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Finding permutation polynomials with low differential and boomeranguniformityis an important topic in S-box designs of many block ciphers. Forexample, AES chooses the inverse function as its S-box, which is differentially4-uniform and boomerang 6-uniform. Also there has been considerable research onmany non-quadratic permutations which are obtained by modifying certain set ofpoints from the inverse function. In this paper, we give a novel approach thatshows that plenty of existing modifications of the inverse function are in factaffine equivalent to permutations of low Carlitz rank and those modificationscannot be APN (almost perfect nonlinear) unless the Carlitz rank is very large.Using nice properties of the permutations of Carlitz form, we present thecomplete list of permutations of Carlitz rank 3 having the boomerang uniformitysix, and also give the complete classification of the differential uniformityof permutations of Carlitz rank 3. We also provide, up to affine equivalence,all the involutory permutations of Carlitz rank 3 having the boomeranguniformity six.

On the Boomerang Uniformity of Permutations of Low Carlitz Rank