Bent functions linear on elements of some classical spreads and presemifields spreads

Abstract

Bent functions are maximally nonlinear Boolean functions with an even number of variables. They have attracted a lot of research for four decades because of their own sake as interesting combinatorial objects, and also because of their relations to coding theory, sequences and their applications in cryptography and other domains such as design theory. In this paper we investigate explicit constructions of bent functions which are linear on elements of spreads. After presenting an overview on this topic, we study bent functions which are linear on elements of presemifield spreads and give explicit descriptions of such functions for known commutative presemifields. A direct connection between bent functions which are linear on elements of the Desarguesian spread and oval polynomials over finite fields was proved by Carlet and the second author. Very recently, further nice extensions have been made by Carlet in another context. We introduce oval polynomials for semifields which are dual to symplectic semifields. In particular, it is shown that from a linear oval polynomial for a semifield one can get an oval polynomial for transposed semifield.

Bent functions and line ovals

“…Calculations in Magma [4] confirm that expressions (4) and (5) represent the same function (moreover, calculations in Magma also confirm formulas (2) and (3)). …”

“…An example of such function G(x) was introduced in [2,11] in a particular case of spreads related to symplectic semifields.…”

“…These functions were throughly studied in [5,6,8,20,17,28] as Niho bent functions. In [2,9,11,31] these investigations were extended to other types of spreads, and bent functions which are affine on the elements of spreads, were studied.…”

The subfield codes of hyperoval and conic codes

“…We mention that hyperovals are related to bent functions [1,6]. Bent functions could be employed to construct linear codes in many ways [7,16,17,18].…”

Equivalence classes of Niho bent functions

“…These functions were thoroughly studied in [11,15,29,32,36] as Niho bent functions. In [3,4,5,18,37] these investigations were extended to other types of spreads, and bent functions which are affine on the elements of spreads, were studied.…”