Integral Cryptanalysis of Reduced-Round SAND-64 Based on Bit-Based Division Property

Conventional Bit-based Division Property (CBDP), as a generalization of integral property, has been a powerful tool for integral cryptanalysis of many block ciphers. Exploiting a Mixed Integral Linear Programming (MILP) optimizer, an alternative approach to searching integral distinguishers was proposed, which has overcome the bottleneck of the cipher block length. The MILP-aided method starts by modeling CBDP propagation by a system of linear inequalities. Then by choosing an appropriate objective function, the problem of searching distinguisher transforms into an MILP problem. As an application of this technique, we focused on a newly proposed lightweight block cipher SAND. SAND is a family of two AND-RX block ciphers SAND-64 and SAND-128, which was designed to overcome the difficulty regarding securityevaluation. For SAND-64, we found a 12-round distinguisher with 23 balanced bits and a data complexity of 263, with the superiority of a higher number of balanced bits than the designers’ one. Furthermore, we applied an integral attack on a 15 and 16-round SAND-64, including the key recovery step which resulted in time complexity of 2105 and 2109.91 and memory complexity of 252 and 285 bytes, respectively.