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On the boomerang uniformity of quadratic permutations

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Abstract

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 S-boxes 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 S-boxes) 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.

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Acknowledgements

The authors thank the anonymous reviewers and the Associate Editor for their detailed comments that highly improved the presentation and the quality of this paper. They also thank Nian Li and Haode Yan for their interesting discussions in Hong Kong. C. Tang was supported by National Natural Science Foundation of China (Grant No. 11871058) and China West Normal University (Grant Nos. 14E013, CXTD2014-4 and the Meritocracy Research Funds). M. Xiong was supported by The Hong Kong Research Grants Council, Project No. N_HKUST619/17.

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Authors and Affiliations

  1. Department of Mathematics, University of Paris VIII, 93526, Saint-Denis, France

    Sihem Mesnager

  2. University Sorbonne Paris Cité, Laboratory Analysis, Geometry and Applications (LAGA), UMR 7539, CNRS, 93430, Villetaneuse, France

    Sihem Mesnager

  3. Telecom Paris, 91120, Palaiseau, France

    Sihem Mesnager

  4. School of Mathematics and Information, China West Normal University, Nanchong, 637002, People’s Republic of China

    Chunming Tang

  5. Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

    Chunming Tang & Maosheng Xiong

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  1. Sihem Mesnager
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  2. Chunming Tang
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  3. Maosheng Xiong
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Correspondence to Chunming Tang.

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Communicated by O. Ahmadi.

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Mesnager, S., Tang, C. & Xiong, M. On the boomerang uniformity of quadratic permutations. Des. Codes Cryptogr. 88, 2233–2246 (2020). https://doi.org/10.1007/s10623-020-00775-2

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