(2026-05) Linear self-equivalence of the known families of APN functions; a unified point of view
2026-05-20
The only known solution to the big APN problem was found by exploring the CCZ-equivalence class of a specific quadratic function, the Kim mapping, which is linearly equivalent to various highly-structured functions. For example, one of these functions has a univariate representation with a specific factorisation highlighting its subspace property and that it is a cyclotomic mapping, while another has a bivariate representation corresponding to a -projective mapping.
In this paper, we show that the properties of this kind all correspond to a type of functions we introduce: multivariate projective mappings. These are multivariate functions whose coordinates are homogeneous. Furthermore, while a function might not have this form, it can still be equivalent to another function that has it. To handle this case, we describe how to identify the presence of a multivariate projective mapping in the linear-equivalence class of a function. We then derive our main result: for almost all known infinite families of APN functions, there exists a multivariate projective mapping, or a function commuting with the Frobenius mapping, that is CCZ-equivalent to them. Despite the widely different initial representations of these families (univariate, bivariate, or trivariate), this pattern holds. We also discuss concrete techniques to detect (or rule out) the presence of a multivariate projective mapping equivalent to a given function.