Paper 2026/1012

Linear self-equivalence of the known families of APN functions: a unified point of view

Abstract

The Kim mapping, which belongs to the only known CCZ-equivalence class in even dimension that contains an APN bijection, and the APN bijections in dimension 9 exhibited by Beierle and Leander, all have a univariate representation corresponding to a cyclotomic mapping (aka Wan-Lidl polynomial), and they are all linearly-equivalent to functions with a highly-structured multivariate representation. In this paper, we show that functions having these properties, which correspond to multivariate projective mappings, i.e. multivariate functions whose coordinates are homogeneous, actually appear in the CCZ-equivalence classes of almost all known infinite families of APN functions. However, this behavior is not inherent to the APN property. Indeed, we prove that the vast majority of the known sporadic examples of APN functions in dimension 6 and 8, including almost all APN functions whose automorphism group is reduced to the automorphisms derived from their affine derivatives, are not CCZ-equivalent to an affinely self-equivalent function.