This thesis investigates the cryptographic security of Schnorr-based signature schemes, centering on the significant and evolving threat posed by the Random Inhomogeneities in a Overdetermined Solvable system of linear equations (ROS) attack. While the linearity of Schnorr signatures is fundamental to the scalability and privacy enhancements introduced in Bitcoin's Taproot upgrade, this same property creates an intrinsic vulnerability that can be exploited to compromise protocol security. The core of the research is a detailed examination of the ROS attack, tracing its cryptanalytic origins and analyzing its modern variants, including recently developed polynomial-time attacks that challenge the foundational assumptions of numerous cryptographic constructions.