The boomerang attack, introduced by Wagner in 1999, is a cryptanalysis technique against block ciphers based on differential cryptanalysis. In particular it takes into consideration two differentials, one for the upper part of the cipher and one for the lower part, and it exploits the dependency of these two differentials. At Eurocrypt’18, Cid et al. introduced a new tool, called the Boomerang Connectivity Table (BCT), that permits to simplify this analysis. Next, Boura and Canteaut introduced an important parameter for cryptographic S-boxes called boomerang uniformity, that is the maximum value in the BCT. Very recently, the boomerang uniformity of some classes of permutations (in particular quadratic functions) have been studied by Li, Qu, Sun and Li, and by Mesnager, Tang and Xiong. In this paper we further study the boomerang uniformity of some non-quadratic differentially 4-uniform functions. In particular, we consider the case of the Bracken-Leander cubic function and three classes of 4-uniform functions constructed by Li, Wang and Yu, obtained from modifying the inverse functions.

Wagner在1999年提出的回旋镖攻击是一种基于差分密码分析的针对分组密码的密码分析技术。特别地，它考虑了两个差分，一个用于密码的上部，一个用于下部，并且利用了这两个差分的相关性。在Eurocrypt'18上，Cid等人。引入了一种称为Boomerang连接表（BCT）的新工具，该工具可以简化此分析。接下来，Boura和Canteaut为密码S盒引入了一个重要参数，称为回旋镖均匀性，即BCT中的最大值。最近，Li，Qu，Sun和Li以及Mesnager，Tang和Xiong研究了某些类置换（特别是二次函数）的回旋镖均匀性。在本文中，我们进一步研究了一些非二次微分4均匀函数的回旋镖均匀性。特别是，我们考虑了Bracken-Leander三次函数以及由Li，Wang和Yu构造的三类4一致函数的情况，这些函数是通过修改逆函数而获得的。