Applying MILP Method to Searching Integral Distinguishers Based on Division Property for 6 Lightweight Block Ciphers

Division property is a generalized integral property proposed by Todo at EUROCRYPT 2015, and very recently, Todo et al. proposed bit-based division property and applied to SIMON32 at FSE 2016. However, this technique can only be applied to block ciphers with block size no larger than 32 due to its high time and memory complexity. In this paper, we extend Mixed Integer Linear Programming (MILP) method, which is used to search differential characteristics and linear trails of block ciphers, to search integral distinguishers of block ciphers based on division property with block size larger than 32. Firstly, we study how to model division property propagations of three basic operations (copy, bitwise AND, XOR) and an Sbox operation by linear inequalities, based on which we are able to construct a linear inequality system which can accurately describe the division property propagations of a block cipher given an initial division property. Secondly, by choosing an appropriate objective function, we convert a search algorithm under Todo's framework into an MILP problem, and we use this MILP problem appropriately to search integral distinguishers. As an application of our technique, we have searched integral distinguishers for SIMON, SIMECK, PRESENT, RECTANGLE, LBlock and TWINE. Our results show that we can find 14-, 16-, 18-, 22- and 26-round integral distinguishers for SIMON32, 48, 64, 96 and 128 respectively. Moreover, for two SP-network lightweight block ciphers PRESENT and RECTANGLE, we found 9-round integral distinguishers for both ciphers which are two more rounds than the best integral distinguishers in the literature [ 22,29]. For LBlock and TWINE, our results are consistent with the best known ones with respect to the longest distinguishers.