International Association
for Cryptologic Research

We introduce a novel class of equations defined over Euclidean domains. These abstract equations establish a unified framework for deriving new, concrete computational problems useful for cryptography. We prove that solving a single such equation is NP-hard. For systems of these equations, we further prove NP-hardness, average-case hardness, random self-reducibility, search-to-decision reducibility, and trapdoorizability. Based on the hardness of solving these systems, we construct various cryptographic primitives. Our results are proved in an abstract, domain-agnostic manner and hold for a wide range of Euclidean domains. This generality allows the framework to accommodate rich mathematical structures, providing both theoretical depth and flexibility for diverse cryptographic applications.