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Cryptanalysis, i.e., the study of methods for breaking cryptographic algorithms, can greatly benefit from hardware acceleration as a key aspect enabling high-performance attacks. This work investigates the new opportunities inherently provided by a particular class of hardware technologies, i.e., reconfigurable hardware devices, addressing the cryptanalysis of the SHA-1 hash function as a case study. We show how hardware reconfiguration enables some unexplored approaches such as algorithm and architecture exploration, as well as on-the-fly system specialization relying on hardware programmability. We also identify some new cryptanalysis methods, including two novel techniques for SHA-1 cryptanalysis called interbit constraints and constraint relaxation. Relying on the proposed approaches, we designed an FPGA-based platform targeting 71- and 75-round versions of SHA-1. Under the same cost budget, the estimated times for a collision achieved by the platform are at least one order of magnitude lower than other solutions based on high-end supercomputing facilities, reaching the highest performance/cost ratio for SHA-1 collision search and providing a striking confirmation of the impact of hardware reconfigurability.