Provably minimum data complexity integral distinguisher based on conventional division property

Division property is an effective method for finding integral distinguishers for block ciphers, performing cube attacks on stream ciphers, and studying the algebraic degree of boolean functions. One of the main problems in this field is how to provably find the smallest input multiset leading to a balanced output. In this paper, we propose a new method, using the division property, to find integral distinguishers for permutation functions and block ciphers, with provably-minimum data complexity, in the conventional division property model. The new method is based on a precise and efficient analysis of the target output bit’s algebraic normal form. We examine the proposed method on LBlock, TWINE, SIMON, Present, Gift, and Clyde-128 block ciphers. Although in most cases, the results are consistent with the distinguishers reported in previous work, their optimality is proved, in the conventional division property model. Moreover, the proposed method can find distinguishers for 8-round Clyde-128 with less data complexity than previously reported. Based on the proposed method, we also develop an algorithm capable of determining the maximum number of balanced output bits for integral distinguishers with a certain number of active bits. Accordingly, for the ciphers under study, we determine the maximum number of balanced bits for integral distinguishers with data complexities set to minimum and slightly higher, resulting in improved distinguishers for Gift-64, Present, and SIMON64, in the conventional model.