Geodesic
On properties of additive differential probabilities of XOR
Prikladnaya Diskretnaya Matematika. Supplement, no. 14 (2021), pp. 46-48

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The additive differential probability of exclusive-or $\mathrm{adp}^{\oplus}(\alpha, \beta, \gamma)$, where $\alpha, \beta, \gamma \in \mathbb{Z}_{2}^{n}$, is studied. It is used in the analysis of symmetric-key primitives that combine XOR and modular addition, such as Addition-Rotation-XOR (ARX) constructions. We focus on the maximal differentials which are helpful when constructing differential trails. It is proven that $\max_{\alpha, \beta} \mathrm{adp}^{\oplus}(\alpha,\beta,\gamma) = \mathrm{adp}^{\oplus}(0,\gamma,\gamma)$. In addition, there exist either $2$ or $8$ distinct pairs ($\alpha$, $\beta$) such that $\mathrm{adp}^{\oplus}(\alpha,\beta,\gamma) = \mathrm{adp}^{\oplus}(0,\gamma,\gamma)$. Also, we obtain a simplified representation of $\mathrm{adp}^{\oplus}(0,\gamma,\gamma)$ and formula for $\min_{\gamma}\mathrm{adp}^{\oplus}(0,\gamma,\gamma)$.
Mots-clés : ARX, XOR
Keywords: modular addition, differential cryptanalysis. 
@article{PDMA_2021_14_a7,
     author = {N. Mouha and N. A. Kolomeets and D. A. Ahtyamov and I. A. Sutormin and M. A. Panferov and K. M. Titova and T. A. Bonich and E. A. Ishchukova and N. N. Tokareva and B. F. Zhantulikov},
     title = {On properties of additive differential probabilities of {XOR}},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {46--48},
     publisher = {mathdoc},
     number = {14},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2021_14_a7/}
}
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N. Mouha; N. A. Kolomeets; D. A. Ahtyamov; I. A. Sutormin; M. A. Panferov; K. M. Titova; T. A. Bonich; E. A. Ishchukova; N. N. Tokareva; B. F. Zhantulikov. On properties of additive differential probabilities of XOR. Prikladnaya Diskretnaya Matematika. Supplement, no. 14 (2021), pp. 46-48. http://geodesic.mathdoc.fr/item/PDMA_2021_14_a7/