New Results about the Boomerang Uniformity of Permutation Polynomials
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In EUROCRYPT 2018, Cid et al. BCT2018 introduced a new concept on the cryptographic property of S-boxes: Boomerang Connectivity Table (BCT for short) for evaluating the subtleties of boomerang-style attacks. Very recently, BCT and the boomerang uniformity, the maximum value in BCT, were further studied by Boura and Canteaut BC2018. Aiming at providing new insights, we show some new results about BCT and the boomerang uniformity of permutations in terms of theory and experiment in this paper. Firstly, we present an equivalent technique to compute BCT and the boomerang uniformity, which seems to be much simpler than the original definition from BCT2018. Secondly, thanks to Carlet's idea Carlet2018, we give a characterization of functions f from F_2^n to itself with boomerang uniformity δ_f by means of the Walsh transform. Thirdly, by our method, we consider boomerang uniformities of some specific permutations, mainly the ones with low differential uniformity. Finally, we obtain another class of 4-uniform BCT permutation polynomials over F_2^n, which is the first binomial.
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