A Relation Between Properties of S-box and Linear Inequalities of DDT

Mixed integer linear programming (MILP)-based tools are widely used to analyze the security of block ciphers against differential attacks. The differential properties of an S-box are represented by the difference distribution table (DDT). The methods based on convex hull and logic minimization are used to represent the DDT through linear inequalities. The impossible transitions in DDT are utilized for the minimization of the number of linear inequalities generated using the convex hull approach. The Boolean logic minimization tools Logic Friday and MILES minimize the truth table of DDT to construct and reduce the set of linear inequalities. In this paper, we construct and compare the number of linear inequalities for S-boxes of size 4 bits used in 42 lightweight block ciphers. We analyze the cryptographic properties of S-boxes and observe an inverse relationship between the boomerang uniformity of S-boxes and the number of linear inequalities constructed. We establish that the number of linear inequalities for an S-box decreases as the value of boomerang uniformity increases.