Paper 2018/1186

MILP Method of Searching Integral Distinguishers Based on Division Property Using Three Subsets

Senpeng Wang, Bin Hu, Jie Guan, Kai Zhang, and Tairong Shi

Abstract

Division property is a generalized integral property proposed by Todo at EUROCRYPT 2015, and then conventional bit-based division property (CBDP) and bit-based division property using three subsets (BDPT) were proposed by Todo and Morii at FSE 2016. The huge time and memory complexity that once restricted the applications of CBDP have been solved by Xiang et al. at ASIACRYPT 2016. They extended Mixed Integer Linear Programming (MILP) method to search integral distinguishers based on CBDP. BDPT can find more accurate integral distinguishers than CBDP, but it can not be modeled efficiently. Thus it cannot be applied to block ciphers with block size larger than 32 bits. In this paper, we focus on the feasibility of applying MILP-aided method to search integral distinguishers based on BDPT. We firstly study how to get the BDPT propagation rules of an S-box. Based on that we can efficiently describe the BDPT propagation of cipher which has S-box. Moreover, we propose a technique called ``fast propagation", which can translate BDPT into CBDP, then the balanced bits based on BDPT can be presented. Together with the propagation properties of BDPT, we can use MILP method based on CBDP to search integral distinguishers based on BDPT. In order to prove the efficiency of our method, we search integral distinguishers on SIMON, SIMECK, PRESENT, RECTANGLE, LBlock, and TWINE. For SIMON64, PRESENT, and RECTANGLE, we find more balanced bits than the previous longest distinguishers. For LBlock, we find a 17-round integral distinguisher which is one more round than the previous longest integral distinguisher, and a better 16-round integral distinguisher with less active bits can be obtain. For other ciphers, our results are in accordance with the previous longest distinguishers.

Note: Some words have been revised.