Vectorial quadratic bent functions as a product of two linearized polynomials

International audienceTo identify and specify trace bent functions of the form T r n 1 (P (x)), where P (x) ∈ GF (2 n)[x], has been an important research topic lately. We show that an infinite class of quadratic vectorial bent functions can be specified in the univariate polynomial form as F (x) = T r^n_k (αx^2^i (x + x^k)), where n = 2k, i = 0,n-1, and α \notin GF(2^k). Most notablyapart from the cases i \in {0,k} for which the polynomial x^2^i(x+x^2^k) is affinely inequivalent tothe monomial x^{2^k+1}, for the remaining indices i the function x^2^i(x+x^2^k) seems to be affinelyinequivalent to x^2^k+1, as confirmed by computer simulations for small n. It is well-knownthat Tr^n_1(x^2^k+1) is Boolean bent for exactly 2^{2k}-2^k values (this is at the same time themaximum cardinality possible) of α \in GF(2n) and the same is true for our class of quadraticbent functions of the form T r^n_k (αx^2^i (x + x^k))though for i > 0 the associated functionsF : GF(2^n) -> GF(2^n) are in general CCZ inequivalent and also have dierent dierentialdistributions