Compositional inverses, complete mappings, orthogonal Latin squares and bent functions

TL;DR

This paper investigates the inverses of permutation polynomials and complete mappings, constructs mutually orthogonal Latin squares, and develops bent vectorial functions, advancing the understanding and construction of combinatorial and cryptographic objects.

Contribution

It provides new methods for computing compositional inverses of linearized binomials and complete mappings, and introduces novel classes of Latin squares and bent functions.

Findings

Derived inverses of linearized binomials permuting the kernel of the trace map.

Constructed a new class of complete mappings improving previous results.

Generated mutually orthogonal Latin squares and bent vectorial functions from the new mappings.

Abstract

We study compositional inverses of permutation polynomials, complete mappings, mutually orthogonal Latin squares, and bent vectorial functions. Recently it was obtained in [33] the compositional inverses of linearized permutation binomials over finite fields. It was also noted in [29] that computing inverses of bijections of subspaces have applications in determining the compositional inverses of certain permutation classes related to linearized polynomials. In this paper we obtain compositional inverses of a class of linearized binomials permuting the kernel of the trace map. As an application of this result, we give the compositional inverse of a class of complete mappings. This complete mapping class improves upon a recent construction given in [34]. We also construct recursively a class of complete mappings involving multi-trace functions. Finally we use these complete mappings to derive a set of mutually orthogonal Latin squares, and to construct a class of $p$-ary bent vectorial functions from the Maiorana-McFarland class.