CRAFT is an substitution-permutation network tweakable block cipher proposed at fast software encryption 2019 by Beierle et al., which is designed to optimize the efficient protection against differential fault analysis (DFA) attacks. In this paper, the full round differential characteristics for CRAFT block cipher are given. A new method on counting the number of differentials by using correlation matrix is given. We can compute the number of all optimal characteristics or suboptimum differentials with the same input difference and output difference by hand. We explore the multiple differential trails and compute the probability of differential characteristic by using the multiplication of correlation matrices. Our work complements automatic search methods for the best differential with a careful manual analysis. Since the automatic search method is stranded by storage and search space limitations, which will cause a computer to crash as the number of search rounds increases. Thanks to the correlation matrix technique, we are able to find differential distinguishers for 9-round of the cipher with the probability of at least $\text{◂◽˙▸}{2}^{-40.68}+\text{◂◽˙▸}{2}^{-48.60}$. Moreover, we can construct differential distinguisher covers more rounds based on the 9-round differential distinguishers. As one of its typical application, we propose the differential characteristics for the full-round CRAFT which ensure that the probability of each round is optimal. Besides, we explore the clustering effect on the full round by exhibiting a class of high probability characteristics for 9-round. In general, we obtain a good understanding of the propagation of differences for CRAFT due to its algebraic structure.

中文翻译：

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使用相关矩阵对 CRAFT 中的微分进行聚类

CRAFT 是 Beierle 等人在 2019 年快速软件加密会议上提出的一种替代置换网络可调整分组密码，旨在优化对差分故障分析 (DFA) 攻击的有效保护。本文给出了CRAFT分组密码的全轮差分特性。给出了一种利用相关矩阵计算微分数的新方法。我们可以手工计算具有相同输入差异和输出差异的所有最优特征或次优微分的数量。我们探索了多个差分轨迹，并通过使用相关矩阵的乘法来计算差分特征的概率。我们的工作通过仔细的手动分析补充了最佳差异的自动搜索方法。由于自动搜索方法受限于存储和搜索空间的限制，随着搜索次数的增加，会导致计算机死机。多亏了相关矩阵技术，我们能够找到 9 轮密码的差分区分器，概率至少为$\text{◂◽˙▸}{\mathrm{2个}}^{-40.68}+\text{◂◽˙▸}{\mathrm{2个}}^{-48.60}$. 此外，我们可以在9轮差分区分器的基础上构造更多轮的差分区分器。作为其典型应用之一，我们提出了整轮CRAFT的差分特性，确保每一轮的概率都是最优的。此外，我们通过展示一类 9 轮的高概率特征来探索整轮的聚类效应。总的来说，由于 CRAFT 的代数结构，我们对 CRAFT 的差异传播有了很好的理解。