On the Maximum Number of Bent Components of Vectorial Functions

Abstract

In this paper, we show that the maximum number of bent component functions of a vectorial function $F:GF(2)^{n}\to GF(2)^{n}$ is $2^{n}-2^{n/2}$ . 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 $F\in GF(2^{n})[x]$ , where $F$ has only a few terms. The only known power functions having such a large number of bent components are $x^{d}$ , where $d=2^{n/2}+1$ . In this paper, we show that the binomials $F^{i}(x)=x^{2^{i}}(x+x^{2^{n/2}})$ also have such a large number of bent components, and these binomials are inequivalent to the monomials $x^{2^{n/2}+1}$ if $0 . In addition, the functions $F^{i}$ have differential properties much better than $x^{2^{n/2}+1}$ . We also determine the complete Walsh spectrum of our functions when $n/2$ is odd and $\gcd (i,n/2)=1$ .