This thesis studies automated methods for analyzing hardness assumptions in generic group models, following ideas of symbolic cryptography. We define a broad class of generic and symbolic group models for different settings---symmetric or asymmetric (leveled) k-linear groups - and prove ''computational soundness'' theorems for the symbolic models. Based on this result, we formulate a master theorem that relates the hardness of an assumption to solving problems in polynomial algebra. We systematically analyze these problems identifying different classes of assumptions and obtain decidability and undecidability results. Then, we develop automated procedures for verifying the conditions of our master theorems, and thus the validity of hardness assumptions in generic group models. The concrete outcome is an automated tool, the Generic Group Analyzer, which takes as input the statement of an assumption, and outputs either a proof of its generic hardness or shows an algebraic attack against the assumption. Structure-preserving signatures are signature schemes defined over bilinear groups in which messages, public keys and signatures are group elements, and the verification algorithm consists of evaluating ''pairing-product equations''. Recent work on structure-preserving signatures studies optimality of these schemes in terms of the number of group elements needed in the verification key and the signature, and the number of pairing-product equations in the verification algorithm. While the size of keys and signatures is crucial for many applications, another aspect of performance is the time it takes to verify a signature. The most expensive operation during verification is the computation of pairings. However, the concrete number of pairings is not captured by the number of pairing-product equations considered in earlier work. We consider the question of what is the minimal number of pairing computations needed to verify structure-preserving signatures. We build an automated tool to search for structure-preserving signatures matching a template. Through exhaustive search we conjecture lower bounds for the number of pairings required in the Type~II setting and prove our conjecture to be true. Finally, our tool exhibits examples of structure-preserving signatures matching the lower bounds, which proves tightness of our bounds, as well as improves on previously known structure-preserving signature schemes.