Classification of quadratic APN functions with coefficients in \(\mathbb{F}_2\) for dimensions up to 9
The classification of APN functions is a very difficult problem, even in a small number of variables \(n\). A complete classification (up to CCZ-equivalence) of APN functions on \(\mathbb{F}_{2^n}\) is known for dimension \(n\le 5\) [\textit{M. Brinkmann} and \textit{G. Leander}, Des. Codes Cryptography 49, No. 1--3, 273--288 (2008; Zbl 1184.94227)]. Further, some partial results for \(n=6\) are known. With a specialization of the construction method of quadratic APN functions via matrices in [\textit{Y. Yu} et al., Des. Codes Cryptography 73, No. 2, 587--600 (2014; Zbl 1320.11122)], by which a huge amount of quadratic APN functions can be generated, the authors present a complete classification (up to CCZ-equivalence) of quadratic APN functions on \(\mathbb{F}_{2^n}\), which have a representation as a polynomial with coefficients in \(\mathbb{F}_2\), up to dimension \(n =9\). Whereas in dimension \(6\le n \le 8\), all such quadratic APN functions are CCZ equivalent to already known APN functions, two new quadratic APN functions (not CCZ-equivalent to any earlier known function) in dimension \(n=9\) are discovered.
scientific article; zbMATH DE number 7312715
scientific article