Representations of Cyclic Diagram Monoids

TL;DR

This paper introduces cyclic diagram monoids, extending classical diagram monoids with internal components for cryptography, classifies their simple representations, and analyzes their exponential growth in resistance to linear cryptographic attacks.

Contribution

It defines cyclic diagram monoids, classifies their simple representations, and explores their potential for cryptographic resistance, a novel extension of classical diagram monoid theory.

Findings

Cyclic diagram monoids have exponentially growing representation gaps.

They possess resistance against linear cryptographic attacks.

Their simple representations are classified and their dimensions computed.

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.