In this paper, we show that the maximum number of bent component functions of a vectorial function F : GF(2)n→ GF(2)nis 2n- 2n/2. We also show that it is very easy to construct such functions. However, it is a much more challenging task to find such functions in polynomial form F ∈ GF(2n)[x], where F has only a few terms. The only known power functions having such a large number of bent components are xd, where d = 2n/2+ 1. In this paper, we show that the binomials Fi(x) = x2i(x + x(2n/2)) also have such a large number of bent components, and these binomials are inequivalent to the monomials x(2n/2+1) if 0n/2+1). We also determine the complete Walsh spectrum of our functions when n/2 is odd and gcd(i, n/2) = 1.

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.

In this paper, we provide necessary and sufficient conditions for a function of the form F(x)=Trk2k(Σi=1taixri(2k-1)) to be bent. Three equivalent statements, all of them providing both the necessary and sufficient conditions, are derived. In particular, one characterization provides an interesting link between the bentness and the evaluation of F on the cyclic group of the (2k+1)th primitive roots of unity in GF(22k). More precisely, for this group of cardinality 2k+1 given by U={u ∈ GF(22k):u2k+1=1}, it is shown that the property of being vectorial bent implies that Im(F)=GF(2k)∪{0}, if F is evaluated on U, that is, F(u) takes all possible values of GF(2k)* exactly once and the zero value is taken twice when u ranges over U. This condition is then reformulated in terms of the evaluation of certain elementary symmetric polynomials related to F, which in turn gives some necessary conditions on the coefficients ai (for binomial trace functions) that can be stated explicitly. Finally, we show that a bent trace monomial of Dillon's type Trk2k(λxr(2k-1)) is never a vectorial bent function.