Geodesic
$\mathsf{XS}$-circuits' properties related to the guaranteed number of activations
Prikladnaya Diskretnaya Matematika. Supplement, no. 15 (2022), pp. 62-66

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The guaranteed number of activations (GNA) is an important characteristic that determines the efficiency of differential cryptanalysis of a given $\mathsf{XS}$-circuit. In the paper, we propose an approach to optimize the known GNA calculation algorithm based on the branch and bound method and the analysis of special matrices that define the $\mathsf{XS}$-circuit. Now, it is possible to compute GNA for more than 30 rounds, which would take significantly longer if the original algorithm were used. The optimized algorithm was used for exhaustive enumeration of low-dimensional $\mathsf{XS}$-schemes. We prove that the canonical forms of the $\mathsf{XS}$-circuit and its dual coincide, which provides a strict connection between the guaranteed number of linear and differential activations. Based on computational experiments, several hypotheses have been proposed. One of the hypotheses is that there are no $\mathsf{XS}$-circuits of dimension greater than two that achieve an optimal GNA in every round.
Keywords: guaranteed number of activations, $\mathsf{XS}$-circuit, differential cryptanalysis, linear cryptanalysis, branch and bound method. 
@article{PDMA_2022_15_a15,
     author = {D. R. Parfenov and A. O. Bakharev and A. V. Kutsenko and A. R. Belov and N. D. Atutova},
     title = {$\mathsf{XS}$-circuits' properties related to the guaranteed number of activations},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {62--66},
     publisher = {mathdoc},
     number = {15},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2022_15_a15/}
}
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D. R. Parfenov; A. O. Bakharev; A. V. Kutsenko; A. R. Belov; N. D. Atutova. $\mathsf{XS}$-circuits' properties related to the guaranteed number of activations. Prikladnaya Diskretnaya Matematika. Supplement, no. 15 (2022), pp. 62-66. http://geodesic.mathdoc.fr/item/PDMA_2022_15_a15/