Search

Generalized semifield spreads are partitions Γ = { U , A 1 , ... , A p k } of F p m × F p m obtained from (pre)semifields with a certain additional property, which generalize semifield spreads. In particular, a generalized semifield spread is a bent partition, i.e., every function f : F p m × F p m → F p , which is constant on every set of Γ , such that every c ∈ F p has the same number p k - 1 of A i in the preimage set, is a bent function. We show that from generalized semifield spreads one obtains not only p-ary and vectorial bent functions, but also bent functions f : F p m × F p m → B for any abelian group of order p s , s ≤ k . We investigate the effect of (pre)semifield isotopisms on generalized semifield spreads. We observe that isotopisms can destroy the bent partition property, and derive conditions on an isotopism between two (pre)semifields such that the corresponding partitions are equivalent bent partitions. Most notably, we show that with some other class of isotopisms, one can obtain inequivalent bent partitions, hence different classes of bent functions. This is in contrast to the situation for classical semifield spreads. The spreads of two isotopic (pre)semifields are always equivalent. Employing the 2-rank of Boolean functions we confirm that generalizations of the Desarguesian spread bent functions, which we call generalized PS ap functions, are in general not in the Maiorana-McFarland class. The generalized PS ap class contains functions which are not Maiorana-McFarland nor partial spread bent functions for any partial spread. Explicitly we determine the 2-rank of some Maiorana-McFarland functions in the generalized PS ap class in terms of Fibonacci numbers. [ABSTRACT FROM AUTHOR]

For a positive integer k and a linearized polynomial L(X), polynomials of the form P (X) = G (X) k − L (X) ∈ F q n [ X ] are investigated. It is shown that when L has a non-trivial kernel and G is a permutation of F q n , then P(X) cannot be a permutation if gcd (k , q n − 1) > 1 . Further, necessary conditions for P(X) to be a permutation of F q n are given for the case that G(X) is an arbitrary linearized polynomial. The method uses plane curves, which are obtained via the multiplicative and the additive structure of F q n , and their number of rational affine points. [ABSTRACT FROM AUTHOR]

A bent4 function is a Boolean function with a flat spectrum with respect to a certain unitary transform T . It was shown previously that a Boolean function f in an even number of variables is bent4 if and only if f + σ is bent, where σ is a certain quadratic function depending on T . Hence bent4 functions are also called shifted bent functions. Similarly, a Boolean function f in an odd number of variables is bent4 if and only if f + σ is a semibent function satisfying some additional properties. In this article, for the first time, we analyse in detail the effect of the shifts on plateaued functions, on partially bent functions and on the linear structures of Boolean functions. We also discuss constructions of bent and bent4 functions from partially bent functions and study the differential properties of partially bent4 functions, unifying the previous work on partially bent functions. [ABSTRACT FROM AUTHOR]

Zhou (J. Combin. Des. 21(12), 563–584, 2013) introduced modified planar functions in order to describe (2n, 2n, 2n, 1) relative difference sets R as a graph of a function on the finite field 픽2n, and pointed out that projections of R are difference sets that can be described by negabent or bent4 functions, which are Boolean functions given in multivariate form. One of the objectives of this paper is to contribute to the understanding of these component functions of modified planar functions. We identify the versions of the Walsh transforms that apply to modified planar functions on 픽2n and their components, obtain a description of modified planar functions by their components which is similar to that of the classical planar functions in odd characteristic as a vectorial bent function, and point out some further properties of these components. Finally we introduce vectorial bent4 functions (over finite fields), which correspond to relative difference sets in certain groups. We give conditions under which Maiorana-McFarland functions are such functions, hence give rise to relative difference sets in ℤ2n/2×ℤ4n/2. [ABSTRACT FROM AUTHOR]