A construction of weakly and non-weakly regular bent functions

In this article a technique for constructing p-ary bent functions from near-bent functions is presented. This technique is then used to obtain both weakly regular and non-weakly regular bent functions. In particular we present the first known infinite class of non-weakly regular bent functions.     

Construction of bent functions via Niho power functions

A Boolean function with an even number n=2k of variables is called bent if it is maximally nonlinear. We present here a new construction of bent functions. Boolean functions of the form f(x)=tr(α₁x^d₁+α₂x^d₂), α₁,α₂,x∈F_2ⁿ, are considered, where the exponents d_i (i=1,2) are of Niho type, i.e. the restriction of x^d_i on F_2^k is linear. We prove for several pairs of (d₁,d₂) that f is a bent function, when α₁ and α₂ fulfill certain conditions. To derive these results we develop a new method to prove that certain rational mappings on F_2ⁿ are bijective.     

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 coefficients of binary bent functions

We prove a 2-adic inequality for the coefficients of binary bent functions in their polynomial representations. The 2-adic inequality implies a family of identities satisfied by the coefficients. The identities also lead to the discovery of some new affine invariants of Boolean functions on .

     

On bent functions with some symmetric properties

This paper discusses a kind of bent functions that have some symmetric properties about some variables. Section 2 mainly discusses the bent functions symmetric about some two variables and gives the necessary and sufficient condition for these functions. Section 3 gives algebraic expressions of some bent functions.     

On the degree of homogeneous bent functions

In this paper, the degree of homogeneous bent functions is discussed. We prove that for any nonnegative integer k, there exists a positive integer N such that for n?N there exist no 2n- variable homogeneous bent functions having degree n-k or more, where N is the least integer satisfying .     

On the ranks of bent functions

The rank of a bent function is the 2-rank of the associated symmetric 2-design. In this paper, it is shown that it is an invariant under the equivalence relation among bent functions. Some upper and lower bounds of ranks of general bent functions, Maiorana–McFarland bent functions and Desarguesian partial spread bent functions are given. As a consequence, it is proved that almost every Desarguesian partial spread bent function is not equivalent to any Maiorana–McFarland bent function.     

p-Ary and q-ary versions of certain results about bent functions and resilient functions

Using the Teichmüller character and Gauss sums, we obtain the following results concerning p-ary bent functions and q-ary resilient functions: (1) a characterization of certain q-ary resilient functions in terms of their coefficients; (2) stronger upper bounds for the degree of p-ary bent functions; (3) determination of all bent functions on ; (4) a characterization of ternary weakly regular bent functions in terms of their coefficients.     

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.     

On homogeneous rotation symmetric bent functions

In this paper, we present partial results towards the conjectured nonexistence of homogeneous rotation symmetric bent functions having degree > 2.     

On the dual of a Coulter–Matthews bent function

Coulter–Matthews (CM) bent functions are from to defined by , where and (α,2n)=1. It is not known if these bent functions are weakly regular in general. In this paper, we show that when n is even and α=n+1 (or n−1), the CM bent function is weakly regular. Moreover, we explicitly determine the dual of the CM bent function in this case. The dual is a bent function not reported previously.     

Strongly regular graphs associated with ternary bent functions

We prove a new characterization of weakly regular ternary bent functions via partial difference sets. Partial difference sets are combinatorial objects corresponding to strongly regular graphs. Using known families of bent functions, we obtain in this way new families of strongly regular graphs, some of which were previously unknown. One of the families includes an example in [N. Hamada, T. Helleseth, A characterization of some {3v₂+v₃,3v₁+v₂,3,3}-minihypers and some [15,4,9;3]-codes with B₂=0, J. Statist. Plann. Inference 56 (1996) 129-146], which was considered to be sporadic; using our results, this strongly regular graph is now a member of an infinite family. Moreover, this paper contains a new proof that the Coulter-Matthews and ternary quadratic bent functions are weakly regular.     

Amorphic association schemes from bent partitions

Bent partitions are partitions of an elementary abelian group, which have similarities to partitions from spreads. In fact, a spread partition is a special case of a bent partition. In particular, bent partitions give rise to a large number of (vectorial) bent functions. Examples of bent partitions, which generalize the Desarguesian spread, have been presented by Anbar, Meidl and Pirsic, 2021, 2022. Bent partitions, which generalize some other classes of (pre)semifield spreads, have been presented by Anbar, Kalaycı, Meidl 2023. In this article, it is shown that these bent partitions induce $(p^k+1)$-class amorphic associations schemes on ${\mathbb{F}}_{p^m}\times {\mathbb{F}}_{p^m}$, where k is a divisor of m with special properties. This implies a construction of amorphic association schemes from some classes of (pre)semifields.     

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.     

Asymptotic regularity of Daubechies' scaling functions

Let , , be Daubechies' scaling function with symbol , and let , be the corresponding Sobolev exponent. In this paper, we make a sharp estimation of , and we prove that there exists a constant independent of such that

This answers a question of Cohen and Daubeschies ( Rev. Mat. Iberoamericana, 12(1996), 527-591) positively.

     

Reconstruction theorems for centered functions and perfect codes

The article addresses the centered functions and perfect codes in the space of all binary n-tuples. We prove that all values of a centered function in a ball of radius k ≤ (n + 1)/2 are uniquely defined from its radial sums with respect to the vertices of the corresponding sphere. We present some theorems of full and partial reconstruction of a centered function from part of its values and derive a new property of the symmetry groups of centered functions.