Abstract
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.
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Acknowledgements
The authors would like to thank the editor for efficiently handling our paper and the reviewers for their careful reading, beneficial comments and constructive suggestions. The first and the third-named authors worked on this paper during visits to the Max Planck Institute for Software Systems in Saarbrücken, Germany in Spring of 2022 and 2023. They thank Professor J. Ouaknine for the invitation and the Institute for hospitality and support. During the final stages of the preparation of this paper, the first-named author was a fellow at the Stellenbosch Institute for Advanced Study. He thanks this Institution for hospitality and support.
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This is an expanded and vastly improved version of the Extended Abstract, which was presented at the 8th International Workshop on Boolean Functions and their Applications (BFA) in Voss in September 2023.
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Luca, F., Sarkar, S. & Stănică, P. Representing the inverse map as a composition of quadratics in a finite field of characteristic 2. Cryptogr. Commun. (2024). https://doi.org/10.1007/s12095-024-00702-5
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DOI: https://doi.org/10.1007/s12095-024-00702-5