关于$F_{2^n}$上（二次）置换的回飞一致性
On the boomerang uniformity of (quadratic) permutations over $F_{2^n}$
摘要：
在Eurocrypt'18上，Cid，Huang，Peyrin，Sasaki和Song介绍了一种名为Boomerang Connectivity Table（BCT）的新工具，用于测量分组密码对Boomerang攻击的抵抗力（这是Wagner在1999年针对分组密码引入的重要密码分析技术）。接下来，Boura和Canteaut为加密Sbox引入了一个重要的参数（与BCT相关），称为回飞棒均匀性。在这种情况下，我们提出了一个简短的国家的最先进的概念，矢量函数（或Sboxes）的回旋镖均匀性，并提供新的结果。更具体地说，我们提出了一个略有不同的（更方便）制定的回飞棒的均匀性，并表明，行总和和列总和的回飞棒连接表可以表示在零的二阶导数的置换或其逆。最重要的是，我们专门研究我们的回旋镖均匀性，二次排列在偶数维和推广的最佳BCT（最佳的意思是，在回旋镖连接表中的最大值等于最低已知的差分均匀性）的二次排列的先前结果。作为我们的一般结果的结果，我们证明了由Bracken，Tan和Tan提出的二项式差分$4$ -均匀排列的回飞去来均匀性等于$4$。这一结果产生了一个新的家庭的最佳Sbox。
英文摘要：
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. In this context, we present a brief state-of-the-art on the notion of boomerang uniformity of vectorial functions (or Sboxes) and provide new results. More specifically, we present a slightly different (and more convenient) formulation of the boomerang uniformity and show that the row sum and the column sum of the boomerang connectivity table can be expressed in terms of the zeros of the second-order derivative of the permutation or its inverse. Most importantly, we specialize our study of boomerang uniformity to quadratic permutations in even dimension and generalize the previous results on quadratic permutation with optimal BCT (optimal means that the maximal value in the Boomerang Connectivity Table equals the lowest known differential uniformity). As a consequence of our general result, we prove that the boomerang uniformity of the binomial differentially $4$-uniform permutations presented by Bracken, Tan, and Tan equals $4$. This result gives rise to a new family of optimal Sboxes.
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