![]() ![]() The new idea is applied to search for optimal differential and linear characteristics for multiple ciphers. With the observations and experience in the tests, a strategy on how to create the sets of bounding conditions that probably achieve extraordinary advances is proposed. Then, we evaluate the accelerating effect of the novel encoding method under different sets of bounding conditions. This approach does not rely on new auxiliary variables and significantly reduces the consumption of clauses for integrating multiple bounding conditions into one SAT problem. Firstly, with the additional encoding variables of the sequential counter circuit for the original objective function in the standard SAT method, we put forward a new encoding method to convert the Matsui’s bounding conditions into Boolean formulas. Compared with the extensive attention on the enhancement for the search with the mixed integer linear programming (MILP) method, few works care for the acceleration of the automatic search with the Boolean satisfiability problem (SAT) or satisfiability modulo theories (SMT) method. However, the performance of the automatic search is not always satisfactory for the search of long trails or ciphers with large state sizes. The introduction of the automatic search boosts the cryptanalysis of symmetric-key primitives to some degree. The result is then fixed by our frameworks. We also perceive that the previous boomerang attacks on LEA are constructed with an incorrect computation of the boomerang connection probability. These are the best distinguishers for them so far. Finally, under these frameworks, we find out the first verifiable 10-round boomerang trail for SPECK32/64 with probability 2−29.15 and a 12-round trail for SPECK48/72 with probability 2−44.15. This is the first time bringing the SAT-aided automatic search techniques into finding boomerang attacks on ARX ciphers. Two automatic search frameworks are also proposed based on these models. ![]() ![]() After rewriting these algorithms with boolean expressions, we construct the corresponding Boolean Satisfiability Problem models. For the boomerang connectivity table, the execution time is 42(n − 1) simple operations while the previous algorithm costs 82(n − 1) simple operations, which generates a smaller model in the searching phase. These algorithms are the most efficient up to now. We provide dynamic programming algorithms to efficiently compute this table and its variants. In this paper, we explore the problem of computing this table for a modular addition and the automatic search of boomerang characteristics for ARX ciphers. In Addition-Rotation-Xor (ARX) ciphers, the large domain size obstructs the application of the boomerang connectivity table. ![]()
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