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

In 1953, Carlitz~\cite{Car53} showed that all permutation polynomials over ${\text{F}}_q$, 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 $0\mathrm{\ne }a,b\in {\text{F}}_q$). 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 $n\le 16$. 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 $n\le 32$. 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.