Properties of the additive differential probability $\mathrm{adp}^{\mathrm{XR}}$ of the composition of bitwise XOR and a bit rotation are investigated, where the differences are expressed using addition modulo $2^n$. This composition is widely used in ARX constructions consisting of additions modulo $2^n$, bit rotations and bitwise XORs. Differential cryptanalysis of such primitives may involve maximums of $\mathrm{adp}^{\mathrm{XR}}$, where some of its input or output differences are fixed. Although there is an efficient way to calculate this probability, many its properties are still unknown. In this work we find maximums of $\mathrm{adp}^{\mathrm{XR}}$, where the rotation is one bit left/right and one of its input differences is fixed. Some symmetries of $\mathrm{adp}^{\mathrm{XR}}$ are obtained as well. Also, we provide all its impossible differentials in terms of regular expression patterns. The number of them is estimated. It turned out to be maximal for the one bit left rotation and noticeably less than the number of impossible differentials of bitwise XOR.

翻译：研究按位异或（XOR）与比特旋转组合的加法差分概率$\mathrm{adp}^{\mathrm{XR}}$的性质，其中差分采用模$2^n$加法表示。该组合广泛应用于由模$2^n$加法、比特旋转和按位异或构成的ARX结构中。对此类原语进行差分密码分析时，可能涉及$\mathrm{adp}^{\mathrm{XR}}$的最大值，其中部分输入或输出差分被固定。尽管存在计算该概率的有效方法，但其许多性质仍未知。本文中，我们发现了当旋转为左移/右移一位且其一个输入差分固定时$\mathrm{adp}^{\mathrm{XR}}$的最大值。同时，获得了$\mathrm{adp}^{\mathrm{XR}}$的若干对称性。此外，我们以正则表达式模式提供了其所有不可能差分，并估计了其数量。结果表明，对于左移一位的旋转，该数量达到最大，且明显小于按位异或的不可能差分数量。