A new class of monomial bent functions

Abstract

We study the Boolean functions f λ : F 2 n → F 2 , n = 6r, of the form f (x) = Tr(λx d ) with d = 2 2r + 2 r + 1 and λ ∈ F 2 n . Our main result is the characterization of those λ for which f λ are bent. We show also that the set of these cubic bent functions contains a subset, which with the constantly zero function forms a vector space of dimension 2r over F 2 . Further we determine the Walsh spectra of some related quadratic functions, the derivatives of the functions f λ .

Hyperbent Functions, Kloosterman Sums, and Dickson Polynomials

“…A number of recent papers are devoted to bent Boolean functions expressed by means of trace-functions [1,2,3,6,11,16,17]. In this paper, we contribute to the knowledge of this fascinating class of functions, by studying a subclass of the so-called PS − class.…”

On Third-Order Nonlinearity of Biquadratic Monomial Boolean Functions

“…Next, if is a vector space over a field F of characteristic 2 and : → F a quadratic form, then dim( ) and dim(E ) have the same parity [21]. The distribution of the WHT values of a quadratic Boolean function ∈ B is given in the following theorem which claims that the weight distribution of the values in the WHS of depends only on the dimension of E .…”

“…Theorem 5 (see [17,21]). Let ∈ B be a quadratic Boolean function and = dim(E ) , where E is defined in (8); then the weight distribution of the WHT values of is given by…”

Construction of a New Class of Bent and Semi-bent Functions

“…Primary and secondary constructions of bent functions are the two kinds of construction of bent functions. In the primary construction, there is no use of previously existing bent functions to construct new ones, while in secondary construction some previously known bent functions are used to construct new bent functions, see [13,14,15,16]. Several constructions of bent functions are discussed in [17,18].…”