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Graph Set-colorings And Hypergraphs In Topological Coding

Bing Yao, Fei Ma

TL;DR

This work develops a unified framework combining graph set-colorings, set-labelings, and hypergraphs within topological coding to produce algebraic encodings via Topcode-matrices $T_{code}$. It shows that every connected graph can be realized as an intersected-graph of a hypergraph and introduces a rich toolkit of graph operations, graph-homomorphisms, and graph/network lattices that support post-quantum cryptographic applications, including homomorphic topology encryption. The paper further develops parameterized, set-based colorings and their matrix representations, and expands to hyperedge-sets, pan-operations, and PWCSC/PSCS algorithms to generate scalable, secure encodings suitable for hypernetwork methods. By integrating hypergraphs with graph colorings and Topcode-matrices, the approach provides a versatile mathematical foundation for constructing and analyzing quantum-resistant topological cryptosystems and hypernetwork-based architectures.

Abstract

In order to make more complex number-based strings from topological coding for defending against the intelligent attacks equipped with quantum computing and providing effective protection technology for the age of quantum computing, we will introduce set-colored graphs admitting set-colorings that has been considerable cryptanalytic significance, and especially related with hypergraphs. We use the set-coloring of graphs to reflect the intersection of elements, and add other constraint requirements to express more connections between sets (as hyperedges). Since we try to find some easy and effective techniques based on graph theory for practical application, we use intersected-graphs admitting set-colorings defined on hyperedge sets to observe topological structures of hypergraphs, string-type Topcode-matrix, set-type Topcode-matrix, graph-type Topcode-matrix, hypergraph-type Topcode-matrix, matrix-type Topcode-matrix \emph{etc}. We will show that each connected graph is the intersected-graph of some hypergraph and investigate hypergraph's connectivity, colorings of hypergraphs, hypergraph homomorphism, hypernetworks, scale-free network generator, compound hypergraphs having their intersected-graphs with vertices to be hypergraphs (for high-dimensional extension diagram). Naturally, we get various graphic lattices, such as edge-coincided intersected-graph lattice, vertex-coincided intersected-graph lattice, edge-hamiltonian graphic lattice, hypergraph lattice and intersected-network lattice. Many techniques in this article can be translated into polynomial algorithms, since we are aiming to apply hypergraphs and graph set-colorings to homomorphic encryption and asymmetric cryptograph.

Graph Set-colorings And Hypergraphs In Topological Coding

TL;DR

This work develops a unified framework combining graph set-colorings, set-labelings, and hypergraphs within topological coding to produce algebraic encodings via Topcode-matrices . It shows that every connected graph can be realized as an intersected-graph of a hypergraph and introduces a rich toolkit of graph operations, graph-homomorphisms, and graph/network lattices that support post-quantum cryptographic applications, including homomorphic topology encryption. The paper further develops parameterized, set-based colorings and their matrix representations, and expands to hyperedge-sets, pan-operations, and PWCSC/PSCS algorithms to generate scalable, secure encodings suitable for hypernetwork methods. By integrating hypergraphs with graph colorings and Topcode-matrices, the approach provides a versatile mathematical foundation for constructing and analyzing quantum-resistant topological cryptosystems and hypernetwork-based architectures.

Abstract

In order to make more complex number-based strings from topological coding for defending against the intelligent attacks equipped with quantum computing and providing effective protection technology for the age of quantum computing, we will introduce set-colored graphs admitting set-colorings that has been considerable cryptanalytic significance, and especially related with hypergraphs. We use the set-coloring of graphs to reflect the intersection of elements, and add other constraint requirements to express more connections between sets (as hyperedges). Since we try to find some easy and effective techniques based on graph theory for practical application, we use intersected-graphs admitting set-colorings defined on hyperedge sets to observe topological structures of hypergraphs, string-type Topcode-matrix, set-type Topcode-matrix, graph-type Topcode-matrix, hypergraph-type Topcode-matrix, matrix-type Topcode-matrix \emph{etc}. We will show that each connected graph is the intersected-graph of some hypergraph and investigate hypergraph's connectivity, colorings of hypergraphs, hypergraph homomorphism, hypernetworks, scale-free network generator, compound hypergraphs having their intersected-graphs with vertices to be hypergraphs (for high-dimensional extension diagram). Naturally, we get various graphic lattices, such as edge-coincided intersected-graph lattice, vertex-coincided intersected-graph lattice, edge-hamiltonian graphic lattice, hypergraph lattice and intersected-network lattice. Many techniques in this article can be translated into polynomial algorithms, since we are aiming to apply hypergraphs and graph set-colorings to homomorphic encryption and asymmetric cryptograph.
Paper Structure (96 sections, 90 theorems, 309 equations, 60 figures)

This paper contains 96 sections, 90 theorems, 309 equations, 60 figures.

Table of Contents

  1. Introduction
  2. Researching background
  3. Information security in the era of quantum computers
  4. Hypergraphs in the era of post-quantum encryption
  5. An example of topology code theory
  6. Main topics in this article
  7. Terminology and notation
  8. Labelings and colorings of graphs
  9. Graph operations
  10. Graph operations by adding or removing vertices and edges
  11. Vertex-splitting and vertex-coinciding operations
  12. Edge-coinciding and edge-splitting operations
  13. $\Omega$-coinciding and $\Omega$-splitting operations
  14. Operations on graph homomorphisms
  15. Colorings And Labelings Based On Sets
  16. ...and 81 more sections

Key Result

Theorem 1

$^*$ The number-based strings generated from the Topcode-matrix $T_{code}$ can be classified into two kinds $S_{public}$ and $S_{private}$, such that each number-based string $s\in S_{public}$ corresponds to a unique number-based string $s\,'\in S_{private}$, and they have the same cardinality $|S_{

Figures (60)

  • Figure 1: Four topological signatures made by the graphs admitting set-colorings based on $H_1$ shown in Fig.\ref{['fig:00-example']}(a).
  • Figure 2: Different topological structures.
  • Figure 3: A scheme for the asymmetric topology encryption, where Alice and Bob are a pair of communication users in a network.
  • Figure 4: The vertex-coinciding and the vertex-splitting operations defined in Definition \ref{['defn:vertex-split-coinciding-operations']}.
  • Figure 5: A scheme for illustrating Problem \ref{['question:trees-same-number-leaves']}.
  • ...and 55 more figures

Theorems & Definitions (330)

  • Example 1
  • Remark 1
  • Theorem 1
  • Remark 2
  • Lemma 2
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • ...and 320 more