A Graph-Based Cryptographic Framework Using Complete Graphs and Lower Triangular Identity Key Matrices

Abstract

In this study, we present a new cryptographic method inspired by the principles of graph theory, offering an innovative way to ensure data security through the use of complete graphs, adjacency matrices, and edge weights. Our approach involves encoding data by mapping it onto weighted complete graphs and performing specific matrix operations on their corresponding adjacency structures. What sets this method apart is the introduction of a lower triangular identity matrix, which serves as a key component in the encryption process. This key design not only strengthens the security of the algorithm but also contributes to its structural simplicity and ease of implementation. The framework we propose takes advantage of the inherent properties of graph theory such as connectivity, symmetry, and weight distribution to provide a mathematically sound and computationally efficient encryption mechanism. Through detailed analysis and a series of practical tests, we demonstrate that the proposed technique is resilient to several well-known cryptographic attacks. Furthermore, its design enables it to be adaptable for various types of digital data, making it a promising tool for modern applications where data protection is critical. By combining theoretical insight with real-world applicability, this paper highlights how classical mathematical concepts can be reimagined to meet the evolving demands of cybersecurity.