Bent, resilient functions and the numerical normal form (Q2717196)

The authors investigate special Boolean (i.e. \(\{0,1\}\)-valued) functions on the vector space \(\mathbb{F}^n_2\), namely bent functions and resilient functions, which are of particular interest in cryptography and coding theory. (\(\mathbb{F}_2\) denotes the 2-element field.)NEWLINENEWLINENEWLINEThe invesigation is done by considering more general real-valued functions on \(\mathbb{F}^n_2\) and so-called Numerical Normal Forms (N.N.F.s) of these functions. The main results are characterizations of bent and resilient functions by means of these N.N.F.s and a characterization of bent functions by means of a quadratic Diophantine equation. Furthermore, an affine invariant related to the N.N.F. (the so-called generalized degree) is considered.NEWLINENEWLINENEWLINEThe paper is a continuation of an earlier work by the same authors [``A new representation of Boolean functions'', Lect. Notes Comput. Sci. 1719, 94-103 (1999; Zbl 0979.94060)], where the first systematic study of the N.N.F. and of relations to other representations used in coding and cryptography were made.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00079].