Hyper-bent Functions Research Articles

Hyperbent functions from hyperovals

Hyperbent functions from hyperovals

Möbius transformations and characterizations of hyper-bent functions from Dillon-like exponents with coefficients in extension fields

In this paper, we are interested in the characterization of the hyper-bentness property of all functions defined over the Galois field $ \mathbb{F}_{2^{2m}} $ in polynomial forms as sums of multiple trace terms obtained via Dillon-like exponents with coefficients in the extension Galois field $ \mathbb{F}_{2^{2m}} $. Characterizations of hyper-bentness were provided firstly by Charpin and Gong fourteen years ago for functions defined over $ \mathbb{F}_{2^{2m}} $, with coefficients in $ \mathbb{F}_{2^{m}} $, in terms of some exponential sums over $ \mathbb{F}_{2^m}{} $ involving Dickson polynomials. Mesnager extended their results in 2011 by considering adding a trace monomial function defined over specific fields of small cardinalities.By employing the Möbius transformation, we succeed in characterizing the hyper-bentness property of functions with Dillon-like exponents with coefficients in the whole $ \mathbb{F}_{2^{2m}} $. We emphasize that this ascension offers more choices on the coefficients (and therefore in exhibiting possible new hyper-bent functions) by keeping the evaluations of the related exponential sums in the $ \mathbb{F}_{2^m}{} $ as well as the computation of the number of rational points on the associate hyperelliptic curves on the same field in the spirit of previous achievements due to Lisoněk (2011) and Flori and Mesnager (2012-2013) in this context. This strategy gives a generalization of the result due to Charpin and Gong, and its effectiveness can be seen from a new infinite family of binomial hyper-bent functions. Their coefficients are in an extension field. Nevertheless, as all the previously found functions, they still lie in the well-known class denoted by $ \mathcal{PS}_{ap}^{\#} $, initially studied in 2006 by Carlet and Gaborit.

New characterizations and construction methods of bent and hyper-bent Boolean functions

In this paper, we first derive a necessary and sufficient condition for a bent Boolean function by analyzing their support set. Next, using this condition and the Pless power moment identities, we propose a construction method of bent functions of 2k variables by a suitable choice of 2k-dimension subspace of F222k−1−2k−1. Further, we extend our results to the so-called hyper-bent functions.

On generalized hyper-bent functions

Hyper-bent Boolean functions were introduced in 2001 by Youssef and Gong (and initially proposed by Golomb and Gong in 1999 as a component of S-boxes) to ensure the security of symmetric cryptosystems but no cryptographic attack has been identified until the one on the filtered LFSRs made by Canteaut and Rotella in 2016. Hyper-bent functions have properties still stronger than the well-known bent functions which were introduced by Rothaus and already studied by Dillon and next by several researchers in more than four decades. Hyper-bent functions are very rare and whose classification is still elusive. Therefore, not only their characterization, but also their generation are challenging problems. Recently, an important direction in the theory of hyper-bent functions was the extension of Boolean hyper-bent functions to whose codomain is the ring of integers modulo a power of a prime, that is, generalized hyper-bent functions. In this paper, we synthesize previous studies on generalized hyper-bent functions in a unified framework. We provide two characterizations of generalized hyper-bent functions in terms of their digits. We establish a complete characterization of a family of generalized hyper-bent functions defined over spreads and establish a link between vectorial hyper-bent functions found recently and that family.

Further Results on Generalized Bent Functions and Their Complete Characterization

This paper contributes to increase our knowledge on generalized bent functions (including generalized bent Boolean functions and generalized $p$ -ary bent functions with odd prime $p$ ) by bringing new results on their characterization and construction in arbitrary characteristic. More specifically, we first investigate relations between generalized bent functions and bent functions by the decomposition of generalized bent functions. This enables us to completely characterize generalized bent functions and $\mathbb Z_{p^{k}}$ -bent functions by some affine space associated with the generalized bent functions. We also present the relationship between generalized bent Boolean functions with an odd number of variables and generalized bent Boolean functions with an even number of variables. Based on the well-known Maiorana-McFarland class of Boolean functions, we present some infinite classes of generalized bent Boolean functions. In addition, we introduce a class of generalized hyperbent functions that can be seen as generalized Dillon’s $PS$ functions. Finally, we solve an open problem related to the description of the dual function of a weakly regular generalized bent Boolean function with an odd number of variables via the Walsh–Hadamard transform of their component functions, and we generalize these results to the case of odd prime.

Decomposing Generalized Bent and Hyperbent Functions

In this paper, we introduce generalized hyperbent functions from $ {\mathbb F}_{2^{n}}$ to $ {\mathbb Z}_{2^{k}}$ , and investigate decompositions of generalized (hyper)bent functions. We show that generalized (hyper)bent functions $f$ from $ {\mathbb F}_{2^{n}}$ to $ {\mathbb Z}_{2^{k}}$ consist of components which are generalized (hyper)bent functions from $ {\mathbb F}_{2^{n}}$ to $ {\mathbb Z}_{2^{k^\prime }}$ for some $k^\prime . For even $n$ , most notably we show that the g-hyperbentness of $f$ is equivalent to the hyperbentness of the components of $f$ with some conditions on the Walsh–Hadamard coefficients. For odd $n$ , we show that the Boolean functions associated to a generalized bent function form an affine space of semibent functions. This complements a recent result for even $n$ , where the associated Boolean functions are bent.

A class of hyper-bent functions and Kloosterman sums

This paper is devoted to the characterization of hyper-bent functions. Several classes of hyper-bent functions have been studied, such as Charpin and Gong's family źrźRTr1n(arxr(2mź1))$\sum \limits _{r\in R}\text {Tr}_{1}^{n} (a_{r}x^{r(2^{m}-1)})$ and Mesnager's family źrźRTr1n(arxr(2mź1))+Tr12(bx2nź13)$\sum \limits _{r\in R}\text {Tr}_{1}^{n}(a_{r}x^{r(2^{m}-1)}) +\text {Tr}_{1}^{2}(bx^{\frac {2^{n}-1}{3}})$ . In this paper, we generalize these results by considering the following class of Boolean functions over źź2n$\mathbb {F}_{2^{n}}$: źrźRźi=02Tr1n(ar,ixr(2mź1)+2nź13i)+Tr12(bx2nź13),$$\sum\limits_{r\in R}\sum\limits_{i=0}^{2}T{r^{n}_{1}}(a_{r,i} x^{r(2^{m}-1)+\frac{2^{n}-1}{3}i}) +T{r^{2}_{1}}(bx^{\frac{2^{n}-1}{3}}), $$where n=2m$n=2m$, m is odd, bźźź4$b\in \mathbb {F}_{4}$, and ar,iźźź2n$a_{r,i}\in \mathbb {F}_{2^{n}}$. With the restriction of ar,iźźź2m$a_{r,i}\in \mathbb {F}_{2^{m}}$, we present a characterization of hyper-bentness of these functions in terms of crucial exponential sums. For some special cases, we provide explicit characterizations for some hyper-bent functions in terms of Kloosterman sums and cubic sums. Finally, we explain how our results on binomial, trinomial and quadrinomial hyper-bent functions can be generalized to the general case where the coefficients ar,i$a_{r,i}$ belong to the whole field źź2n$\mathbb {F}_{2^{n}}$.

The bent and hyper-bent properties of a class of Boolean functions

This paper considers the bent and hyper-bent properties of a class of Boolean functions. For one case, we present a detailed description for them to be hyper-bent functions, and give a necessary condition for them to be bent functions for another case. Keywords—Boolean functions, bent functions, hyper-bent

Vectorial Hyperbent Trace Functions From the \(\mathcal {PS}_{\rm ap}\) Class—Their Exact Number and Specification

To identify and specify trace bent functions of the form \(Tr(P(x))\) , where \(P(x) \in {\mathbb F} _{2^n}[x]\) , has been an important research topic lately. We characterize a class of vectorial (hyper)bent functions of the form \(F(x)=Tr_k^n(\sum _{i=0}^{2^k}a_ix^{i(2^k-1)})\) , where \(n=2k\) , in terms of finding an explicit expression for the coefficients \(a_i\) so that \(F\) is vectorial hyperbent. These coefficients only depend on the choice of the interpolating polynomial used in the Lagrange interpolation of the elements of \(\mathcal {U}\) and some prespecified outputs, where \(\mathcal {U}\) is the cyclic group of \((2^{n/2}+1)\) th roots of unity in \( {\mathbb F}_{2^n}\) . We show that these interpolation polynomials can be chosen in exactly \((2^k+1)!2^{k-1}\) ways and this is the exact number of vectorial hyperbent functions of the form \(Tr^n_k(\sum _{i=0}^{2^k}a_ix^{i(2^k-1)})\) . Furthermore, a simple optimization method is proposed for selecting the interpolation polynomials that give rise to trace polynomials with a few nonzero coefficients.

Hyperbent Functions via Dillon-Like Exponents

<?Pub Dtl=""?> This paper is devoted to hyperbent functions with multiple trace terms (including binomial functions) via Dillon-like exponents. We show how the approach developed by Mesnager to extend the Charpin–Gong family, which was also used by Wang and coworkers to obtain another similar extension, fits in a much more general setting. To this end, we first explain how the original restriction for Charpin–Gong criterion can be weakened before generalizing the Mesnager approach to arbitrary Dillon-like exponents. Afterward, we tackle the problem of devising infinite families of extension degrees for which a given exponent is valid and apply these results not only to reprove straightforwardly the results of Mesnager and Wang and coworkers, but also to characterize the hyperbentness of several new infinite classes of Boolean functions. We go into full details only for a few of them, but provide an algorithm (and the corresponding software) to apply this approach to an infinity of other new families. Finally, we compare the asymptotic and practical performances of different characterizations, including these in terms of hyperelliptic curves, and actually build hyperbent functions in cases which could not be attained through naive computations of exponential sums.

An Effective Construction of a Class of Hyper-Bent Functions

An Effective Construction of a Class of Hyper-Bent Functions

An efficient characterization of a family of hyper-bent functions with multiple trace terms

Lisoněk recently reformulated the characterization of Charpin and Gong of a large class of hyperbent functions in terms of cardinalities of curves. In this note, we show that such a reformulation can be naturally extended to a distinct family of functions proposed by Mesnager. Doing so, a polynomial time and space test is obtained to test the hyperbentness of functions in this family. Finally we show how this reformulation can be transformed to obtain a more efficient test.

Bent and Hyper-Bent Functions in Polynomial Form and Their Link With Some Exponential Sums and Dickson Polynomials

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) and give some conjectures that lead to constructions of new hyper-bent functions.

An Efficient Characterization of a Family of Hyperbent Functions

Charpin and Gong recently characterized a large class of hyperbent functions defined on fields of order 2n, which include the well-known monomial functions with the Dillon exponent as a special case. We give a reformulation of the Charpin-Gong criterion in terms of the number of rational points on certain hyperelliptic curves. We present two applications of our result: The time needed to check the hyperbentness of a specific function is now polynomial in n , and hyperbent functions with subfield coefficients can be constructed.

A construction of hyperbent functions with polynomial trace form

This paper presents a construction of infinite classes of binary and p-ary hyperbent functions of polynomial trace form, based on finding a zero of a Kloosterman sum.

Hyperbent Functions, Kloosterman Sums, and Dickson Polynomials

This paper is devoted to the study of hyperbent functions in n variables, i.e., bent functions which are bent up to a change of primitive roots in the finite field GF(2n). Our main purpose is to obtain an explicit trace representation for some classes of hyperbent functions. We first exhibit an infinite class of monomial functions which is not hyperbent. This result indicates that Kloosterman sums on F2 m cannot be zero at some points. For functions with multiple trace terms, we express their spectra by means of Dickson polynomials. We then introduce a new tool to describe these hyperbent functions. The effectiveness of this new method can be seen from the characterization of a new class of binomial hyperbent functions.

Bent and hyper-bent functions over a field of 2ℓ elements

We study the parameters of bent and hyper-bent (HB) functions in n variables over a field $$ P = \mathbb{F}_q $$ with q = 2? elements, ? > 1. Any such function is identified with a function F: Q ? P, where $$ P 1 is obtained if instead of the nonlinearity degree of a function one considers its binary nonlinearity index (in the case ? = 1 these parameters coincide). We construct a class of HB functions that generalize binary HB functions found in [1]; we indicate a set of parameters q and n for which there are no other HB functions. We introduce the notion of the period of a function and establish a relation between periods of (hyper-)bent functions and their frequency characteristics.

Generalized hyper-bent functions over [formula omitted

In this paper, we extend the concept of binary hyper-bent functions introduced by Carlet to functions defined over GF ( p ) . We show that such functions must be quadratic. We also provide the necessary and sufficient conditions on the symmetric coefficient matrix corresponding to the quadratic form of f : Z p n → Z p that guarantee that f is a hyper-bent function.

Approximation of Boolean functions by monomial ones

Every Boolean function of n variables is identified with a function F : Q → P , where Q = GF (2 n ), P = GF (2). A. Youssef and G. Gong showed that for n = 2 λ there exist functions F which have equally bad approximations not only by linear functions (that is, by functions tr ( μx ), where μ ∈ Q * and tr: Q → P is the trace function), but also by proper monomial functions (functions tr( μx δ ), where ( δ , 2 n − 1) = 1). Such functions F were called hyper-bent functions ( HB functions, HBF ), and for any n = 2 λ a non-empty class of HBF having the property F (0) = 0 was constructed. This class consists of the functions F ( x ) = such that the equation F ( x ) = 1 has exactly (2 λ − 1)2 λ −1 solutions in Q . In the present paper, we give some essential restrictions on the parameters of an arbitrary HBF showing that the class of HBF is far less than that of bent functions. In particular, we show that any HBF is a bent function having the degree of nonlinearity λ , and for some n (for instance, if λ &gt; 2 and 2 λ − 1 is prime, or λ ∈ {4,9,25,27}) the class of HBF is exhausted by the functions F ( x ) = described by A. Youssef and G. Gong. For n = 4, in addition to 10 HBF listed above there exist 18 more HBF with property F (0) = 0. The question of whether there exist other hyper-bent functions for n &gt; 4 remains open.

Hyper-bent functions and cyclic codes

Bent functions are those Boolean functions whose Hamming distance to the Reed–Muller code of order 1 equal 2 n - 1 - 2 n / 2 - 1 (where the number n of variables is even). These combinatorial objects, with fascinating properties, are rare. Few constructions are known, and it is difficult to know whether the bent functions they produce are peculiar or not, since no way of generating at random bent functions on 8 variables or more is known. The class of bent functions contains a subclass of functions whose properties are still stronger and whose elements are still rarer. Youssef and Gong have proved the existence of such hyper-bent functions, for every even n. We prove that the hyper-bent functions they exhibit are exactly those elements of the well-known PS ap class, introduced by Dillon, up to the linear transformations x ↦ δ x , δ ∈ F 2 n * . Hyper-bent functions seem still more difficult to generate at random than bent functions; however, by showing that they all can be obtained from some codewords of an extended cyclic code H n with small dimension, we can enumerate them for up to 10 variables. We study the non-zeroes of H n and we deduce that the algebraic degree of hyper-bent functions is n / 2 . We also prove that the functions of class PS ap are some codewords of weight 2 n - 1 - 2 n / 2 - 1 of a subcode of H n and we deduce that for some n, depending on the factorization of 2 n - 1 , the only hyper-bent functions on n variables are the elements of the class PS ap # , obtained from PS ap by composing the functions by the transformations x ↦ δ x , δ ≠ 0 , and by adding constant functions. We prove that non- PS ap # hyper-bent functions exist for n = 4 , but it is not clear whether they exist for greater n. We also construct potentially new bent functions for n = 12 .