Bent Functions Research Articles

New nonexistence results on (m, n)-generalized bent functions

In this paper, we present some new nonexistence results on (m, n)-generalized bent functions, which improved recent results. More precisely, we derive new nonexistence results for general n and m odd or $$m \equiv 2 \pmod {4}$$, and further explicitly prove nonexistence of (m, 3)-generalized bent functions for all integers m odd or $$m \equiv 2 \pmod {4}$$. The main tools we utilized are certain exponents of minimal vanishing sums from applying characters to group ring equations that characterize (m, n)-generalized bent functions.

The research on the properties of Fourier matrix and bent function

<p style='text-indent:20px;'>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.

SAO 1-Resilient Functions With Lower Absolute Indicator in Even Variables

In 2018, Tang and Maitra presented a class of balanced Boolean functions in $n$ variables with the absolute indicator $\Delta _{f} and the nonlinearity $NL(f)> 2^{n-1}-2^{n/2}$ , that is, $f$ is SAO (strictly almost optimal), for $n=2k\equiv 2\,\,({\mathrm {mod}\,\,}4)$ and $n\geq 46$ in [IEEE Ttans. Inf. Theory 64 (1) : 393-402, 2018]. However, there is no evidence to show that the absolute indicator of any 1-resilient function in $n$ variables can be strictly less than $2^{\lfloor ({n+1})/{2}\rfloor }$ , and the previously best known upper bound of which is $5\cdot 2^{n/2}-2^{n/4+2}+4$ . In this paper, we concentrate on two directions. Firstly, to complete Tang and Maitra’s work for $k$ being even, we present another class of balanced functions in $n$ variables with the absolute indicator $\Delta _{f} and the nonlinearity $NL(f)> 2^{n-1}-2^{n/2}$ for $n\equiv 0~({\mathrm {mod}~}4)$ and $n\geq 48$ . Secondly, we obtain two new classes of 1-resilient functions possessing very high nonlinearity and very low absolute indicator, from bent functions and plateaued functions, respectively. Moreover, one class of them achieves the currently known highest nonlinearity $2^{n-1}-2^{n/2-1}-2^{n/4}$ , and the absolute indicator of which is upper bounded by $2^{n/2}+2^{n/4+1}$ that is a new upper bound of the minimum of absolute indicator of 1-resilient functions, as it is clearly optimal than the previously best known upper bound $5\cdot 2^{n/2}-2^{n/4+2}+4$ .

Asymptotically Optimal Codebooks Derived From Generalised Bent Functions

Codebooks are required to have small inner-product correlation in many practical applications, such as direct spread code division multiple access communications, space-time codes and compressed sensing. In general, it is difficult to construct optimal codebooks. In this paper, two kinds of codebooks are presented and proved to be optimally optimal with respect to the Welch bound. Additionally, the constructed codebooks in this paper have new parameters.

A Construction of Bent Functions on a Finite Group

In this paper, we discuss when a class function on a finite group is a bent function. We have found a necessary condition for a class function on a finite abelian group to be bent. Also, we have found a necessary and sufficient condition for a class function on a finite cyclic group to be bent.

Generic constructions of $$\mathbb {Z}$$-bent functions

$$\mathbb {Z}$$-bent functions, mappings from $$\mathbb {F}_2^n$$ to a subset of $$\mathbb {Z}$$, were introduced by Dobbertin and Leander (Des Codes Cryptogr 49:3–22, 2008) as an attempt to capture the origin of standard bent functions and in particular to understand better a recursive construction framework of bent functions. Nevertheless, many questions have been left open in Dobbertin and Leander (2008) such as efficient construction methods of $$\mathbb {Z}$$-bent functions of different levels, where these levels specify precisely a subset of $$\mathbb {Z}$$ containing both the image values of f and its normalized Fourier coefficients. In this article, using different design techniques, we provide several generic construction methods of $$\mathbb {Z}$$-bent functions of arbitrary levels, thereby solving an open problem posed in Dobbertin and Leander (2008). On the other hand, apart from an independent theoretical interest in these objects, our rigor treatment of the so-called gluing technique reveals that this approach is equivalent to a classical concept of concatenation. More precisely, gluing four suitable n-variables $$\mathbb {Z}$$-bent functions of level one to obtain an $$(n+2)$$-variable bent function directly corresponds to a concatenation of four suitable n-variable Boolean functions. Nevertheless, the recursive framework based on $$\mathbb {Z}$$-bent functions remains to be investigated further in this context.

On Two-to-One Mappings Over Finite Fields

Two-to-one ($2$-to-$1$) mappings over finite fields play an important role in symmetric cryptography. In particular they allow to design APN functions, bent functions and semi-bent functions. In this paper we provide a systematic study of two-to-one mappings that are defined over finite fields. We characterize such mappings by means of the Walsh transforms. We also present several constructions, including an AGW-like criterion, constructions with the form of $x^rh(x^{(q-1)/d})$, those from permutation polynomials, from linear translators and from APN functions. Then we present $2$-to-$1$ polynomial mappings in classical classes of polynomials: linearized polynomials and monomials, low degree polynomials, Dickson polynomials and Muller-Cohen-Matthews polynomials, etc. Lastly, we show applications of $2$-to-$1$ mappings over finite fields for constructions of bent Boolean and vectorial bent functions, semi-bent functions, planar functions and permutation polynomials. In all those respects, we shall review what is known and provide several new results.

Polyphase zero correlation zone sequences from generalised bent functions

Sequence families with zero correlation zone (ZCZ) have been extensively studied in recent years due to their important applications in quasi-synchronous code-division multiple-access (QS-CDMA) systems. To accommodate multiuser environments, multiple ZCZ sequence sets with low inter-set cross-correlation are expected. In this paper, we propose a construction of polyphase ZCZ sequences based on generalised bent functions. Moreover, multiple polyphase ZCZ sequence sets with good inter-set cross-correlation are presented. Each generated ZCZ sequence set is optimal with respect to the Tang-Fan-Matsufuji bound.

Relationship between homogeneous bent functions and Nagy graphs

Relationship between homogeneous bent functions and Nagy graphs

Composition of Boolean functions: An application to the secondary constructions of bent functions

Bent functions are optimal combinatorial objects and have been attracted their research for four decades. Secondary constructions play a central role in constructing bent functions since a complete classification of this class of functions seems to be elusive. This paper is devoted to establishing a relationship between the secondary constructions and the composition of Boolean functions. We firstly prove that some well-known secondary constructions of bent functions, can be described by the composition of a plateaued Boolean function and some bent functions. Then their dual functions can be calculated by the Lagrange interpolation formula. By following this observation, two secondary constructions of bent functions are presented. We show that they are inequivalent to the known ones, and may generate bent functions outside the primary classes M and PS. These results show that the method we present in this paper is genetic and unified and therefore can be applied to the constructions of Boolean functions with other cryptographical criteria.

Linear Codes From Perfect Nonlinear Functions Over Finite Fields

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.

Ramanujan graphs and expander families constructed from p-ary bent functions

We present a method for constructing an infinite family of non-bipartite Ramanujan graphs. We mainly employ p-ary bent functions of $$(p-1)$$-form for this construction, where p is a prime number. Our result leads to construction of infinite families of expander graphs; this is due to the fact that Ramanujan graphs play as base expanders for constructing further expanders. For our construction we directly compute the eigenvalues of the Ramanujan graphs arsing from p-ary bent functions. Furthermore, we establish a criterion on the regularity of p-ary bent functions in m variables of $$(p-1)$$-form when m is even. Finally, using weakly regular p-ary bent functions of $$\ell $$-form, we find (amorphic) association schemes in a constructive way; this resolves the open case that $$\ell = p-1$$ for $$p >2$$ for finding (amorphic) association schemes.

Bent Vectorial Functions, Codes and Designs

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.

Boolean Functions with Multiple‐Valued Walsh Spectra

Compared with the method by Zhang in 2017, an extended one for constructing Boolean functions with multiple-valued Walsh spectra is given, which is derived from a bent function by complementing the values at some points. Based on which, new classes of Boolean functions with four-valued and five-valued Walsh spectra are presented, and some of their cryptographic properties are studied.

On relationship between the parameters characterizing nonlinearity and nonhomomorphy of vector spaces transformation

Abstract We find a relation between the W-intersection matrix (which characterizes the degree of “nonhomomorphy”) of a transformation and the difference distribution table and the correlation matrix (which characterize the degree nonlinearity of a transformation). An upper estimate for the dimension of a subspace invariant under almost bent functions is put forward. A formula for evaluation of the W-intersection matrix of a composition of two transformations is obtained.

Relationship Between Homogeneous Bent Functions and Nagy Graphs

We study the relationship between homogeneous bent functions and some intersection graphs of a special type that are called Nagy graphs and denoted by Γ(n,k). The graph Γ(n,k) is the graph whose vertices correspond to (nk) unordered subsets of size k of the set 1,..., n. Two vertices of Γ(n,k) are joined by an edge whenever the corresponding k-sets have exactly one common element. Those n and k for which the cliques of size k + 1 are maximal in Γ(n,k) are identified. We obtain a formula for the number of cliques of size k + 1 in Γ(n,k) for n = (k + 1)k/2. We prove that homogeneous Boolean functions of 10 and 28 variables obtained by taking the complement to the cliques of maximal size in Γ(10,4) and Γ(28,7) respectively are not bent functions.

A Note on Rotation Symmetric S-boxes

This paper is twofold. The first is devoted to study a class of quadratic rotation symmetric S-boxes (RSSBs) which was presented by Gao G, et al., Constructions of quadratic and cubic rotation symmetric bent functions, IEEE Transactions on Information Theory, vol. 58, no. 7, pp. 4908–4913, 2012, by decomposing a class of cubic rotation symmetric bent functions. The authors obtain its nonlinearity and differential uniformity of such class of S-boxes. In particular, the compositional inversion of the class of rotation symmetric S-boxes is also presented. Then the authors introduce a steepest-descent-like search algorithm for the generation of RSSBs. The algorithm finds 5,6,7,8-bit RSSBs with very good cryptographic properties which can be applied in designing cryptographical algorithms.

Metrical properties of self-dual bent functions

In this paper we study metrical properties of Boolean bent functions which coincide with their dual bent functions. We propose an iterative construction of self-dual bent functions in $$n+2$$ variables through concatenation of two self-dual and two anti-self-dual bent functions in n variables. We prove that minimal Hamming distance between self-dual bent functions in n variables is equal to $$2^{n/2}$$. It is proved that within the set of sign functions of self-dual bent functions in $$n\geqslant 4$$ variables there exists a basis of the eigenspace of the Sylvester Hadamard matrix attached to the eigenvalue $$2^{n/2}$$. Based on this result we prove that the sets of self-dual and anti-self-dual bent functions in $$n\geqslant 4$$ variables are mutually maximally distant. It is proved that the sets of self-dual and anti-self-dual bent functions in n variables are metrically regular sets.

Strongly regular graphs arising from non-weakly regular bent functions

In this paper, we study two special subsets of a finite field of odd characteristics associated with non-weakly regular bent functions. We show that those subsets associated to non-weakly regular even bent functions in the GMMF class (see Cesmelioglu et al. Finite Fields Appl. 24, 105–117 2013) are never partial difference sets (PDSs), and are PDSs if and only if they are trivial subsets. Moreover, we analyze the two known sporadic examples of non-weakly regular ternary bent functions given in Helleseth and Kholosha (IEEE Trans. Inf. Theory 52(5), 2018–2032 2006, Cryptogr. Commun. 3(4), 281–291 2011). We observe that corresponding subsets are non-trivial partial difference sets. We show that they are the union of some cyclotomic cosets and so correspond to 2-class fusion schemes of a cyclotomic scheme. We also present a further construction giving non-trivial PDSs from certain p-ary functions which are not bent functions.

Designing Plateaued Boolean Functions in Spectral Domain and Their Classification

The design of plateaued functions over $GF(2)^n$, also known as 3-valued Walsh spectra functions (taking the values from the set $\{0, \pm 2^{\lceil \frac{n+s}{2} \rceil}\}$), has been commonly approached by specifying a suitable algebraic normal form which then induces this particular Walsh spectral characterization. In this article, we consider the reversed design method which specifies these functions in the spectral domain by specifying a suitable allocation of the nonzero spectral values and their signs. We analyze the properties of trivial and nontrivial plateaued functions (as affine inequivalent distinct subclasses), which are distinguished by their Walsh support $S_f$ (the subset of $GF(2)^n$ having the nonzero spectral values) in terms of whether it is an affine subspace or not. The former class exactly corresponds to partially bent functions and admits linear structures, whereas the latter class may contain functions without linear structures. A simple sufficient condition on $S_f$, which ensures the nonexistence of linear structures, is derived and some generic design methods of nontrivial plateaued functions without linear structures are given. The extended affine equivalence of plateaued functions is also addressed using the concept of dual of plateaued functions. Furthermore, we solve the problem of specifying disjoint spectra (non)trivial plateaued functions of maximal cardinality whose concatenation can be used to construct bent functions in a generic manner. This approach may lead to new classes of bent functions due to large variety of possibilities to select underlying duals that define these disjoint spectra plateaued functions. An additional method of specifying affine inequivalent plateaued functions, obtained by applying a nonlinear transform to their input domain, is also given.