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

In 1953, Carlitz showed that all permutation polynomials over \({\mathbb F}_q\) ${\mathbb{F}}_q$ , where \(q>2\) $q>2$ is a power of a prime, are generated by the special permutation polynomials \(x^{q-2}\) $x^{q-2}$ (the inversion) and \( ax+b\) $ax+b$ (affine functions, where \(0\ne a, b\in {\mathbb F}_q\) $0\ne a,b\in {\mathbb{F}}_q$ ). Recently, Nikova, Nikov and Rijmen (2019) proposed an algorithm (NNR) to find a decomposition of the inverse function in quadratics, and computationally covered all dimensions \(n\le 16\) $n\le 16$ . Petrides (2023) theoretically 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 \(n\le 32\) $n\le 32$ . In this paper, we extend Petrides’ result, as well as we propose a new number theoretical approach, which allows us to easily cover all (surely, odd) exponents up to 250, at least.