Some permutation polynomials via linear translators

TL;DR

This work advances explicit constructions of permutation polynomials over finite fields by generalizing linear translators to $(b,A)$-linear translators and leveraging them with additive-permutation maps. The main method yields explicit permutation polynomials of the form $\phi(x)=L(x)+L(\gamma)(g(f(x))+A^{-1}(f(x)/b))$, with a closed-form compositional inverse, under the condition that $g(x)=x+b h(x)$ permutes the base field. The paper broadens this framework to multi-translator and bent-function settings, providing a Bent-H dual and extending to multi-term forms $F(x)=x+\sum_i\gamma_i h_i(f_i(x))$ with rank-based and Frobenius variants, thereby generating large families of explicit permutation polynomials and inverses. Special cases include characteristic-2 involutions and Frobenius-based translators, expanding the toolkit for designing PP with potential applications in coding and cryptography.

Abstract

Permutation polynomials with explicit constructions over finite fields have long been a topic of great interest in number theory. In recent years, by applying linear translators of functions from $\mathbb{F}_{q^n}$ to $\mathbb{F}_q$, many scholars constructed some classes of permutation polynomials. Motivated by previous works, we first naturally extend the notion of linear translators and then construct some permutation polynomials.

Theorems & Definitions (19)

• Theorem 1.1
• Definition 2.1
• Remark 2.1
• Proposition 2.2
• Remark 2.2
• Theorem 2.1
• Proposition 2.3
• Proposition 2.4
• Theorem 3.1
• Corollary 3.2