Data Security: A New Symmetric Cryptosystem based on Graph Theory

Abstract

Sharing private data in an unsecured channel is extremely critical, as unauthorized entities can intercept it and could break its privacy. The design of a cryptosystem that fulfills the security requirements in terms of confidentiality, integrity and authenticity of transmitted data has therefore become an unavoidable imperative. Indeed, a lot of work has been carried out in this regard. Although many cryptosystems have been proposed in the published literature, it has been found that their robustness and performance vary relatively from one to another. Adopting this reflection, we address in this paper the concept of block cipher, which is a major cryptographic solution to guarantee confidentiality, by involving the properties of graph theory to represent the plaintext message. Our proposal is in fact a new symmetric encryption block cipher that proceeds by representing plaintext messages using disjoint Hamiltonian circuits and then dealing with them as an adjacency matrix in a pre-encryption phase. The proposed system relies on a particular sub-key generator that has been carefully designed to produce the encryption keys according to the specifications of the system. The obtained experimental results demonstrate that our proposed cryptosystem is robust against statistical attacks, particularly the DIEHARD test, and presents both good confusion and good diffusion.

A Hamilton Path-Based Approach to DNA Sequence Analysis

“…Simply put, think of a graph as a collection of nodes or vertices, and connect these vertices with edges. A Hamilton path is a route that starts at one vertex, visits all other vertices exactly once, and ends at another vertex [1]. It essentially loops through the entire chart without repeating.…”

“…In a complete graph with n vertices there are edge disjoint Hamiltonian paths if n is an even number greater than 4 [1]. Proof: we need to demonstrate two key concepts: (1) Hamiltonian paths exist in complete graphs, and (2) there can be at most edge disjoint Hamiltonian paths in such a graph.Hamiltonian Paths in complete graphs: A Hamiltonian path is a path that visits every vertex of a graph exactly once[7][8].…”

“…Proof: we need to demonstrate two key concepts: (1) Hamiltonian paths exist in complete graphs, and (2) there can be at most edge disjoint Hamiltonian paths in such a graph.Hamiltonian Paths in complete graphs: A Hamiltonian path is a path that visits every vertex of a graph exactly once[7][8]. In a complete graph, every vertex is connected to every other vertex by an edge[1]. Therefore, it is possible to construct a Hamiltonian path that covers all n vertices by visiting each vertex exactly once.Number of edge disjoints Hamiltonian paths:In an even complete graph with n vertices, every vertex has n-1 degree since it is connected to every other vertex.…”