Permutations via linear translators

TL;DR

This paper introduces new classes of permutations over finite fields constructed via linear translators, characterizes functions with such translators, and explores their inverses and connections to complete permutations.

Contribution

It provides a comprehensive characterization of functions with linear translators and constructs numerous new permutation classes, extending previous results and introducing new analytical tools.

Findings

Several new infinite classes of permutations are constructed.

Explicit formulas for the compositional inverses of these permutations are provided.

New tools are proposed for studying permutations of specific algebraic forms.

Abstract

We show that many infinite classes of permutations over finite fields can be constructed via translators with a large choice of parameters. We first charac- terize some functions having linear translators, based on which several families of permutations are then derived. Extending the results of [10], we give in several cases the compositional inverse of these permutations. The connection with complete permutations is also utilized to provide further infinite classes of permutations. Moreover, we propose new tools to study permutations of the form x is mapped to x+(x^(p^m) - x+ lambda)^s and a few infinite classes of permutations of this form are proposed.