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

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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: Fₚₙ → Fₚₖ, where n = rk, are of the form f(x + uϕ) - f(x) = uᵖⁱb, for a fixed b ∈ Fₚₖ and all u ∈ Fₚₖ, 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 f₁, f₂, f₃ such that their sum f₄ is bent and that the sum of their duals f₁* +f₂* +f₃* +f₄* = 1. Finally, we also specify two huge families of permutations over Fₚₙ related to the condition that G(y) = -L(y)+(y+δ)ˢ -(y+δ)pᵏs permutes the set S ={β∈ Fₚₙ: Trⁿₖ(β) = 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].