On additive differentials that go through ARX transfromation with high probability

In the paper, we consider additive differential probabilities of the function $(x \oplus y) \lll r$, where $x, y \in \mathbb{Z}_2^n$ and $1 \leq r < n$. They are interesting in the context of differential cryptanalysis of ciphers that use addition modulo $2^n$, bitwise XOR ($\oplus$) and bit rotations ($\lll r$) as basic operations. All differentials up to argument symmetries whose probability exceeds $1/4$ are obtained. The possible values of their probabilities are $1/3 + 4^{2 - i} / 6$ for $i \in \{1, \dots, n\}$, which coincide with the differentials probabilities of the function $x \oplus y$. We describe differentials with each of these probabilities and calculate the number of them. It is proven that the number of all considered differentials is equal to $48n - 68$ for $n \geq 2$.