Graph Set-colorings And Hypergraphs In Topological Coding

TL;DR

This paper explores set-colorings of graphs and hypergraphs to enhance topological coding for quantum-resistant cryptography, introducing new graph-theoretic techniques and structures for secure communication.

Contribution

It introduces novel set-coloring methods for hypergraphs and their applications in topological coding, with potential for cryptographic algorithms resistant to quantum attacks.

Findings

Hypergraph set-colorings reflect complex set intersections.

Connected graphs can be represented as intersected-graphs of hypergraphs.

Various graph lattices and hypernetwork structures are constructed.

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.