Class Of Bent Functions Research Articles

Further investigations on permutation based constructions of bent functions

Constructing bent functions by composing a Boolean function with a permutation was introduced by Hou and Langevin in 1997. The approach appears simple but heavily depends on the construction of desirable permutations. In this paper, we further study this approach by investigating the exponential sums of certain monomials and permutations. We propose several classes of bent functions from quadratic permutations and permutations with (generalized) Niho exponents, and also a class of bent functions from a generalization of the Maiorana-McFarland class. The relations among the proposed bent functions and the known families of bent function are studied. Numerical results show that our constructions include bent functions that are not contained in the completed Maiorana-McFarland class M#, the class PSap or the class H.

Several classes of bent functions over finite fields

Inspired by the works of Mesnager (IEEE Trans Inf Theory 60(7):4397–4407, 2014) and Tang et al. (IEEE Trans Inf Theory 63(10):6149–6157, 2017), we study a class of bent functions of the form $$f(x)=g(x)+F(f_1(x),f_2(x),\ldots ,$$ $$f_{\tau }(x))$$ , where g(x) is a function from $$\mathbb {F}_{p^{n}}$$ to $${\mathbb {F}}_{p}$$ , $$f_i(x)=\prod _{j=1}^{\kappa _i}{\text {Tr}}(u_{ij}x)$$ for $$1\le i\le \tau $$ , $$\kappa _i\ge 1$$ and $$F(x_1,\ldots ,x_{\tau })$$ is a reduced polynomial in $${\mathbb {F}}_{p}[x_1,\ldots ,x_{\tau }]$$ . As a consequence, we obtain a generic result on the Walsh transform of f(x) in terms of g(x) and characterize the bentness of f(x) for the cases $$F(x_1,\ldots ,x_{\tau })$$ without and with restrictions respectively, which enables us to generalize some earlier works and derive new bent functions from known ones. In addition, we study the construction of bent functions f(x) when g(x) is not bent for the first time and present a class of bent functions from non-bent Gold functions.

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.

Modifying Maiorana--McFarland Type Bent Functions for Good Cryptographic Properties and Efficient Implementation

Very recently, a class of cryptographically significant Boolean functions were constructed by Tang and Maitra [IEEE Trans. Inform. Theory, 64 (2018), pp. 393--402] by modifying the $\mathcal{PS}_{ap}$ class of bent functions. The basic ideas used in Tang--Maitra construction were derived from a modification of a subclass of bent functions which is defined over the finite field, and a concern was raised in the same paper whether the implementation of such functions will be as efficient as that of Maiorana--McFarland type bent functions. In this paper, we look at the concrete realization of such functions over a vector space and answer the question positively. The first part of this paper investigates how the finite field implementation of the functions can be viewed as simple truth tables. Next, we present a completely new construction that itself starts from Maiorana--McFarland bent functions which are straightforward concatenations of linear functions.

Bent functions from nonlinear permutations and conversely

This work extends the idea introduced by Hou and Langevin (J. Combin. Theory, Ser. A, 80:232–246, 1997) of applying nonlinear permutations to (a portion of) the input variable space of a given Boolean function so that the resulting function is bent. Applying such a permutation to a bent function that can be represented in a suitable form then gives an affine inequivalent bent function which potentially does not belong to the same class as the original one. While Hou and Langevin only provided two sporadic examples of bent functions that can be turned into affine inequivalent ones, in this article we identify two generic families of bent functions suitable for generating such affine inequivalent counterparts. The same method when applied to the Marioana-McFarland class of bent functions, depending on the subset of inputs to which a nonlinear action is applied, either lead to bent functions that are provably within the same class or to bent functions that are potentially outside this class. The problem of finding suitable permutations that act nonlinearly on more than two input variables of the initial function and ensure the bentness of the resulting function appears to be generally hard. In this direction, we only slightly extend the approach of Hou and Langevin by identifying suitable permutations that act nonlinearly on three input variabl es. Most notably, the existence of nonlinear permutations that act without strict separation of the input space in terms of linear and nonlinear action is also confirmed. Finally, we show a direct correspondence between (some classes of) bent functions and permutations by providing an efficient method to define permutations using the derivatives of a given bent function. This not only gives a relationship between two seemingly different algebraic objects, but also provides us with a new infinite family of permutations over finite fields.

An Analysis of the 𝒞 Class of Bent Functions

Two (so-called C;D) classes of permutation-based bent Boolean functions were introduced by Carlet two decades ago, but without specifying some explicit construction methods for their construction (apart from the subclass D0). In this article, we look in more detail at theC class, and derive some existence and nonexistence results concerning the bent functions in the C class for many of the known classes of permutations over F2n. Most importantly, the existence results induce generic methods of constructing bent functions in classC which possibly do not belong to the completed Maiorana-McFarland class. The question whether the specic permutations and related subspaces we identify in this article indeed give bent functions outside the completed Maiorana-McFarland class remains open.

A note on two classes of Boolean functions with optimal algebraic immunity

Tu and Deng proposed a class of bent functions which are of optimal algebraic immunity under the assumption of a combinatorial conjecture. In this paper, the authors compute the dual of the Tu-Deng functions and then show that they are still of optimal algebraic immunity under the assumption of the same conjecture. For another class of Boolean functions constructed by Tang, et al. which are of optimal algebraic immunity with similar forms to Tu-Deng functions, the authors show that they are not bent functions by using some basic properties of binary complete Kloosterman sums.

Generalized Maiorana–McFarland class and normality of p-ary bent functions

A class of bent functions which contains bent functions with various properties like regular, weakly regular and not weakly regular bent functions in even and in odd dimension, is analyzed. It is shown that this class includes the Maiorana–McFarland class as a special case. Known classes and examples of bent functions in odd characteristic are examined for their relation to this class. In the second part, normality for bent functions in odd characteristic is analyzed. It turns out that differently to Boolean bent functions, many – also quadratic – bent functions in odd characteristic and even dimension are not normal. It is shown that regular Coulter–Matthews bent functions are normal.

On the Walsh Transform of a Class of Functions From Niho Exponents

In this paper, a class of functions from Niho exponents with four-valued Walsh transform is obtained for any prime by a uniform method, and the distribution of the Walsh transform values is also completely determined. In particular, this class of functions is proven to be bent for a special case. Although it is shown that the obtained bent functions are equivalent to the Leander-Kholosha's class of bent functions, a direct and much simpler proof for the bentness of this kind of Niho functions is provided.

Systematic Authentication Codes Based on a Class of Bent Functions and the Gray Map on a Galois Ring

We use a class of bent functions on a Galois ring of characteristic $p^2$ and the Gray map on this ring to construct new classes of systematic authentication codes.

О совпадении класса бент-функций с классом функций,минимально близких к линейным

О совпадении класса бент-функций с классом функций,минимально близких к линейным

Bent functions on a Galois ring and systematic authentication codes

A class of bent functions on a Galois ring is introduced and based on these functions systematic authentication codes are presented. These codes generalize those appearing in [4] for finite fields.

The Lower Bound on the Second-Order Nonlinearity for a Class of Bent Functions

The Lower Bound on the Second-Order Nonlinearity for a Class of Bent Functions

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.

Constructions of 1‐resilient Boolean functions on odd number of variables with a high nonlinearity

ABSTRACTIn this paper, we concentrate on the design of 1‐resilient Boolean functions with desirable cryptographic properties. Firstly, we put forward a novel secondary construction to obtain 1‐resilient functions. Next, we present the relationships between the properties of these constructed 1‐resilient functions and that of the initial functions. Based on the construction and a class of bent functions on n variables, we can obtain a class of (n + 3)‐variable 1‐resilient non‐separable cryptographic functions with a high algebraic immunity, whose nonlinearity is equal to the bent concatenation bound 2n + 2 − 2(n + 2)/2. Furthermore, we propose a set of 1‐resilient non‐separable functions on odd number of variables with an optimal algebraic degree, a high algebraic immunity, and a high nonlinearity. Copyright © 2011 John Wiley & Sons, Ltd.

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.

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 .

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.

A Characterization of Binary Bent Functions

A recent paper by Carlet introduces a general class of binary bent functions on (GF(2))n(neven) whose elements are expressed by means of characteristic functions (indicators) of (n/2)-dimensional vector-subspaces of (GF(2))n. An extended version of this class is introduced in the same paper; it is conjectured that this version is equal to the whole class of bent functions. In the present paper, we prove that this conjecture is true.

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>