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In this paper we give three constructions of cyclic self-orthogonal codes over Z 2 k , for k ≥ 3 , from Boolean functions on n variables. The first construction for each k, 3 ≤ k ≤ n , yields a self-orthogonal Z 2 k -code of length 2 n + 2 with all Euclidean weights divisible by 2 k + 1. In the remaining two constructions, for each even n and k ≥ 3 , we generate a self-orthogonal Z 2 k -code of length 2 n + 1. All Euclidean weights in the constructed code are divisible by 2 2 k - 1 or 2 k + 1 , depending on which of the two constructions is used. [ABSTRACT FROM AUTHOR]

In this paper, we study the value distributions of perfect nonlinear functions, i.e., we investigate the sizes of image and preimage sets. Using purely combinatorial tools, we develop a framework that deals with perfect nonlinear functions in the most general setting, generalizing several results that were achieved under specific constraints. For the particularly interesting elementary abelian case, we derive several new strong conditions and classification results on the value distributions. Moreover, we show that most of the classical constructions of perfect nonlinear functions have very specific value distributions, in the sense that they are almost balanced. Consequently, we completely determine the possible value distributions of vectorial Boolean bent functions with output dimension at most 4. Finally, using the discrete Fourier transform, we show that in some cases value distributions can be used to determine whether a given function is perfect nonlinear, or to decide whether given perfect nonlinear functions are equivalent. [ABSTRACT FROM AUTHOR]

Let m be an even positive integer. A Boolean bent function ƒ on F2m-1 × F2 is called a cyclic bent function if for any a ≠ b ∈ F2m-1 and ε ∈ F2, ƒ(ax1, x2)+ ƒ(bx1, x2 + ε) is always bent, where x1 ∈ F2m-1, x2 ∈ F2. Cyclic bent functions look extremely rare. This paper focuses on cyclic bent functions on F2m-1 × F2 and their applications. The first objective of this paper is to establish a link between quadratic cyclic bent functions and a special type of prequasifields, and construct a class of quadratic cyclic bent functions from the Kantor-Williams prequasifields. The second objective is to use cyclic bent functions to construct families of optimal sequences. The results of this paper show that cyclic bent functions have nice applications in several fields such as coding theory, symmetric cryptography, and CDMA communication. [ABSTRACT FROM AUTHOR]

Perfect (quantum) state transfer has been proved to be an effective model for quantum information processing. In this paper, we give a characterization of cubelike graphs having perfect edge state transfer (PEST, in short). By using a lifting technique, we show that every bent function, and some semi-bent functions as well, can produce some graphs having PEST. Some concrete constructions of such graphs are provided. Notably, using our method, one can obtain some classes of infinite graphs possessing PEST. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Bent functions, semi-bent functions and other Boolean functions with a few Walsh spectra have important applications in coding theory, cryptography and sequence design. In this paper, motivated by the work of Tang et al.(IEEE Trans. Inf. Theory 63(10), 6149-6157, 2017), we provide several infinite families of bent, semi-bent functions and Boolean functions of $ n = 2m $ ($ m\ge 4 $ an integer) variables with five to thirteen-valued Walsh spectra by Kasami types, Gold-like types and Maiorana-MacFarland types. The conditions for these functions to be bent and semi-bent are sufficient and necessary. Furthermore, the dual functions of the new bent functions are given. The algebraic degree of the new functions range from $ 4 $ to $ m $ (or $ m+2 $ except for bent and semi-bent functions) and all the functions possess high nonlinearity. [ABSTRACT FROM AUTHOR]

Boolean functions, coding theory and $t$ -designs have close connections and interesting interplay. A standard approach to constructing $t$ -designs is the use of linear codes with certain regularity. The Assmus-Mattson Theorem and the automorphism groups are two ways for proving that a code has sufficient regularity for supporting $t$ -designs. However, some linear codes hold $t$ -designs, although they do not satisfy the conditions in the Assmus-Mattson Theorem and do not admit a $t$ -transitive or $t$ -homogeneous group as a subgroup of their automorphisms. The major objective of this paper is to develop a theory for explaining such codes and obtaining such new codes and hence new $t$ -designs. To this end, a general theory for punctured and shortened codes of linear codes supporting $t$ -designs is established, a generalized Assmus-Mattson theorem is developed, and a link between 2-designs and differentially $\delta $ -uniform functions and 2-designs is built. With these general results, binary codes with new parameters and explicit weight distributions are obtained, new 2-designs and Steiner system $S(2, 4, 2^{n})$ are produced in this paper. [ABSTRACT FROM AUTHOR]

In this paper, a class of $p$ -ary 3-weight linear codes and a class of binary 2-weight linear codes are proposed respectively by virtue of the properties of the perfect nonlinear functions over $\mathbb {F}_{p^{m}}$ and $(m,s)$ -bent functions from $\mathbb {F}_{2^{m}}$ to $\mathbb {F}_{2^{s}}$ , where $p$ is an odd prime and $m, s$ are positive integers. The weight distributions are completely determined by the sign of the Walsh transform of weakly regular bent functions and the size of the preimage of the employed $(m,s)$ -bent functions at the zero point, respectively. As a special case, a class of optimal linear codes meeting Griesmer bound is obtained from our construction. [ABSTRACT FROM AUTHOR]

Bent functions are a kind of Boolean functions which have the maximum Hamming distance to linear and affine functions, they have some interesting applications in combinatorics, coding theory, cryptography and sequences. However, generally speaking, how to find new bent functions is a hard work and is a hot research project during the past decades. A subclass of bent functions that has received attention since Dillon's seminal thesis (1974) is the subclass of those Boolean functions that are equal to their dual (or Fourier transform in Dillon's terminology): the so-called self dual bent functions. In this paper, we propose a construction of involutions from linear translators, and provide two methods for constructing new involutions by utilizing some given involutions. With the involutions presented in this paper, several new classes of self-dual bent functions are produced. [ABSTRACT FROM AUTHOR]

Bent functions, or equivalently, Hadamard difference sets in the elementary Abelian group (${\mathrm {GF}}(2^{2m}), $ +), have been employed to construct symmetric and quasi-symmetric designs having the symmetric difference property. The main objective of this paper is to use bent vectorial functions for a construction of a two-parameter family of binary linear codes that do not satisfy the conditions of the Assmus–Mattson theorem, but nevertheless hold 2-designs. A new coding-theoretic characterization of bent vectorial functions is presented. [ABSTRACT FROM AUTHOR]

In this paper, we study further three kinds of bent functions (binary, p-ary and gbent) related to spreads. We characterize those p-ary bent functions whose restrictions to the elements of the Dillon spread are affine. A technique for using finite pre-quasifield spreads (from finite geometry) to construct p-ary bent functions and generalized bent functions is studied and applied. [ABSTRACT FROM AUTHOR]

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 (M ) class and so-called class H of bent functions. For cubic bent functions of the form T r 1 t (x y 2 i + 1) in 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 H are EA-inequivalent. [ABSTRACT FROM AUTHOR]

In this paper, for any even integer n = 2m ≥ 4, a new construction of n-variable rotation symmetric bent function with maximal algebraic degree m is given as f(x0,x1⋯,xn−1) = m−1⨁i=0(xixm+i)⊕n−1⨁i=0(xixi+1⋯xi+m−2xi+m), whose dual function is ƒ(x0,x1⋯,xn−1) = m−1⨁i=0(xixm+i)⊕n−1⨁i=0(xixi+1⋯xi+m−2xi+n−2), where xi = xi ⊕ 1 and the subscript of x is modulo n. [ABSTRACT FROM AUTHOR]

In this paper, we discover that univariate form of a Niho bent function is a sum of functions having the form of a Leander–Kholosha bent function taken with particular coefficients from \mathbb F2^{n}^{*} for every term. We know that the Niho bent functions are related to o-polynomials. The power terms in the univariate Niho bent function can be derived by working, in a first step, on each monomial of the corresponding o-polynomial separately, and in a second step, adding them to obtain the global expression. This allows, knowing the monomials in an o-polynomial, to obtain the power terms of the polynomial representing corresponding bent function. However, the coefficients are not calculated explicitly. The explicit form is given for the bent functions obtained from quadratic and cubic o-polynomials. We also calculate the algebraic degree of any bent function in the Leander–Kholosha class. [ABSTRACT FROM AUTHOR]

Linear codes with few weights have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, several classes of p-ary linear codes with two or three weights are constructed from quadratic Bent functions over the finite field $${\mathbb {F}}_p$$ , where p is an odd prime. They include some earlier linear codes as special cases. The weight distributions of these linear codes are also determined. [ABSTRACT FROM AUTHOR]

In this paper, we first present a novel secondary construction of bent functions (building new bent functions from two already defined ones). Furthermore, the algebraic degree and algebraic immunity of the constructed functions are analysed. Finally, we apply the construction using as initial functions some specific bent functions and then specify sufficient conditions for the resulting bent functions not to be contained in the completed Maiorana-McFarland class. In the second part of the paper, we present a corrigendum of 'Constructions of bent-negabent functions and their relation to the completed Maiorana-McFarland Class' (IEEE Trans Inf Theory 61(3):1496-1506, 2015). [ABSTRACT FROM AUTHOR]

In the predecessor to this paper Dempwolff (Comm Algebra 34(3):1077–1131, 2006), group-theoretic methods were used to solve automorphism and equivalence questions for (certain) ordinary bent functions, i.e., bent functions over F 2 . Here, we consider the same problems for p-ary bent functions, p an odd prime and solve these questions for functions analogous to those which appear in Dempwolff (Comm Algebra 34(3):1077–1131, 2006). Although our general analysis is similar to the approach of Dempwolff (Comm Algebra 34(3):1077–1131, 2006), it turns out that the odd characteristic leads to simplifications: Often, the double derivative can be computed (cf. Lemma 2.10) and factorizations of the automorphism group (cf. Lemma 2.3) can be established resulting in restrictions for automorphisms and equivalence maps. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

An S-box is the most important part of a symmetric encryption algorithm. Various schemes are put forward by using chaos theory. In this paper, a construction method of S-boxes with good cryptographic properties is proposed. The output of an S-box can be regarded as a group of Boolean functions. Therefore, we can use the different properties of chaos and Bent functions to generate a random Bent function with a high nonlinearity. By constructing a set of Bent functions as the output of an S-box, we can create an S-box with good cryptological properties. The nonlinearity, differential uniformity, strict avalanche criterion and the independence criterion of output bits are then analyzed and tested. A security analysis shows that the proposed S-box has excellent cryptographic properties. [ABSTRACT FROM AUTHOR]

In this paper, we first discuss the bentness of a large class of quadratic Boolean functions in polynomial form f (x) = ∑ i = 1 n / 2 - 1 Tr 1 n (c i x 1 + 2 i ) + Tr 1 n / 2 (c n / 2 x 1 + 2 n / 2 ) , where n is even, c i ∈ GF (2 n) for 1 ≤ i ≤ n / 2 - 1 and c n / 2 ∈ GF (2 n / 2) . The bentness of these functions can be connected with linearized permutation polynomials. Hence, methods for constructing quadratic bent functions are given. Further, we consider a subclass of quadratic Boolean functions of the form f (x) = ∑ i = 1 m / 2 - 1 Tr 1 n (c i x 1 + 2 ei ) + Tr 1 n / 2 (c m / 2 x 1 + 2 n / 2 ) , where n = e m , m is even, and c i ∈ GF (2 e) . The bentness of these functions is characterized and some methods for deriving new quadratic bent functions are given. Finally, when m and e satisfy some conditions, we determine the number of these quadratic bent functions. [ABSTRACT FROM AUTHOR]

It has been an active research issue for many years to construct new bent functions. For k odd with \gcd (n, k)=1 and a\in \mathbb F3^{n}^{*} , the function f(x)=Tr1^{n}(ax^{(({3^{k}+1})/{2})}) is weakly regular bent over \mathbb F3^{n} , where Tr1^{n}(\cdot) is the trace function from \mathbb F3^{n} to \mathbb F3 . This is the well-known Coulter–Matthews bent function. In this paper, we determine the dual function of f(x) completely. As a consequence, we find many classes of ternary bent functions not reported in the literature previously. Such bent functions are not quadratic if k>1 and have (((1+\sqrt 5)/2)^w+1-((1-\sqrt 5)/2)^w+1)/\sqrt 5 or (((1+\sqrt 5)/2)^n-w+1-((1-\sqrt 5)/2)^n-w+1)/\sqrt 5 trace terms, where 0

Let F q denote the finite field with q elements. Permutation polynomials and complete permutation polynomials over finite fields have been widely investigated in recent years due to their applications in cryptography, coding theory and combinatorial design. In this paper, several classes of (complete) permutation polynomials with the form (Tr m n (x) k + δ) s + L (x) and (Tr m n (x) k 1 + δ 1) s 1 + (Tr m n (x) k 2 + δ 2) s 2 + L (x) are proposed based on the AGW criterion and some techniques in solving equations over the finite field F 2 n , where L (x) = a Tr m n (x) + b x , a ∈ F 2 n and b ∈ F 2 m ⁎. We also determine the compositional inverse of these polynomials in some special cases. Besides, Mesnager (2014) [13] proposed a construction of bent functions by finding some triples of permutation polynomials satisfying a particular property named (A n). With the help of this approach, several classes of bent functions are presented. [ABSTRACT FROM AUTHOR]

Three classes of (balanced) Boolean functions with few Walsh transform values derived from bent functions, Gold functions and the product of linearized polynomials are obtained in this paper. Further, the value distributions of their Walsh transform are also determined by virtue of the property of bent functions, the Walsh transform property of Gold functions and the -tuple balance property of trace functions respectively. [ABSTRACT FROM AUTHOR]

Rotation symmetric bent functions are a special class of Boolean functions, and their construction is of theoretical and practical interest. In this paper, we propose a generic construction of rotation symmetric bent functions by modifying the support of a known class of quadratic rotation symmetric bent functions, which generalizes some earlier works. Moreover, many infinite classes of rotation symmetric bent functions with maximal algebraic degree can be easily obtained from our construction. [ABSTRACT FROM AUTHOR]

Rotation symmetric bent functions and their generation two-rotation symmetric bent functions are two classes of cryptographically significant Boolean functions. However, few constructions have been presented in the literature, which either have restriction on integer $n$ or have algebraic degree no more than 4. In this paper, for any even integer $n\ge 4$ , three classes of bent functions are presented respectively. Most notably, the proposed $n$ -variable rotation symmetric bent functions and two-rotation symmetric bent functions can have any possible algebraic degree ranging from 2 to $n/2$ . Besides, we obtain bent idempotent functions with the maximal algebraic degree $n/2$ . [ABSTRACT FROM PUBLISHER]

For any bent function, it is very interesting to determine its dual function, because the dual function is also bent in certain cases. For k odd and \gcd (n, k)=1 , it is known that the Coulter–Matthews bent function f(x)=Tr\left(ax^\frac 3^k+12\right) is weakly regular bent over \mathbb F3^{n} , where a\in \mathbb F3^{n}^{*} , and Tr(\cdot ):\mathbb F3^{n}\rightarrow \mathbb F3 is the trace function. In this paper, we investigate the dual function of $f(x)$ , aiming to determine a universal formula. In particular, for two cases, we determine the formula explicitly: for the case of n=3t+1$ and k=2t+1$ with t\geq 2n=3t+2$ and k=2t+1$ with t\geq 2f(x)$ is regular bent. [ABSTRACT FROM PUBLISHER]

In this paper, several classes of Boolean functions with few Walsh transform values, including bent, semi-bent and five-valued functions, are obtained by adding the product of two or three linear functions to some known bent functions. Numerical results show that the proposed class contains cubic bent functions that are affinely inequivalent to all known quadratic ones. [ABSTRACT FROM AUTHOR]

Inspired by a recent work of Mesnager, we present several infinite families of quadratic ternary bent, near-bent and 2-plateaued functions from some known quadratic ternary bent functions. Meanwhile, the distributions of the Walsh spectrum of two classes of 2-plateaued functions obtained in this paper are completely determined. [ABSTRACT FROM AUTHOR]

Bent functions can be classified into regular bent functions, weakly regular but not regular bent functions, and non-weakly regular bent functions. Regular and weakly regular bent functions always appear in pairs, since their duals are also bent functions. In general, this does not apply to non-weakly regular bent functions. However, the first known construction of non-weakly regular bent functions by Çeşmelioğlu et al. yields bent functions for which the dual is also bent. In this paper, the first construction of non-weakly regular bent functions for which the dual is not bent is presented. We call such functions non-dual-bent functions. Until now, only sporadic examples found via computer search were known. We then show that with the direct sum of bent functions and with the construction by Çeşmelioğlu et al., one can obtain infinitely many non-dual-bent functions once one example of a non-dual-bent function is known. [ABSTRACT FROM PUBLISHER]

This paper first gives out basic background and some definitions and propositions for Fourier matrix and bent function. Secondly we construct an standard orthogonal basis by the eigenvectors of the corresponding Fourier matrix. At last the diagonalization work of Fourier matrix is completed and some theorems about them are proved. [ABSTRACT FROM AUTHOR]

In the mid‐sixties, Rothaus introduced the notion of bent function and later presented a secondary construction of bent functions (building new bent functions from already defined ones), called Rothaus' construction. In Zhang et al. 2017 ('Constructing bent functions outside the Maiorana–Mcfarland class using a general form of Rothaus,' IEEE Transactions on Information Theory, 2017, vol. 63, no. 8, pp. 5336–5349.') provided two constructions of bent functions using a general form of Rothaus and showed that the obtained classes lie outside the completed Maiorana–McFarland (MM) class. In this study, the authors propose two similar methods for constructing bent functions outside the completed MM class but with significantly simplified sufficient conditions compared to those in Zhang et al. 2017. These simplified conditions do not induce any serious restrictions on the choice of permutations used in the construction apart from a simple requirement on their algebraic degree and the request that the component functions of one permutation do not admit linear structures. This enables us to generate a huge class of bent functions lying outside the completed MM class. Even more importantly, they prove that the new classes of bent functions are affine inequivalent to the bent functions in Zhang et al. 2017. [ABSTRACT FROM AUTHOR]

The problem of constructing bent–negabent functions that do not belong to the completed Maiorana–McFarland class emerges implicitly through a series of construction methods proposed recently. These approaches manage to optimize the algebraic degree of bent–negabent functions, but all of the constructed bent–negabent functions belong to the completed Maiorana–McFarland class. In this paper, we use the indirect sum construction (proposed by Carlet in 2004) for constructing the bent–negabent functions that are not provably contained in this class, which is the first significant attempt in this direction. To achieve this, we first provide a class of bent functions with certain desirable properties that does not belong to this class and demonstrate the existence of the class members. Then, embedding these functions in the framework of the indirect sum construction, we are able to specify sufficient conditions for bent–negabent functions not being contained in the completed Maiorana–McFarland class. [ABSTRACT FROM PUBLISHER]

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 PS a p or the class H. [ABSTRACT FROM AUTHOR]

Let n = 2 m. In the present paper, we study the binomial Boolean functions of the form f a , b (x) = Tr 1 n (a x 2 m − 1) + Tr 1 2 (b x 2 n − 1 3 ) , where m is an even positive integer, a ∈ F 2 n ⁎ and b ∈ F 4 ⁎. We show that f a , b is a bent function if the Kloosterman sum K m (a 2 m + 1) = 1 + ∑ x ∈ F 2 m ⁎ (− 1) Tr 1 m (a 2 m + 1 x + 1 x) equals 4, thus settling an open problem of Mesnager. The proof employs tools including computing Walsh coefficients of Boolean functions via multiplicative characters, divisibility properties of Gauss sums, and graph theory. [ABSTRACT FROM AUTHOR]

The nonlinearity and additive nonlinearity of a function are defined as the Hamming distances, respectively, to the set of all affine mappings and to the set of all mappings having nontrivial additive translators. On the basis of the revealed relation between the nonlinearities and the Fourier coefficients of the characters of a function, convenient formulas for nonlinearity evaluation for practically important classes of functions over an arbitrary finite field are found. In the case of a field of even characteristic, similar results were obtained for the additive nonlinearity in terms of the autocorrelation coefficients. The formulas obtained made it possible to present specific classes of functions with maximal possible and high nonlinearity and additive nonlinearity. [ABSTRACT FROM AUTHOR]