Design a secure multi-party computation protocol enabling privacy-preserving linear regression on vertically partitioned datasets. Specify an explicit threat model (e.g., honest-but-curious or malicious, collusion limits), choose appropriate cryptographic primitives and system architecture, provide formal correctness and privacy proofs (simulation- or game-based), propose concrete efficiency optimizations (computation, communication, and storage), and outline an empirical evaluation plan (datasets, metrics, baselines, and scalability experiments).