Linearly equivalent S-boxes and the division property

Division property is a cryptanalysis method that proves to be very efficient on block ciphers. Computer-aided techniques such as MILP have been widely and successfully used to study various cryptanalysis techniques, and it especially led to many new results for the division property. Nonetheless, we claim that the previous techniques do not consider the full search space. We show that even if the previous techniques fail to find a distinguisher based on the division property over a given function, we can potentially find a relevant distinguisher over a linearly equivalent function. We show that the representation of the block cipher heavily influences the propagation of the division property, and exploiting this, we give an algorithm to efficiently search for such linear mappings. As a result, we exhibit a new distinguisher over 10 rounds of RECTANGLE , while the previous best was over 9 rounds, and rule out such a distinguisher over more than 9 rounds of PRESENT . We also give some insight about the construction of an S-box to strengthen a block cipher against our technique. We prove that using an S-box satisfying a certain criterion is optimal in term of resistance against classical division property. Accordingly, we exhibit stronger variants of RECTANGLE and PRESENT , improving the resistance against division property based distinguishers by 2 rounds.