On more bent functions from Dillon exponents

Bent functions are maximally nonlinear Boolean functions with an even number of variables. They were introduced by Rothaus in 1976. For their own sake as interesting combinatorial objects, but also because of their relations to coding theory (Reed-Muller codes) and applications in cryptography (design of stream ciphers), they have attracted a lot of research, specially in the last 15 years. The class of bent functions contains a subclass of functions, introduced by Youssef and Gong in 2001, the so-called hyper-bent functions whose properties are still stronger and whose elements are still rarer than bent functions. Bent and hyper-bent functions are not classified. A complete classification of these functions is elusive and looks hopeless. So, it is important to design constructions in order to know as many of (hyper)-bent functions as possible. This paper is devoted to the constructions of bent and hyper-bent Boolean functions in polynomial forms. We survey and present an overview of the constructions discovered recently. We extensively investigate the link between the bentness property of such functions and some exponential sums (involving Dickson polynomials).

This paper aims at the identification of classes of bent functions in order to allow their construction without searching or sieving. In order to reach this aim, we studied first the relationship between bent functions and complexity classes defined by the Specific Normal Forms of all Boolean functions. As result of this exploration we found classes of bent functions which are embedded in different complexity classes defined by the Specific Normal Form. In the second step to reach our global aim, we utilized the found classes of bent functions in order to express bent functions in terms of derivative operations of the Boolean Differential Calculus. In detail, we studied bent functions of two and four variables. This exploration leads finally to Boolean differential equations that will allow the direct calculation of all bent functions of two and four variables. A given generalization allows to calculate subsets of bent functions for each even number of Boolean variables.

Binary bent functions have a strictly specified number of non-zero values. In the same way, ternary bent functions satisfy certain requirements on the elements of their value vectors. These requirements can be used to specify six classes of ternary bent functions. Classes are mutually related by encoding of function values. Functions within a class are mutually related by permutation of elements in their function vectors. Given a basic ternary bent function, other functions in the same class can be constructed by permutation matrices having a block structure similar to that of the factor matrices appearing in the Good-Thomas decomposition of Cooley-Tukey Fast Fourier transform and related algorithms. Conversion of function vectors into matrices or equivalent matrix-valued vectors of smaller length leads to the redistribution of space complexity of related manipulation algorithms. In this matrix representations, construction of other bent functions from a given known bent function is performed by using either properties of bent functions or by manipulation of such representations in terms of FFT-like permutation matrices of dimensions smaller than the length of the function vector of the initial bent functions.

Bent functions are a class of discrete functions which exhibit the highest degree of nonlinearity. As such bent functions form an essential part of cryptographic systems. Original concept of bent functions defined in GF(2) can be extended to multiple-valued case. Multiple-valued bent functions are defined in therms of properties of their Vilenkin-Chrestenson spectra. Decision diagrams are a method of compact representation of discrete functions. Special types of decision diagrams have been introduced for various types of discrete functions. In this paper we demonstrate how Vilenkin-Chrestenson decision diagrams can be used for efficient representation of multiple-valued bent functions.

Boolean functions are fundamental building blocks in both discrete mathematics and computer science, with applications spanning from cryptography to coding theory. Bent functions, a subset of Boolean functions with maximal nonlinearity, are particularly valuable in cryptographic applications. This study introduces a novel equivalence relation among all Boolean functions and presents an algorithm to generate bent functions based on this relation. We systematically generated a collection of 10,000 bent functions over eight variables, all originating from the same equivalence class, and analyzed their structural complexity through rank determination. Our findings revealed the presence of at least five distinct affine classes of bent functions within this collection. By employing this construction, we devised an algorithm to generate a filter function capable of combining Boolean functions. This filter function can be dynamically adjusted based on a key, offering potential applications in symmetric cipher design, such as enhancing security or improving efficiency.

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.

This article is a survey of several recent applications of methods from analytic number theory to research in coding theory, including results on Kloosterman codes, binary Goppa codes, and prime phase shift sequences. The mathematical methods focus on exponential sums, in particular Kloosterman sums. The interrelationships with the Weil–Carlitz–Uchiyama bound, results on Hecke operators, theorems of Bombieri and Deligne and the Eichler–Selberg trace formula are reviewed.

Hyper-bent functions as a subclass of bent functions attract much interest and it is elusive to completely characterize hyper-bent functions. Most of known hyper-bent functions are Boolean functions with Dillon exponents and they are often characterized by special values of Kloosterman sums. In this paper, we present a method for characterizing hyper-bent functions with Dillon exponents. A class of hyper-bent functions with Dillon exponents over \(\mathbb {F}_{2^{2m}}\) can be characterized by a Boolean function over \(\mathbb {F}_{2^m}\), whose Walsh spectrum takes the same value twice. Further, we show several classes of hyper-bent functions with Dillon exponents characterized by Kloosterman sum identities and the Walsh spectra of some common Boolean functions.

Let L be an even lattice of odd rank with discriminant group L ′ / L {L^{\prime}/L} , and let α , β ∈ L ′ / L {\alpha,\beta\in L^{\prime}/L} . We prove the Weil bound for the Kloosterman sums S α , β ⁢ ( m , n , c ) {S_{\alpha,\beta}(m,n,c)} of half-integral weight for the Weil Representation attached to L. We obtain this bound by proving an identity that relates a divisor sum of Kloosterman sums to a sparse exponential sum. This identity generalizes Kohnen’s identity for plus space Kloosterman sums with the theta multiplier system.

Kloosterman sum identities and low-weight codewords in a cyclic code with two zeros

Divisibility properties of classical binary Kloosterman sums

Mean value of some exponential sums and applications to Kloosterman sums

Recently in Çeşmelioğlu, Meidl (Adv. Math. Commun.,18, 2024), the study of EA-equivalence and CCZ-equivalence for functions from {mathbb {V}}_n^{(p)} to the cyclic group {mathbb {Z}}_{p^k} has been initiated, where {mathbb {V}}_n^{(p)} denotes an n-dimensional vector space over {mathbb {F}}_p. Amongst others it has been shown that there exist functions from {mathbb {V}}_n^{(2)} to {mathbb {Z}}_4 which are CCZ-equivalent but not EA-equivalent. We extend these results to larger classes of functions from {mathbb {V}}_n^{(p)} to {mathbb {Z}}_{p^k}. We then discuss constructions of generalized bent functions from {mathbb {V}}_n^{(p)} to {mathbb {Z}}_{p^k}, p odd or p=2 and n is even, which correspond to large affine spaces of bent functions. In particular we employ versions of the direct sum, the semi-direct sum and of a recent secondary bent function construction in Wang et. al., (IEEE Trans. Inform. Theory69, 2023), to generate large affine spaces of bent functions. Finally we present a solution for constructing generalized bent functions from {mathbb {V}}_n^{(2)} to {mathbb {Z}}_{2^k}, n odd, from arbitrary generalized bent functions from {mathbb {V}}_{n-1}^{(2)} to {mathbb {Z}}_{2^{k-1}}.

The bent functions are the Boolean functions of an even number of variables which are at the maximum possible distance from the set of all affine functions. In this paper, it is shown that each isometric mapping of the set of Boolean functions of n variables to itself preserving the class of bent functions is a combination of an affine transformation of coordinates and a shift by an affine function. It is proved that the affine functions are precisely all Boolean functions which are at the maximum possible distance from the class of bent functions.

The p-ary function f(x) mapping GF(p4k) to GF(p) and given by f(x)=Tr4k(xp3k+p2k-pk+1+x2) is proven to be a weakly regular bent function and the exact value of its Walsh transform coefficients is found. This is the first proven infinite class of nonquadratic generalized bent functions over the fields of an arbitrary odd characteristic. The proof is based on a few new results in the area of exponential sums and polynomials over finite fields that may also be interesting as independent problems.