On cross-correlation properties of S-boxes and their design using semi-bent functions

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 f ,g of two bent functions f and g achieve its lowest possible value 2 n/2. More precisely, it is sufficient that f + g is 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 Zhou et al., claiming that if f is bent, then f ,g = 2 n/2 if and only if g is 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, 1. A sufficient condition that the absolute indicator of two bent functions achieves its lowest value is derived. 2. A construction of S-boxes with good auto-correlation properties from vectorial bent functions is given. 3. Two classes of nonlinear vectorial semi-bent functions with good auto-correlation properties are proposed. Copyright © 2014 John Wiley & Sons, Ltd.