Automatic Search for Related-Key Differential Trails in SIMON-like Block Ciphers Based on MILP

Abstract

In this paper, we revisit the relationship between the probability of differential trails and the input difference of each round for SIMON-like block ciphers. The key observation is that not only the Hamming weight but also the positions of active bits of the input difference have effect on the probability. Based on this, our contributions are mainly twofold. Firstly, we rebuild the MILP model for SIMON-like block ciphers without quadratic constraints. Accordingly, we give the accurate objective function and reduce its degree to one by adding auxiliary variants to make the model easy to solve. Secondly, we search for optimal differential trails for SIMON and SIMECK based on this model. To the best of our knowledge, this is the first time that related-key differential trails have been obtained. Besides, we not only recover the single-key results in [11], but also obtain impossible differentials through this method.