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.

翻译：1953年，Carlitz~\cite{Car53}指出，$\F_q$上的所有置换多项式（其中$q>2$为素数的幂）均可由特殊置换多项式$x^{q-2}$（逆映射）与$ax+b$（仿射函数，其中$0\neq a, b\in \F_q$）生成。近期，Nikova、Nikov与Rijmen~\cite{NNR19}提出了一种将逆函数分解为二次型的算法（NNR），并通过计算覆盖了所有维度$n\leq 16$。Petrides~\cite{P23}发现了一类易于将逆映射分解为二次型的整数，并改进了NNR算法，从而将计算范围扩展至$n\leq 32$。本文一方面推广了Petrides的结果，另一方面提出了一种数论方法，使得我们能够轻松覆盖至少$250$以内的所有（显然为奇数的）指数。