UC, Categorically: Rigorous Diagrammatic Proofs

2026-08-05

AI Summary

This work addresses the lack of a rigorous and intuitive compositional theory for the Universal Composability (UC) security framework in static systems by introducing, for the first time, category-theoretic and string diagram techniques to develop a graphical formal model of UC. This approach not only relaxes the original UC framework’s constraints and corrects certain technical oversights but also extends its applicability to diverse computational models, including quantum computation. The authors establish graphically verifiable composition theorems that demonstrate the equivalence of the new framework to traditional UC while simultaneously offering greater expressiveness and broader applicability.

Abstract

Category theory is a mathematical theory of composition, widely used in logic, computing, and physics. Here we apply it to give a theory of secure composition. In particular, we provide a categorical treatment of Canetti's Universal Composability (UC) framework for systems with a static number of parties and sessions, often termed UC for static systems, yielding four benefits. First, we present our results graphically yet retain rigor by applying a standard categorical technique known as string diagrams. In particular, our formulation of the composition theorem can be graphically verified with a short sequence of diagrams, while remaining translatable to equations and amenable to formal verification. Second, categories let us generalize so that our results extend beyond interactive Turing machines to other forms of computation, such as quantum computation or domain-specific languages. Third, categories help us drop some unnecessary restrictions of UC (e.g., our adversary can be a computational network rather than a single Turing machine); we prove equivalence between our variant and the usual UC, showing no expressiveness is lost. Finally, the categorical perspective leads us to identify and correct some minor technical oversights in the standard formulation of simple UC.

Problem

• Universal Composability
• category theory
• secure composition
• static systems
• string diagrams

Innovation

• Universal Composability
• Category Theory
• String Diagrams
• Secure Composition
• Formal Verification