Parametric Iteration in Resource Theories

TL;DR

This paper introduces a parametric iteration framework within resource theories, enabling abstract reasoning about asymptotic negligibility in cryptography using diagrammatic methods.

Contribution

It develops a general parametric iteration construction in resource theories and applies it to probabilistic circuits to formalize negligibility.

Findings

Captured negligibility via asymptotic equivalence in a compositional manner

Proved cryptographic theorems using diagrammatic reasoning

Applied the framework to probabilistic Boolean circuits

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.