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Representations of Cyclic Diagram Monoids

Jason Liu

TL;DR

It is shown 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.

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.

Representations of Cyclic Diagram Monoids

TL;DR

It is shown 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.

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.

Paper Structure

This paper contains 20 sections, 63 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Results
  4. Future directions
  5. Background
  6. Green's relations
  7. Simple representations
  8. Representation gap
  9. Sandwich cellular algebras
  10. Cyclic Temperley--Lieb monoids
  11. Definitions
  12. As platform monoid
  13. Cell structure
  14. Trivial extensions
  15. Dimensions of simple representations
  16. ...and 5 more sections

Key Result

Proposition 1

The idempotent $H$-cell $\mathcal{H}(e)$ is a group with identity $e$.

Theorems & Definitions (149)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Proposition 2
  • Definition 4
  • Proposition 3
  • Definition 5
  • Lemma 1
  • Proposition 4
  • ...and 139 more