  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.
