Automatic MILP-based Impossible Differential Security Cryptanalysis of RAIN-64

Impossible differential cryptanalysis is an efficient cryptanalysis method in the field of block ciphers, and Mixed Integer Linear Programming (MILP) is widely used in the automatic search for impossible differentials (IDs) characteristics. In this paper, we study the two-dimensional binary impossible differential model which proposed by Liu et al. Based on this model, we add a new state difference to obtain an improved version. The new model can provide a more detailed process of the changes in state differences in the impossible differential trail. At the same time, in terms of applicability, our model is suitable for AES-like ciphers with SPN structures in block ciphers. RAIN is a lightweight block cipher with a SP-network structure that ensures security while also considering implementation at both the software and hardware levels. Compared to the previous model's search for impossible differential characteristics of RAIN-64, our improved model identified one additional round of impossible differential characteristics. In the new model, we determined the 6-round and 7-round impossible differential characteristics of RAIN-64, finding a total of 23,468 characteristics for 6 rounds and 152 characteristics for 7 rounds. Additionally, utilizing the key schedule and key-bridging techniques of RAIN-64, We proposed a 10-round impossible differential attack (IDA) on RAIN-64, where the time complexity and data complexity of the attack are 2<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">124.07</sup> and 2<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">52</sup>. Therefore, we have concluded that RAIN-64 is secure under the 10-round impossible differential attack. This is currently the longest rounds of distinguisher and IDA for RAIN-64.