On the boomerang uniformity of permutations of low Carlitz rank

Finding permutation polynomials with low differential and boomerang uniformity is 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 researc...

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Bibliographic Details
Published in:Finite fields and their applications 2022-08, Vol.81, p.102033, Article 102033
Main Authors: Jeong, Jaeseong, Koo, Namhun, Kwon, Soonhak
Format: Article
Language:English
Subjects:
APN
Boolean function
Boomerang uniformity
Differential uniformity
Permutation
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Summary:Finding permutation polynomials with low differential and boomerang uniformity is 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 modifications of the inverse function. In this paper, we give a novel approach which 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. We also present the complete list of permutations of Carlitz rank 3 having the boomerang uniformity six, and give the complete classification of the differential uniformities of permutations of Carlitz rank 3. As an application, we provide all the involutions of Carlitz rank 3 having the boomerang uniformity six.
ISSN:1071-5797
1090-2465
DOI:10.1016/j.ffa.2022.102033