On the Maximum Number of Bent Components of Vectorial Functions

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