International Association
for Cryptologic Research

We propose the first adaptation of Matsui's algorithm for finding the best differential and linear trails to the class of ARX ciphers. It is based on a branch-and-bound search strategy, does not use any heuristics and returns optimal results. The practical application of the new algorithm is demonstrated on reduced round variants of block ciphers from the \textsc{Speck} family. More specifically, we report the probabilities of the best differential trails for up to 10, 9, 8, 7, and 7 rounds of \textsc{Speck32}, \textsc{Speck48}, \textsc{Speck64}, \textsc{Speck96} and \textsc{Speck128} respectively, together with the exact number of differential trails that have the best probability. The new results are used to compute bounds, under the Markov assumption, on the security of \textsc{Speck} against single-trail differential cryptanalysis. Finally, we propose two new ARX primitives with provable bounds against single-trail differential and linear cryptanalysis -- a long standing open problem in the area of ARX design.