Piecewise Constructions of Bent and Almost Optimal Boolean Functions

The first aim of this work was to generalize the techniques used in MacWilliams’ and Sloane’s presentation of the Kerdock code and develop a theory of piecewise quadratic Boolean functions. This generalization led us to construct large families of potentially new bent and almost optimal functions from quadratic forms in this piecewise fashion. We show how our motivating example, the Kerdock code, fits into this setting. These constructions were further generalized to non-quadratic bent functions. The resulting constructions design n-variable bent (resp. almost optimal) functions from n-variable bent or almost optimal functions. Communicated by: T. Helleseth     

Duality between bent functions and affine functions

A Boolean function in an even number of variables is called bent if it is at the maximal possible Hamming distance from the class of all affine Boolean functions. We prove that there is a duality between bent functions and affine functions. Namely, we show that affine function can be defined as a Boolean function that is at the maximal possible distance from the set of all bent functions.     

Normal Boolean functions

Dobbertin (Construction of bent functions and balanced Boolean functions with high nonlinearity, in: Fast Software Encryption, Lecture Notes in Computer Science, Vol. 1008, Springer, Berlin, 1994, pp. 61–74) introduced the normality of bent functions. His work strengthened the interest for the study of the restrictions of Boolean functions on k-dimensional flats providing the concept of k-normality. Using recent results on the decomposition of any Boolean functions with respect to some subspace, we present several formulations of k-normality. We later focus on some highly linear functions, bent functions and almost optimal functions. We point out that normality is a property for which these two classes are strongly connected. We propose several improvements for checking normality, again based on specific decompositions introduced in Canteaut et al. (IEEE Trans. Inform. Theory, 47(4) (2001) 1494), Canteaut and Charpin (IEEE Trans. Inform. Theory). As an illustration, we show that cubic bent functions of 8 variables are normal.     

On Partial Classes Containig All Monotone and Zero-Preserving Total Boolean Functions

We describe sets of partial Boolean functions being closed under the operations of superposition. For any class A of total functions we define the set ??(A) consisting of all partial classes which contain precisely the functions of A as total functions. The cardinalities of such sets ??(A) can be finite or infinite. We state some general results on ??(A). In particular, we describe all 30 closed sets of partial Boolean functions which contain all monotone and zero-preserving total Boolean functions.     

Finding nonnormal bent functions

The question if there exist nonnormal bent functions was an open question for several years. A Boolean function in n variables is called normal if there exists an affine subspace of dimension n/2 on which the function is constant. In this paper we give the first nonnormal bent function and even an example for a nonweakly normal bent function. These examples belong to a class of bent functions found in [J.F. Dillon, H. Dobbertin, New cyclic difference sets with Singer parameters, in: Finite Fields and Applications, to appear], namely the Kasami functions. We furthermore give a construction which extends these examples to higher dimensions. Additionally, we present a very efficient algorithm that was used to verify the nonnormality of these functions.     

On the confusion and diffusion properties of Maiorana–McFarland's and extended Maiorana–McFarland's functions

A practical problem in symmetric cryptography is finding constructions of Boolean functions leading to reasonably large sets of functions satisfying some desired cryptographic criteria. The main known construction, called Maiorana–McFarland, has been recently extended. Some other constructions exist, but lead to smaller classes of functions. Here, we study more in detail the nonlinearities and the resiliencies of the functions produced by all these constructions. Further we see how to obtain functions satisfying the propagation criterion (among which bent functions) with these methods, and we give a new construction of bent functions based on the extended Maiorana–McFarland's construction.     

Relation between o-equivalence and EA-equivalence for Niho bent functions

Boolean functions, and bent functions in particular, are considered up to so-called EA-equivalence, which is the most general known equivalence relation preserving bentness of functions. However, for a special type of bent functions, so-called Niho bent functions there is a more general equivalence relation called o-equivalence which is induced from the equivalence of o-polynomials. In the present work we study, for a given o-polynomial, a general construction which provides all possible o-equivalent Niho bent functions, and we considerably simplify it to a form which excludes EA-equivalent cases. That is, we identify all cases which can potentially lead to pairwise EA-inequivalent Niho bent functions derived from o-equivalence of any given Niho bent function. Furthermore, we determine all pairwise EA-inequivalent Niho bent functions arising from all known o-polynomials via o-equivalence.     

MacWilliams duality and a Gleason-type theorem on self-dual bent functions

We prove that the MacWilliams duality holds for bent functions. It enables us to derive the concept of formally self-dual Boolean functions with respect to their near weight enumerators. By using this concept, we prove the Gleason-type theorem on self-dual bent functions. As an application, we provide the total number of (self-dual) bent functions in two and four variables obtaining from formally self-dual Boolean functions.     

Certain properties of generalized α-formulas

The problem of realization of Boolean functions by generalized α-formulas is considered. The notion of a universal set of generalized α-formulas is introduced for a given set of Boolean functions. Universal sets of generalized α-formulas are constructed for the set of constant-preserving Boolean functions.     

Boolean functions with MacWilliams duality

We introduce a new class of Boolean functions for which the MacWilliams duality holds, called MacWilliams-dual functions, by considering a dual notion on Boolean functions. By using the MacWilliams duality, we prove the Gleason-type theorem on MacWilliams-dual functions. We show that a collection of MacWilliams-dual functions contains all the bent functions and all formally self-dual functions. We also obtain the Pless power moments for MacWilliams-dual functions. Furthermore, as an application, we prove the nonexistence of bent functions in 2n variables with minimum degree n?k for any nonnegative integer k and n ≥ N with some positive integer N under a certain condition.     

Bent函数的一般构造法

本文用概率方法给出小项表示的布尔函数谱的性质,据此得到了Ｂｅｎｔ函数的特征矩阵的等价刻画,原则上给出了Ｂｅｎｔ函数的一般构造法,并为Ｂｅｎｔ函数的计数问题提供了一个模型。文中还提出了Ｂｅｎｔ矩阵的概念,考察了Ｂｅｎｔ矩阵的性质,并借助Ｂｅｎｔ矩阵得到由已知Ｂｅｎｔ函数构造新的Ｂｅｎｔ函数构造新的Ｂｅｎｔ函数的方法。     

Bent and generalized bent Boolean functions

In this paper, we investigate the properties of generalized bent functions defined on ${\mathbb{Z}_2^n}$ with values in ${\mathbb{Z}_q}$ , where q ≥ 2 is any positive integer. We characterize the class of generalized bent functions symmetric with respect to two variables, provide analogues of Maiorana–McFarland type bent functions and Dillon’s functions in the generalized set up. A class of bent functions called generalized spreads is introduced and we show that it contains all Dillon type generalized bent functions and Maiorana–McFarland type generalized bent functions. Thus, unification of two different types of generalized bent functions is achieved. The crosscorrelation spectrum of generalized Dillon type bent functions is also characterized. We further characterize generalized bent Boolean functions defined on ${\mathbb{Z}_2^n}$ with values in ${\mathbb{Z}_4}$ and ${\mathbb{Z}_8}$ . Moreover, we propose several constructions of such generalized bent functions for both n even and n odd.     

Generalizations of bent functions. A survey

Bent functions (Boolean functions with extreme nonlinearity properties) are actively studied for their numerous applications in cryptography, coding theory, and other fields. New statements of problems lead to a large number of generalizations of the bent functions many of which remain little known to the experts in Boolean functions. In this article, we offer a systematic survey of them.     

Linking systems of difference sets

A linking system of difference sets is a collection of mutually related group difference sets, whose advantageous properties have been used to extend classical constructions of systems of linked symmetric designs. The central problems are to determine which groups contain a linking system of difference sets, and how large such a system can be. All previous constructive results for linking systems of difference sets are restricted to 2‐groups. We use an elementary projection argument to show that neither the McFarland/Dillon nor the Spence construction of difference sets can give rise to a linking system of difference sets in non‐2‐groups. We make a connection to Kerdock and bent sets, which provides large linking systems of difference sets in elementary abelian 2‐groups. We give a new construction for linking systems of difference sets in 2‐groups, taking advantage of a previously unrecognized connection with group difference matrices. This construction simplifies and extends prior results, producing larger linking systems than before in certain 2‐groups, new linking systems in other 2‐groups for which no system was previously known, and the first known examples in nonabelian groups.     

Towards the classification of self-dual bent functions in eight variables

In this paper, we classify quadratic and cubic self-dual bent functions in eight variables with the help of computers. There are exactly four and 45 non-equivalent self-dual bent functions of degree two and three, respectively. This result is achieved by enumerating all eigenvectors with ± 1 entries of the Sylvester Hadamard matrix with an integer programming algorithm based on lattice basis reduction. The search space has been reduced by breaking the symmetry of the problem with the help of additional constraints. The final number of non-isomorphic self-dual bent functions has been determined by exploiting that EA-equivalence of Boolean functions is related to the equivalence of linear codes.     

A conjecture about binary strings and its applications on constructing Boolean functions with optimal algebraic immunity

In this paper, a combinatorial conjecture about binary strings is proposed. Under the assumption that the proposed conjecture is correct, two classes of Boolean functions with optimal algebraic immunity can be obtained. The functions in first class are bent, and then it can be concluded that the algebraic immunity of bent functions can take all possible values except one. The functions in the second class are balanced, and they have optimal algebraic degree and the best nonlinearity up to now.     

Boolean functions with a simple certificate for CNF complexity

In this paper we study relationships between CNF representations of a given Boolean function $f$ and essential sets of implicates of $f$. It is known that every CNF representation and every essential set must intersect. Therefore the maximum number of pairwise disjoint essential sets of $f$ provides a lower bound on the size of any CNF representation of $f$. We are interested in functions, for which this lower bound is tight, and call such functions coverable. We prove that for every coverable function there exists a polynomially verifiable certificate (witness) for its minimum CNF size. On the other hand, we show that not all functions are coverable, and construct examples of non-coverable functions. Moreover, we prove that computing the lower bound, i.e. the maximum number of pairwise disjoint essential sets, is $NP$-hard under various restrictions on the function and on its input representation.     

A novel algorithm enumerating bent functions

Based on the relationship between the Walsh spectra of a Boolean function at partial points and the Walsh spectra of its subfunctions, and on the binary Möbius transform, a novel algorithm is developed, which can theoretically construct all bent functions. Practically we enumerate all bent functions in 6 variables. With the restriction on the algebraic normal form, the algorithm is also efficient in more variables case. For example, enumeration of all homogeneous bent functions of degree 3 in 8 variables can be done in one minute with a P4 1.7 GHz computer; the nonexistence of homogeneous bent functions in 10 variables of degree 4 is computationally proved.     

Exclusive and essential sets of implicates of Boolean functions

In this paper we study relationships between CNF representations of a given Boolean function f and certain sets of implicates of f. We introduce two definitions of sets of implicates which are both based on the properties of resolution. The first type of sets, called exclusive sets of implicates, is shown to have a functional property useful for decompositions. The second type of sets, called essential sets of implicates, is proved to possess an orthogonality property, which implies that every CNF representation and every essential set must intersect. The latter property then leads to an interesting question, to which we give an affirmative answer for some special subclasses of Horn Boolean functions.