Paper 2019/079

New Results about the Boomerang Uniformity of Permutation Polynomials

Kangquan Li, Longjiang Qu, Bing Sun, and Chao Li


In EUROCRYPT 2018, Cid et al. 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. 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. Secondly, thanks to Carlet's idea, we give a characterization of functions $f$ from $\mathbb{F}_{2}^n$ to itself with boomerang uniformity $\delta_{f}$ 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 $\mathbb{F}_{2^n}$, which is the first binomial.

Available format(s)
Publication info
Preprint. MINOR revision.
Finite FieldBoomerang Connectivity TableBoomerang UniformityPermutation Polynomial
Contact author(s)
likangquan11 @ nudt edu cn
2019-01-28: received
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Creative Commons Attribution


      author = {Kangquan Li and Longjiang Qu and Bing Sun and Chao Li},
      title = {New Results about the Boomerang Uniformity of Permutation Polynomials},
      howpublished = {Cryptology ePrint Archive, Paper 2019/079},
      year = {2019},
      note = {\url{}},
      url = {}
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