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Cryptographic problems can often be reduced to solving Boolean polynomial systems, whose equivalent logical formulas can be treated using SAT solvers. Given the algebraic nature of the problem, the use of the logical XOR operator is common in SAT-based cryptanalysis. Recent works have focused on advanced techniques for handling parity (XOR) constraints, such as the Gaussian Elimination technique. First, we propose an original XOR-reasoning SAT solver, named WDSat (Weil Descent SAT solving), dedicated to a specific cryptographic problem. Secondly, we show that in some cases Gaussian Elimination on SAT instances does not work as well as Gaussian Elimination on algebraic systems. We demonstrate how this oversight is fixed in our solver, which is adapted to read instances in algebraic normal form (ANF). Finally, we propose a novel preprocessing technique based on the Minimal Vertex Cover Problem in graph theory. This preprocessing technique is, within the framework of multivariate Boolean polynomial systems, used as a DLL branching selection rule that leads to quick linearization of the underlying algebraic system. Our benchmarks use a model obtained from cryptographic instances for which a significant speedup is achieved using the findings in this paper. We further explain how our preprocessing technique can be used as an assessment of the security of a cryptographic system.