Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Representation Gap of the Motzkin Monoid

Katharina Arms

TL;DR

The paper addresses the security risk posed by linear decomposition attacks in non-classical algebraic structures by analyzing the Motzkin monoid via cellular methods. It introduces and leverages Green's cell theory and Gram-matrix techniques to bound the dimensions of simple and semisimple representations, proving an exponential growth of the representation gap with base $3$. By truncating to submonoids based on the number of through strands and examining idempotent diagrams and faithfulness, it shows that both semisimple and simple gaps scale as $ heta(n^{-3/2} 3^n)$ while maintaining favorable gap ratios. These results position the Motzkin monoid as a promising candidate for cryptographic constructions resilient to linear-decomposition attacks, linking diagrammatic topology with finite monoid representation theory and potentially impacting quantum-topological contexts.

Abstract

The linear decomposition attack reveals a vulnerability in encryption algorithms operating within groups or monoids with excessively small representations. The representation gap, defined as the size of the smallest non-trivial representation, therefore serves as a metric to assess the security of these algorithms. This paper will demonstrate that the diagrammatic Motzkin monoids exhibit a large representation gap, positioning them as promising candidates for robust encryption algorithms.

Representation Gap of the Motzkin Monoid

TL;DR

The paper addresses the security risk posed by linear decomposition attacks in non-classical algebraic structures by analyzing the Motzkin monoid via cellular methods. It introduces and leverages Green's cell theory and Gram-matrix techniques to bound the dimensions of simple and semisimple representations, proving an exponential growth of the representation gap with base . By truncating to submonoids based on the number of through strands and examining idempotent diagrams and faithfulness, it shows that both semisimple and simple gaps scale as while maintaining favorable gap ratios. These results position the Motzkin monoid as a promising candidate for cryptographic constructions resilient to linear-decomposition attacks, linking diagrammatic topology with finite monoid representation theory and potentially impacting quantum-topological contexts.

Abstract

The linear decomposition attack reveals a vulnerability in encryption algorithms operating within groups or monoids with excessively small representations. The representation gap, defined as the size of the smallest non-trivial representation, therefore serves as a metric to assess the security of these algorithms. This paper will demonstrate that the diagrammatic Motzkin monoids exhibit a large representation gap, positioning them as promising candidates for robust encryption algorithms.

Paper Structure

This paper contains 24 sections, 31 theorems, 78 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Cell Theory
  4. Classification of Simple Representations
  5. Cells and Semisimple Representation Gaps
  6. Cells and Gram Matrices
  7. Cell Submonoids
  8. The Motzkin Monoid and Its Properties
  9. Defining the Motzkin Monoid
  10. Representations Using Cell Theory
  11. Sub-quotients: Truncating the Motzkin Monoid.
  12. Trivial Extensions
  13. Representation Gap of the Motzkin Monoid
  14. Semisimple Representation Gap
  15. Representation Gap
  16. ...and 9 more sections

Key Result

Theorem 1

For a well-connected monoid $\mathcal{S}$ with $H^1(\mathcal{G}, \mathbb{K}) \cong 0$ the representation gap as defined above matches KST-monoidal-crypto.

Figures (5)

  • Figure 1: Number of diagrams existing for $k \in\{0,\text{...},20\}$ over $n=20$ nodes. (Thus illustrating the semisimple dimensions.) The second plot uses a logarithmic scale on the y-axis.
  • Figure 2: Value within summation of $\operatorname{ssdim}(\mathcal{M}o_{600})$ for varying values of $t \in [0,500]$. The second plot uses a logarithmic scale on the y-axis.
  • Figure 3: Value of $t$ which yielded the highest output, and the line $t=\frac{n}{3}$. The second plot uses a logarithmic scale on the y-axis.
  • Figure 4: Growth of $n^{th}$ root of our lower bound for $\operatorname{gap}_{\mathcal{S}}(\mathcal{M}o_n)$.
  • Figure 5: Semisimple and simple dimensions of the Motzkin monoid over $n$ and $k$.

Theorems & Definitions (74)

  • Theorem 1
  • proof
  • Definition 1
  • Definition 2
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 3
  • ...and 64 more