On the Boomerang Uniformity of Permutations of Low Carlitz Rank

TL;DR

This paper investigates permutations with low boomerang uniformity, especially those of low Carlitz rank, providing classifications and showing many are affine equivalent to known low-uniformity permutations.

Contribution

It introduces a novel approach linking inverse function modifications to permutations of low Carlitz rank and classifies permutations of Carlitz rank 3 with low boomerang uniformity.

Findings

Complete list of Carlitz rank 3 permutations with boomerang 6-uniformity.

Classification of differential uniformity for Carlitz rank 3 permutations.

Identification of involutory permutations of Carlitz rank 3 with boomerang 6-uniformity.

Abstract

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.