A note on APN permutations in even dimension

TL;DR

This paper investigates APN permutations in even dimensions, establishing new theoretical restrictions, proving their non-existence in dimension 4, and providing constraints for dimension 6, advancing understanding in cryptographic design.

Contribution

It provides the first proof of non-existence of APN permutations in dimension 4 and derives new constraints for dimension 6, with insights into their component properties.

Findings

No quadratic components in APN permutations in even dimensions.

Existence of a component with many balanced derivatives if all components are cubic.

Proof of non-existence of APN permutations in dimension 4.

Abstract

APN permutations in even dimension are vectorial Boolean functions that play a special role in the design of block ciphers. We study their properties, providing some general results and some applications to the low-dimension cases. In particular, we prove that none of their components can be quadratic. For an APN vectorial Boolean function (in even dimension) with all cubic components we prove the existence of a component having a large number of balanced derivatives. Using these restrictions, we obtain the first theoretical proof of the non-existence of APN permutations in dimension 4. Moreover, we derive some contraints on APN permutations in dimension 6.