Paper 2019/277

On the boomerang uniformity of quadratic permutations

Sihem Mesnager, Chunming Tang, and Maosheng Xiong


At Eurocrypt'18, Cid, Huang, Peyrin, Sasaki, and Song introduced a new tool called Boomerang Connectivity Table (BCT) for measuring the resistance of a block cipher against the boomerang attack which is an important cryptanalysis technique introduced by Wagner in 1999 against block ciphers. Next, Boura and Canteaut introduced an important parameter related to the BCT for cryptographic Sboxes called boomerang uniformity. The purpose of this paper is to present a brief state-of-the-art on the notion of boomerang uniformity of vectorial Boolean functions (or Sboxes) and provide new results. More specifically, we present a slightly different but more convenient formulation of the boomerang uniformity and prove some new identities. Moreover, we focus on quadratic permutations in even dimension and obtain general criteria by which they have optimal BCT. {As a consequence of the new criteria}, two previously known results can be derived, and many new quadratic permutations with optimal BCT (optimal means that the maximal value in the Boomerang Connectivity Table equals the lowest known differential uniformity) can be found. In particular, we show that the boomerang uniformity of the binomial differentially $4$-uniform permutations presented by Bracken, Tan, and Tan equals $4$. Furthermore, we show a link between the boomerang uniformity and the nonlinearity for some special quadratic permutations. {Finally, we present a characterization of quadratic permutations with boomerang uniformity $4$. With this characterization, we show that the boomerang uniformity of a quadratic permutation with boomerang uniformity $4$ is preserved by the extended affine (EA) equivalence.

Available format(s)
Secret-key cryptography
Publication info
Preprint. MINOR revision.
Vectorial functionsBlock ciphersBoomerang uniformityBoomerang Connectivity TableBoomerang attackSymmetric cryptography
Contact author(s)
tangchunmingmath @ 163 com
2019-09-04: revised
2019-03-12: received
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Creative Commons Attribution


      author = {Sihem Mesnager and Chunming Tang and Maosheng Xiong},
      title = {On the boomerang uniformity of quadratic permutations},
      howpublished = {Cryptology ePrint Archive, Paper 2019/277},
      year = {2019},
      note = {\url{}},
      url = {}
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