Frobenius linear translators giving rise to new infinite classes of permutations and bent functions

We show the existence of many infinite classes of permutations over finite fields and bent functions by extending the notion of linear translators, introduced by Kyureghyan [12]. We call these translators Frobenius translators since the derivatives of $f : F_{p^n} \rightarrow F_{p^k}$, where $n = rk$, are of the form $f(x + u\phi) - f(x) = u^{p^i}b$, for a fixed $b \in F_{p^k}$ and all $u \in F_{p^k}$, rather than considering the standard case corresponding to $i = 0$. This considerably extends a rather rare family {f} admitting linear translators of the above form. Furthermore, we solve a few open problems in the recent article [4] concerning the existence and an exact specification of $f$ admitting classical linear translators, and an open problem introduced in [9] of finding a triple of bent functions $f_1, f_2, f_3$ such that their sum $f_4$ is bent and that the sum of their duals $f_1* +f_2* +f_3* +f_4* = 1$. Finally, we also specify two huge families of permutations over $F_{p^n}$ related to the condition that $G(y) = -L(y)+(y+\delta)^s -(y+\delta)^{p^ks}$ permutes the set $S =\{\beta \in F_{p^n} : Tr^n_k(\beta) = 0\}$, where $n = 2k$ and $p > 2$. Finally, we offer generalizations of constructions of bent functions from [16] and described some new bent families using the permutations found in [4].