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 (J. Combin. Theory Ser. A 118(3), 1052–1061, 2011). We call these translators Frobenius translators since the derivatives of \(f:{\mathbb F}_{p^{n}} \rightarrow {\mathbb F}_{p^{k}}\), where n = rk, are of the form \(f(x+u\gamma )-f(x)=u^{p^{i}}b\), for a fixed \(b \in {\mathbb F}_{p^{k}}\) and all \(u \in {\mathbb F}_{p^{k}}\), rather than considering the standard case corresponding to i = 0. It turns out that Frobenius translators correspond to standard linear translators of an exponentiated version of f, namely to \(f^{p^{k-i}}\) with respect to \(b^{p^{k-i}}\). Nevertheless, this concept turns out to be useful for providing further explicit specification of a rather rare family {f} of quadratic polynomials (especially sparse ones) admitting linear translators. In this direction, we solve a few open problems in the recent article (Cepak et al., Finite Fields Appl. 45, 19–42, 2017) concerning the existence and an exact specification of f admitting classical linear translators. In addition, an open problem introduced in Hodžić et al. (2018), of finding a triple of bent functions f₁,f₂,f₃ such that their sum f₄ is bent and that the sum of their duals satisfies \(f_{1}^{*}+f_{2}^{*}+f_{3}^{*}+f_{4}^{*}=1\), is also resolved. We also specify two huge families of permutations over \({\mathbb F}_{p^{n}}\) related to the condition that \(G(y)=-L(y)+(y+\delta )^{s}-(y+\delta )^{p^{k}s}\) permutes the set \({\mathcal S}=\{\beta \in {\mathbb F}_{p^{n}}: T{r_{k}^{n}}(\beta )=0\}\), where n = 2k and p > 2. Finally, we give some generalizations of constructions of bent functions in Mesnager et al. (2017) and describe some new bent families using the permutations found in Cepak et al. (Finite Fields Appl. 45, 19–42, 2017).