An Unrestricted Notion of the Finite Factorization Property

Overview

Our modern digital security rests on the difficulty and structure of prime factorization—a principle that has powered number theory for centuries. To understand the strengths and weaknesses of factorization-based cryptography, mathematicians can study how such factorizations behave in general algebraic systems through finiteness conditions: properties that guarantee controlled factorizations. Despite decades of study, however, the precise boundaries between key finiteness conditions remains unresolved, and a more natural, fundamental version of a core property has never been proposed. In this work, I introduce the unrestricted finite factorization (U–FF) property, a condition more intuitive than its classical predecessors, extending the work on unrestricted UFDs carried out by Coykendall and Zafrullah in 2004. Through novel and technically demanding constructions, I rigorously establish the independence of longstanding finiteness conditions and precisely situate the U–FF property within this landscape, resolving open problems and strengthening classical results dating back to Anderson, Anderson, and Zafrullah's seminal 1990 paper. Most significantly, I tackle a generalization of a famously difficult open problem first posed by Anderson et al.—one which took mathematicians over thirty years to fully resolve. By discovering analogous theory for the U–FF property, my work introduces new ways to rapidly compound the complexity of a number system while preserving controlled factorization—progress that could lead to greatly enhanced security in cryptographic systems. Together, these results fill longstanding gaps in the literature and equip mathematicians with powerful new tools for understanding the algebraic structures underlying modern number theory and cryptography.

Competition history

• CSEF 2026 Mathematical Sciences (Senior Division) · Entry S-14-09

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