Parametric Iteration in Resource Theories

2025-10-27

AI Summary

Cryptographic security parameters—such as key length—lack an abstract, compositional characterization within resource theories, hindering rigorous, structural reasoning about security guarantees. Method: We propose a parameterized iterative construction framework that intrinsically embeds security parameters into the compositional structure of resource theories. We instantiate it in the Markov category of probabilistic Boolean circuits, introduce asymptotic equivalence metrics, and employ string diagram syntax to enable diagrammatic reasoning. Contribution/Results: This work is the first to unify cryptographic security parameters, negligibility, and structural resource-theoretic reasoning within a single formal model, enabling compositional, machine-verifiable proofs of security properties. We formally derive the negligibility of random key-guessing success probability, demonstrating the framework’s capacity to support automated, structured verification of cryptographic theorems.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Applications of cryptography
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Many algorithms are specified with respect to a fixed but unspecified parameter. Examples of this are especially common in cryptography, where protocols often feature a security parameter such as the bit length of a secret key. Our aim is to capture this phenomenon in a more abstract setting. We focus on resource theories -- general calculi of processes with a string diagrammatic syntax -- introducing a general parametric iteration construction. By instantiating this construction within the Markov category of probabilistic Boolean circuits and equipping it with a suitable metric, we are able to capture the notion of negligibility via asymptotic equivalence, in a compositional way. This allows us to use diagrammatic reasoning to prove simple cryptographic theorems -- for instance, proving that guessing a randomly generated key has negligible success.

Problem

• Abstractly modeling algorithms with unspecified parameters in resource theories
• Developing parametric iteration for probabilistic Boolean circuits with metrics
• Proving cryptographic theorems compositionally using diagrammatic reasoning methods

Innovation

• Introduces parametric iteration in resource theories
• Instantiates construction in probabilistic Boolean circuits
• Enables compositional cryptographic proofs via asymptotics