The guaranteed number of activations (GNA) is an important characteristic to determine the effectiveness of differential cryptanalysis of a given \(\textsf{XS}\) -circuit. In this paper, we propose an approach to optimize the known algorithm for GNA computation based on the branch and bound method. We also analyze special matrices that define \(\textsf{XS}\) -circuit. The experiments show that the proposed algorithm significantly outperforms the existing approach. In this paper, we prove that canonical forms of \(\textsf{XS}\) -circuit and its dual coincide, providing the strict connection between the guaranteed number of linear and differential activations. The circuits with the extremal values of GNA are studied. We made several hypotheses based on computational experiments. One of the hypotheses is that there are no \(\textsf{XS}\) -circuits of dimension greater than 2, which achieve an optimal GNA on every round.
