Towards Finding the Best Characteristics of Some Bit-oriented Block Ciphers and Automatic Enumeration of (Related-key) Differential and Linear Characteristics with Predefined Properties

In this paper, we investigate the Mixed-integer Linear Programming (MILP) modelling of the differential and linear behavior of a wide range of block ciphers. We point out that the differential behavior of an arbitrary S-box can be exactly described by a small system of linear inequalities. Based on this observation and MILP technique, we propose an automatic method for finding high probability (related-key) differential or linear characteristics of block ciphers. Compared with Sun et al.’s heuristic method presented in Asiacrypt 2014, the new method is exact for most ciphers in the sense that every feasible 0-1 solution of the MILP model generated by the new method corresponds to a valid characteristic, and therefore there is no need to repeatedly add valid cutting-off inequalities into the MILP model as is done in Sun et al.’s method; the new method is more powerful which allows us to get the exact lower bounds of the number of differentially or linearly active S-boxes; and the new method is more efficient which allows to obtain characteristic with higher probability or covering more rounds of a cipher (sometimes with less computational effort). Further, by encoding the probability information of the differentials of an S-boxes into its differential patterns, we present a novel MILP modelling technique which can be used to search for the characteristics with the maximal probability, rather than the characteristics with the smallest number of active S-boxes. With this technique, we are able to get tighter security bounds and find better characteristics. Moreover, by employing a type of specially constructed linear inequalities which can remove exactly one feasible 0-1 solution from the feasible region of an MILP problem, we propose a method for automatic enumeration of all (related-key) differential or linear characteristics with some predefined properties, e.g., characteristics with given input or/and output difference/mask, or with a limited number of active S-boxes. Such a method is very useful in the automatic (related-key) differential analysis, truncated (related-key) differential analysis, linear hull analysis, and the automatic construction of (related-key) boomerang/rectangle distinguishers. The methods presented in this paper are very simple and straightforward, based on which we implement a Python framework for automatic cryptanalysis, and extensive experiments are performed using this framework. To demonstrate the usefulness of these methods, we apply them to SIMON, PRESENT, Serpent, LBlock, DESL, and we obtain some improved cryptanalytic results.