Prime Duality: Decoupling Computational Utility and Cryptographic Security

The use of prime numbers for both canonical data encoding and cryptographic security has created a fundamental, long-term conflict in computer science. The migration to post-quantum standards, driven by the threat of quantum algorithms, reveals an implicit but unformalized design principle: selecting for mathematical problems that are hard but lack the broad “constructive utility” of integer factorization. This work provides a formal framework to classify computational hardness assumptions based on their intrinsic algebraic and geometric properties. We argue that a problem’s potential for constructive utility is a direct function of its underlying algebraic structure, such as supporting unique, canonical decompositions in an abelian group. By analyzing the properties of integer factorization versus those of lattice-based problems, we formalize the “hardness without utility” principle, providing a theoretical explanation for the ongoing cryptographic transition. The proposed taxonomy offers a constructive methodology for the design and evaluation of future cryptographic systems, ensuring security is decoupled from general computational utility.