Description
To identify and specify trace bent functions of the form *Tr_*1^*n *(*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*) = *Tr_**k^n *(*αx^{*2^*i}*(*x *+ *x^{*2^*k}*)), where *n *= 2*k*, *i *= 0*, . . . , n **− *1, and *α not in **GF*(2^*k*). Most notably, apart from the cases *i **∈ {*0*,k**} *for which the polynomial *x^{*2^*i}*(*x*+*x^{*2^*k}*) is affinely equivalent 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 confirmed by computer simulations for small *n*. It is well-known that *Tr_*1^*n *(*αx^{*2^*k}*+1) is Boolean bent for exactly 2^{2^*k} **− *2^*k *values (this is at the same time the maximum cardinality possible) of *α **∈ **GF* (2^*n*) and the same is true for our class of quadratic bent functions of the form *Tr_*1^ *n*(*αx^{*2^*i}*(*x *+ *x^{*2^*k}*)), though for *i > *0 the associated functions *F *: *GF*(2^*n*) *→ **GF*(2^*n*) are in general CCZ inequivalent and also have different differential distributions.
Primary author
Dr
Samed Bajrić
(University of Primorska)
