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 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<i<n/2 . 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 $\gcd (i,n/2)=1$ . [ABSTRACT FROM PUBLISHER]