Approximation of Boolean functions by monomial ones

Abstract

Every Boolean function of n variables is identified with a function F W Q ! P , where Q D GF.2 n /, P D GF.2/. A. Youssef and G. Gong showed that for n D 2 there exist functions F which have equally bad approximations not only by linear functions (that is, by functions tr. x/, where 2 Q and trW Q ! P is the trace function), but also by proper monomial functions (functions tr.

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.…”