Representing the inverse map as a composition of quadratics in a finite field of characteristic $2$

TL;DR

This paper explores representing the inverse map over finite fields of characteristic 2 as a composition of quadratic functions, extending previous algorithms and covering a broader range of exponents with a new number theoretical approach.

Contribution

It extends existing algorithms for decomposing inverses into quadratics and introduces a number theoretical method to handle larger exponents efficiently.

Findings

Extended decomposition algorithms up to dimension 32.

Proposed a number theoretical approach covering exponents up to 250.

Enhanced understanding of inverse map representations in finite fields.

Abstract

In 1953, Carlitz~\cite{Car53} showed that all permutation polynomials over $\F_q$, where $q>2$ is a power of a prime, are generated by the special permutation polynomials $x^{q-2}$ (the inversion) and $ ax+b$ (affine functions, where $0\neq a, b\in \F_q$). Recently, Nikova, Nikov and Rijmen~\cite{NNR19} proposed an algorithm (NNR) to find a decomposition of the inverse function in quadratics, and computationally covered all dimensions $n\leq 16$. Petrides~\cite{P23} found a class of integers for which it is easy to decompose the inverse into quadratics, and improved the NNR algorithm, thereby extending the computation up to $n\leq 32$. Here, we extend Petrides' result, as well as we propose a number theoretical approach, which allows us to cover easily all (surely, odd) exponents up to~$250$, at least.