  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.
