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.

中文翻译：

---

关于二次排列的回旋镖均匀性

在 Eurocrypt'18 上，Cid、Huang、Peyrin、Sasaki 和 Song 推出了一种名为 Boomerang Connectivity Table (BCT) 的新工具，用于测量分组密码对回旋镖攻击的抵抗力，这是 Wagner 于 1999 年引入的一项重要的密码分析技术，用于对抗回旋镖攻击。块密码。接下来，Boura 和 Canteaut 引入了一个与加密 S 盒的 BCT 相关的重要参数，称为回旋镖均匀性。本文的目的是简要介绍向量布尔函数（或 S 盒）的回旋镖均匀性概念的最新技术，并提供新的结果。更具体地说，我们提出了一个稍微不同但更方便的回旋镖均匀性公式，并证明了一些新的身份。而且，我们专注于偶数维度的二次排列，并获得它们具有最佳 BCT 的一般标准。作为新标准的结果，可以推导出两个先前已知的结果，并且可以找到许多具有最佳 BCT（最佳意味着 Boomerang Connectivity Table 中的最大值等于最低已知差分均匀性）的新二次排列。特别是，我们证明了 Bracken、Tan 和 Tan 提出的二项式微分 4-均匀排列的回旋镖均匀性等于 4。此外，我们展示了回旋镖均匀性与某些特殊二次排列的非线性之间的联系。最后，我们提出了具有回旋镖均匀度 4 的二次排列的特征。有了这个特征，