily to generalized Boolean functions. We first provide a framework for employing the Gowers <inline-formula><tex-math id="M5">\begin{document}$ U_2 $\end{document}</tex-math></inline-formula> norm in the context of generalized Boolean functions with cryptographic significance, in particular, we give a recurrence rule for the Gowers <inline-formula><tex-math id="M6">\begin{document}$ U_2 $\end{document}</tex-math></inline-formula> norms, and an evaluation of the Gowers <inline-formula><tex-math id="M7">\begin{document}$ U_2 $\end{document}</tex-math></inline-formula> norm of functions that are affine over spreads. We also give an introduction to <inline-formula><tex-math id="M8">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-bent functions, as proposed by Dobbertin and Leander [<xref ref-type="bibr" rid="b8">8</xref>], to provide a recursive framework to study bent functions. In the second part of the paper, we concentrate on <inline-formula><tex-math id="M9">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-bent functions and their <inline-formula><tex-math id="M10">\begin{document}$ U_2 $\end{document}</tex-math></inline-formula> norms. As a consequence of one of our results, we give an alternate proof to a known theorem of Dobbertin and Leander, and also find necessary and sufficient conditions for a function obtained by <i>gluing</i> <inline-formula><tex-math id="M11">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-bent functions to be bent, in terms of the Gowers <inline-formula><tex-math id="M12">\begin{document}$ U_2 $\end{document}</tex-math></inline-formula> norms of its components.