XS-cir-cuits' properties related to the guaranteed number of activations

The guaranteed number of activations (GNA) is an important characteristic that determines the efficiency of differential cryptanalysis of a given 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 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 XS-schemes. We prove that the canonical forms of the 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 XS-circuits of dimension greater than two that achieve an optimal GNA in every round.

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