  Finding permutation polynomials with low differential and boomerang
uniformityis an important topic in S-box designs of many block ciphers. For
example, AES chooses the inverse function as its S-box, which is differentially
4-uniform and boomerang 6-uniform. Also there has been considerable research on
many non-quadratic permutations which are obtained by modifying certain set of
points from the inverse function. In this paper, we give a novel approach that
shows that plenty of existing modifications of the inverse function are in fact
affine equivalent to permutations of low Carlitz rank and those modifications
cannot be APN (almost perfect nonlinear) unless the Carlitz rank is very large.
Using nice properties of the permutations of Carlitz form, we present the
complete list of permutations of Carlitz rank 3 having the boomerang uniformity
six, and also give the complete classification of the differential uniformity
of permutations of Carlitz rank 3. We also provide, up to affine equivalence,
all the involutory permutations of Carlitz rank 3 having the boomerang
uniformity six.
