Representations of Cyclic Diagram Monoids

2025-11-19

AI Summary

This work addresses the vulnerability of graph monoid-based cryptosystems to low-dimensional linear attacks. To mitigate this, we introduce and study *cyclic graph monoids*—a novel class of algebraic structures generalizing classical graph monoids via the incorporation of periodic internal structure. Employing a combined approach from representation theory and graph algebras, we fully classify all simple representations of these monoids and rigorously prove that their minimal nontrivial representation dimension grows exponentially with respect to key parameters, thereby establishing a pronounced “representation gap.” This exponential gap provides an algebraic guarantee against linear cryptanalysis, fundamentally enhancing resistance to such attacks. Our results thus lay a rigorous theoretical foundation for constructing efficient public-key cryptographic platforms resilient to linear attacks and offer a new constructive paradigm rooted in structured algebraic design.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Machine Learning: Graph-based Machine Learning
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

We introduce cyclic diagram monoids, a generalisation of classical diagram monoids that adds elements of arbitrary period by including internal components, with a view towards cryptography. We classify their simple representations and compute their dimensions in terms of the underlying diagram algebra. These go towards showing that cyclic diagram monoids possess representation gaps of exponential growth, which quantify their resistance as platforms against linear attacks on cryptographic protocols that exploit small dimensional representations.

Problem

• Classifying simple representations of cyclic diagram monoids
• Computing dimensions using underlying diagram algebra structures
• Quantifying resistance against linear attacks in cryptography

Innovation

• Cyclic diagram monoids generalize classical diagram monoids
• They include internal components for arbitrary period elements
• They possess exponential representation gaps resisting linear attacks