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On the boomerang uniformity of quadratic permutations

机译：论二次排列的旋转亮均匀性

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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.
机译：在Eurocrypt'18，CID，黄，佩林，萨卡基和歌曲介绍了一个名为Boomerang连接表（BCT）的新工具，用于测量块密码的阻力，这是1999年瓦格纳推出的重要密码分析技术阻止密码。接下来，BARRA和CoTeaut引入了与BCT相关的一个重要参数，用于加密S箱称为Boomerang均匀性。本文的目的是对矢量布尔函数（或S盒）的Boomerang均匀性的概念展示了一个简短的最先进的最先进性，并提供了新的结果。更具体地说，我们略有不同但更方便的Boomerang统一的制定，并证明了一些新的身份。此外，我们专注于甚至维度的二次偏移，并获得它们具有最佳BCT的一般标准。由于新标准的结果，可以找到两个先前已知的结果，并且可以找到具有最佳BCT的许多新的二次排列（最佳意味着旋转局连接表中的最大值等于最低已知的差分均匀性）。特别是，我们表明，由Bracken，Tan和Tan呈现的二项式差异4连制的释放均匀性等于4.此外，我们展示了传球统一性与某些特殊二次排列的非线性之间的联系。最后，我们展示了旋转悬浮均匀性的二次排列的表征。通过这种表征，我们表明，通过延长的仿射（EA）等价保持二次偏置的二次置换的旋转亮均匀性。

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