Improved Integral Cryptanalysis of Block Ciphers BORON and Khudra

Integral cryptanalysis is one of the frequently-used cryptanalytic methods of symmetric-key primitives. With the help of division property and the adoption of the automatic tool Mixed Integer Linear Programming (MILP), integral distinguishers can be found more efficiently. This paper uses MILP models to find integral distinguishers based on bit-based division property for block ciphers BORON and Khudra. It is worth noting that we used a combined technique to generate the according inequality set when describing the available division property propagation through the non-linear operation S-box. For one thing, we generate a larger inequality set based on the original set generated by the convex hull computation method. For another, we select a small but sufficient inequality subset from the larger set in the previous step. The numbers of linear constraints that describe the available division property propagation through S-boxes of BORON and Khudra are both reduced from 11 to 7 by our methods. Besides, the best 7-round integral distinguisher for BORON, and the best 9-round integral distinguisher with the smallest data complexity for Khudra are found based on the smaller scale of the whole MILP searching model.