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 \({\mathbb F}_q\), 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 \(0\ne a, b\in {\mathbb F}_q\)). 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 \(n\le 16\). 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 \(n\le 32\). 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.