Vectorial quadratic bent functions as a product of two linearized polynomials
Résumé
To 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 notably
apart from the cases i \in {0,k} for which the polynomial x^2^i
(x+x^2^k) is affinely inequivalent to
the monomial x^{2^k+1}, for the remaining indices i the function x^2^i
(x+x^2^k) seems to be affinely
inequivalent to x^2^k+1, as confi rmed by computer simulations for small n. It is well-known
that Tr^n_1( x^2^k+1) is Boolean bent for exactly 2^{2k}-2^k values (this is at the same time the
maximum cardinality possible) of α \in GF(2n) and the same is true for our class of quadratic
bent functions of the form
T r^n_k (αx^2^i (x + x^k))
though for i > 0 the associated functions
F : GF(2^n) -> GF(2^n) are in general CCZ inequivalent and also have di erent di erential
distributions.
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