A note on APN permutations in even dimension
This work addresses a specific problem in cryptography for designing secure block ciphers, but it is incremental as it builds on existing knowledge with new theoretical results.
The paper tackled the problem of APN permutations in even dimension, which are important for block cipher design, by proving that none of their components can be quadratic and providing constraints for dimensions 4 and 6, including the first theoretical proof of non-existence in dimension 4.
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.