Geodesic
On the number of impossible differentials of some ARX transformation
Prikladnaya Diskretnaya Matematika. Supplement, no. 16 (2023), pp. 47-50

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The additive differential probabilities of the function $(x \oplus y) \lll r$ are considered, where $x, y \in \mathbb{Z}_2^{n}$ and $1 \leq r n$. They are interesting in the context of differential cryptanalysis of ciphers whose schemes consist of additions modulo $2^n$, bitwise XORs ($\oplus$) and bit rotations ($\lll r$). We calculate the number of all impossible differentials, i.e. differentials with probability $0$, for all possible $r$ and $n$. The limit of the ratio of this number to the number of all differentials as $r$ and $n-r$ tend to $\infty$ equals $38/245$. We also compare the given numbers and the number of impossible differentials for the function $x \oplus y$.
Mots-clés : ARX, XOR, bit rotations
Keywords: differential probabilities, modular addition, impossible differentials. 
@article{PDMA_2023_16_a11,
     author = {N. A. Kolomeec},
     title = {On the number of impossible differentials of some {ARX} transformation},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {47--50},
     publisher = {mathdoc},
     number = {16},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2023_16_a11/}
}
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N. A. Kolomeec. On the number of impossible differentials of some ARX transformation. Prikladnaya Diskretnaya Matematika. Supplement, no. 16 (2023), pp. 47-50. http://geodesic.mathdoc.fr/item/PDMA_2023_16_a11/