A mining-based approach for efficient enumeration of algebraic structures

Algebraic structures are well-studied mathematical structures in abstract algebra with applications in many fields of computer security such as cryptography and authentication. Generating such structures is computationally very expensive because of the huge number of permutations. Also, many of these permutations are redundant as they are symmetrically equivalent. The symmetry breaking (finding symmetrically equivalent structures) is also a computationally challenging task. In this paper, we present a mining-based approach for symmetry breaking in algebraic structures. The approach reduces the number of redundant structures by identifying rules based on recurring patterns in the previously known structures. These rules are then used as constraints in a constraint solver. The proposed approach is applied to IP loop, a special class of algebraic structures, and the deduced rules have eliminated a large number of redundant solutions resulting in significant time improvement.