Bent and hyper-bent functions over a field of 2ℓ elements

Unique Aspects of Usage of the Quadratic Cryptanalysis Method to the GOST 28147-89 Encryption Algorithm

“…A large number of works are devoted to the study of various classes of approximating functions and to the construction of functions that are most difficult to such approximations. In these papers, bent functions (Logachev et al, 2004;Dobbertin & Leander, 2004;Chee et al, 1994) are considered, which are Boolean functions from an even number of variables that are maximally distant from the set of all linear functions in the Hamming metric, as well as their generalizations: semi-bent functions (Dobbertin & Leander, 2005), partially bent functions (Qu et al, 2000), Z−bent functions (Pfitzmann, 2003), homogeneous bent functions (Kuzmin et al, 2006), hyper best functions (Carlet & Gaborit, 2006;Youssef, 2007;Kuz'min et al, 2008;Knudsen & Robshaw, 1996). The main idea of using linear cryptanalysis of nonlinear approximations (Knudsen & Robshaw, 1996) is to enrich the class of approximating functions (of m variables) with nonlinear functions and increase the quality of approximation due to this.…”

Generalizations of bent functions. A survey

“…For more on this topic, see [9,61]. A. V. Ivanov also studied [3,4] the monomial approximations of Boolean functions.…”

“…For q = 2, Definitions 14 and 15 coincide. A detailed study of these generalized hyper-bent functions appears in [7]. Let us present here only one construction of them.…”

“…S. Kuz'min et al [7,8] generalize the concept of a hyper-bent function: from Boolean functions they pass to functions over an arbitrary finite field of characteristic 2:…”

“…The problem of approximating arbitrary function from F n q to F q (as above, it is identified with a function f : F q n → F q ) by functions of some bounded class A is considered in [8]. In order to estimate the efficiency of approximation of f by a function g ∈ A, the agreement parameter ∇(f, g) was related to the proximity function of Solodovnikov (4) as…”