On the Maximum Number of Bent Components of Vectorial Functions
Alexander Pott, Enes Pašalić, Amela Muratović-Ribić, Samed Bajrić · IEEE Transactions on Information Theory · 2017
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.