On Known APNs
Valérie Gillot ad Philippe Langevin
TL;DR
The paper attacks the problem of completeness for 6-bit APN mappings by leveraging new invariants and an APN-extendibility criterion combined with backtracking. It formalizes Boolean and vectorial function invariants, analyzes component and counting functions, and studies switching and extension phenomena to map the 6-bit APN landscape. The results show that the known 14 CCZ-classes are closed under 1-switching and, with a backtracking framework, under 2-switching for many subclasses, while two-level quartic-cubic constructions predominantly yield CCZ-equivalent to the Dublin permutation. Overall, the work provides strong numerical and structural evidence supporting the conjecture that no additional 6-bit APN CCZ-classes exist beyond the 14 known ones, albeit leaving open the full resolution of the two-level; its techniques offer practical discriminants and scalable procedures for exploring APN extensions. The study advances understanding of APN mappings through precise invariants, extension tests, and computational strategies applicable to finite-field cryptography and the broader theory of APN functions.
Abstract
We present new invariants, APN-extendibility criterion and a backtracking approach to identify several numerical facts supporting the conjecture that the set of 6-bit \APN functions is limited to 14 CCZ-classes.