On cross‐correlation properties of S‐boxes and their design using semi‐bent functions
Enes Pašalić, Samed Bajrić, Milan Djordjević · Security and Communication Networks · 2014
Abstract In this paper, several methods for constructing substitution boxes (S‐boxes) with good cross‐correlation properties are proposed. We firstly analyze the cross‐correlation properties of bent functions and derive a sufficient condition that the absolute indicator Δf,gof two bent functionsfandgachieve its lowest possible value 2n ∕ 2. More precisely, it is sufficient thatf + gis also a bent function, which then implies that the absolute indicator of vectorial bent functions equals to 2n ∕ 2. This indicates an erroneous conclusion in by Zhouet al., claiming that iffis bent, then Δf,g = 2n ∕ 2if and only ifgis an affine function, which is not true. Furthermore, because of a strong relationship between the cross‐correlation properties and disjoint spectra semi‐bent functions, two classes of highly nonlinear vectorial semi‐bent functions with very good cross‐correlation properties are proposed. In particular, the first class of vectorial semi‐bent functions introduced here compares favorably to other methods in terms of the cross‐correlation properties of its component functions. In addition, A sufficient condition that the absolute indicator of two bent functions achieves its lowest value is derived. A construction of S‐boxes with good auto‐correlation properties from vectorial bent functions is given. Two classes of nonlinear vectorial semi‐bent functions with good auto‐correlation properties are proposed. Copyright © 2014 John Wiley & Sons, Ltd.