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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 specialpermutation polynomials $x^{q-2}$ (the inversion) and $ ax+b$ (affinefunctions, where $0\neq a, b\in \F_q$). Recently, Nikova, Nikov andRijmen~\cite{NNR19} proposed an algorithm (NNR) to find a decomposition of theinverse function in quadratics, and computationally covered all dimensions$n\leq 16$. Petrides~\cite{P23} found a class of integers for which it is easyto 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 tocover easily all (surely, odd) exponents up to~$250$, at least.

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