CCZ-equivalence of bent vectorial functions and related constructions

We observe that the CCZ-equivalence of bent vectorial functions over ${{\bf F}_2^n}$ (n even) reduces to their EA-equivalence. Then we show that in spite of this fact, CCZ-equivalence can be used for constructing bent functions which are new up to EA-equivalence and therefore to CCZ-equivalence: applying CCZ-equivalence to a non-bent vectorial function F which has some bent components, we get a function F?? which also has some bent components and whose bent components are CCZ-inequivalent to the components of the original function F. Using this approach we construct classes of nonquadratic bent Boolean and bent vectorial functions.     

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.     

Relating three nonlinearity parameters of vectorial functions and building APN functions from bent functions

We survey the properties of two parameters introduced by C. Ding and the author for quantifying the balancedness of vectorial functions and of their derivatives. We give new results on the distribution of the values of the first parameter when applied to F + L, where F is a fixed function and L ranges over the set of linear functions: we show an upper bound on the nonlinearity of F by means of these values, we determine then the mean of these values and we show that their maximum is a nonlinearity parameter as well, we prove that the variance of these values is directly related to the second parameter. We briefly recall the known constructions of bent vectorial functions and introduce two new classes obtained with Gregor Leander. We show that bent functions can be used to build APN functions by concatenating the outputs of a bent (n, n/2)-function and of some other (n, n/2)-function. We obtain this way a general infinite class of quadratic APN functions. We show that this class contains the APN trinomials and hexanomials introduced in 2008 by L. Budaghyan and the author, and a class of APN functions introduced, in 2008 also, by Bracken et al.; this gives an explanation of the APNness of these functions and allows generalizing them. We also obtain this way the recently found Edel?CPott cubic function. We exhibit a large number of other sub-classes of APN functions. We eventually design with this same method classes of quadratic and non-quadratic differentially 4-uniform functions.     

Nonexistence of two classes of generalized bent functions

We obtain some new nonexistence results of generalized bent functions from \({\mathbb {Z}}^n_q\) to \({\mathbb {Z}}_q\) (called type [n, q]) in the case that there exist cyclotomic integers in \( {\mathbb {Z}}[\zeta _{q}]\) with absolute value \(q^{\frac{n}{2}}\). This result generalizes two previous nonexistence results \([n,q]=[1,2\times 7]\) of Pei (Lect Notes Pure Appl Math 141:165–172, 1993) and \([3,2\times 23^e]\) of Jiang and Deng (Des Codes Cryptogr 75:375–385, 2015). We also remark that by using a same method one can get similar nonexistence results of GBFs from \({\mathbb {Z}}^n_2\) to \({\mathbb {Z}}_m\).     

Partial difference sets and amorphic group schemes from pseudo-quadratic bent functions

We present new abelian partial difference sets and amorphic group schemes of both Latin square type and negative Latin square type in certain abelian p-groups. Our method is to construct what we call pseudo-quadratic bent functions and use them in place of quadratic forms. We also discuss a connection between strongly regular bent functions and amorphic group schemes.     

The relationship between the nonexistence of generalized bent functions and diophantine equations

Two new results on the nonexistence of generalized bent functions are presented by using properties of the decomposition law of primes in cyclotomic fields and properties of solutions of some Diophantine equations, and examples satisfying our results are given.     

A relationship between the nonexistence of generalized bent functions and class groups

A new result on the nonexistence of generalized bent functions is presented by using properties of the decomposition law of primes in cyclotomic fields and properties of solutions of some Diophantine equations. At the same time,a method is given which can be used to simplify the known results. Then we give the bounds and the meaning in algebraic number theory of the parameters in our results.     

New results on non-existence of generalized bent functions (Ⅱ)

Several new results on non-existence of generalized bent functions are presented by using the class group of related imaginary abelian number fields.     

New results on non-existence of generalized bent functions (II)

Several new results on non-existence of generalized bent functions are presented by using the class group of related imaginary abelian number fields.     

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.     

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.     

Construction of bent functions from near-bent functions

We give a construction of bent functions in dimension 2m from near-bent functions in dimension 2m−1. In particular, we give the first ever examples of non-weakly-normal bent functions in dimensions 10 and 12, which demonstrates the significance of our construction.     

Generalized bent functions and their properties

Jet J_q^m denote the set of m-tuples over the integers modulo q and set $\text{i=}\text{?1}$, $\text{w = e}{}^{\mathrm{<mtext>i(</mtext><mtext>2\pi </mtext><mtext>q</mtext><mtext>)</mtext>}}$. As an extension of Rothaus' notion of a bent function, a function f, f: J_q^m → J_q¹ is called bent if all the Fourier coefficients of w^f have unit magnitude. An important feature of these functions is that their out-of-phase autocorrelation value is identically zero. The nature of the Fourier coefficients of a bent function is examined and a proof for the non-existence of bent functions over J_q^m, m odd, is given for many values of q of the form q = 2 (mod 4). For every possible value of q and m (other than m odd and q = 2 (mod 4)), constructions of bent functions are provided.     

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.     

A construction of bent functions from plateaued functions

In this presentation, a technique for constructing bent functions from plateaued functions is introduced and analyzed. This generalizes earlier techniques for constructing bent from near-bent functions. Using this construction, we obtain a big variety of inequivalent bent functions, some weakly regular and some non-weakly regular. Classes of bent functions having some additional properties that enable the construction of strongly regular graphs are formed, and explicit expressions for bent functions with maximal degree are presented.     

Vectorial bent functions and partial difference sets

Designs, Codes and Cryptography - The objective of this article is to broaden the understanding of the connections between bent functions and partial difference sets. Recently, the first two...     

Generalized penalization and maximization of vectorial nonsmooth functions

We use directional Lipschitz concepts and a minimal time function with respect to a set of directions in order to derive generalized penalization results for Pareto minimality in set-valued constrained optimization. Then, we obtain necessary optimality conditions for maximization in constrained vector optimization in terms of generalized differentiation objects. To the latter aim, we deduce first some enhanced calculus rules for coderivatives of the difference of two mappings. All the main results of this paper are tailored to model directional features of the optimization problem under study.     

Equivalence classes of Niho bent functions

Equivalence classes of Niho bent functions are in one-to-one correspondence with equivalence classes of ovals in a projective plane. Since a hyperoval can produce several ovals, each hyperoval is associated with several inequivalent Niho bent functions. For all known types of hyperovals we described the equivalence classes of the corresponding Niho bent functions. For some types of hyperovals the number of equivalence classes of the associated Niho bent functions are at most 4. In general, the number of equivalence classes of associated Niho bent functions increases exponentially as the dimension of the underlying vector space grows. In small dimensions the equivalence classes were considered in detail.