Abstract: We introduce the notion of k-bent function, i.e., a Boolean function with even number m of variables ν1,..., νm which can be approximated with all functions of the form 〈u, v〉j ⊕ a in the equally poor manner, where u ∈2 ℤ2m, a ∈ ℤ2, and 1 ≤ j ≤ k. Here 〈·, ·〉j is an analog of the inner product of vectors; k changes from 1 to m/2. The operations 〈·, ·〉k, 1 ≤ k ≤ m/2, are defined by using the special series of binary Hadamard-like codes Amk of length 2m. Namely, the vectors of values for the functions 〈u, v〉k ⊕ are codewords of the code Amk. The codes Amk are constructed using subcodes of the ℤ4-linear Hadamard-like codes of length 2m+1, which were classified by D. S. Krotov (2001). At that the code Am1 is linear and Am1,..., Amm/2 are pairwise nonequivalent. On the codewords of a code Amk, we define a group operation • coordinated with the Hamming metric. That is why we can consider k-bent functions as functions that are maximal nonlinear in k distinct senses of linearity at the same time. Bent functions in usual sense coincide with 1-bent functions. For k > ℓ ≥ 1, the class of k-bent functions is a proper subclass of the class of ℓ-bent functions. In the paper, we give methods for constructing k-bent functions and study their properties. It is shown that there exist k-bent functions with an arbitrary algebraic degree d, where 2 ≤ d ≤ max {2, m/2 - k + 1}. Given an arbitrary k, we define the subset Fmk Fmk of the set Fm Fmk Amk Bmk of all Boolean functions in m variables with the following property: on this subset k-bent functions and 1-bent functions coincide.

Cite: Токарева Н.Н.
Bent functions with stronger nonlinear properties: K-bent functions
Journal of Applied and Industrial Mathematics. 2008. Т.2. №4. С.566-584. DOI: 10.1134/S1990478908040133 Scopus OpenAlex

Identifiers:

Scopus:	2-s2.0-57549095939
OpenAlex:	W2158851786

Citing:

DB	Citing
Scopus	2
OpenAlex	5

Altmetrics: