Structural Collapse and the Complexity Gap in Semiring-Based Cryptography

The distinctive algebraic properties of tropical semirings and their connections to computationally difficult problems have made tropical cryptography a subject of growing interest. However, a growing body of cryptanalytic results has shown that many such schemes are vulnerable to efficient attacks. In this paper, we develop a unified mathematical framework for analyzing this phenomenon through the notion of structural collapse. Structural collapse occurs when the reconstruction problem on protocol-generated instances reduces to a problem that is polynomial-time solvable on the resulting instances and whose solutions yield a valid reconstruction of the cryptographic target. We examine recurring structural reduction mechanisms, including linearization, periodicity and monotonicity, optimization and inequality-based reformulations, and residual separability, and distinguish structural reduction from structural collapse. We further formalize the complexity gap between established worst-case hardness of unrestricted algebraic problems and polynomial-time reconstruction on structured protocol-generated families. We show that structural vulnerabilities identified in a broad class of tropical and semiring-based cryptographic protocols are not caused by insufficient algebraic richness, but by structural properties that permit reductions to computationally simpler problems. The framework provides a basis for analyzing the computational structure of semiring-based cryptographic protocols and for identifying structural properties relevant to future protocol design.