Properties Of Bent Functions Research Articles

О свойствах бент-функций, построенных по некоторой бентфункции с помощью подпространств

О свойствах бент-функций, построенных по некоторой бентфункции с помощью подпространств

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 Δf,g of two bent functions f and g achieve its lowest possible value 2ni¾?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 2ni¾?2. This indicates an erroneous conclusion in by Zhou et al., claiming that if f is bent, then Δf,g=2ni¾?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, 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.

Some Properties of Bent Functions

First, this paper discusses and sums up some properties of a pair of functions p(x), q(x) that makes (y +1)p(x)+yq(x) into a bent function. Then it discusses the properties of bent functions. Also, the upper and lower bounds of the number of bent functions on GF(2)2k are discussed.

Results on Bent Functions

In this paper, we present three results on bent functions: a construction, a restriction, and a characterization. Starting with a single bent function, in a simple but very effective way, the construction produces a large number of new bent functions in the same number of variables. The restriction imposes new conditions on the directional derivatives of bent functions. Certain non-existence results that were previously obtained through computer search follow easily from these conditions. The characterization describes bent functions as certain solutions of a system of quadratic equations. Interesting new properties of bent functions are obtained using the characterization.