Block encryption and decryption of a sentence using decomposition of Generalized Peterson graph

Abstract

In the modern digital era data breaches and cyberattacks threaten the integrity of global communications more frequently. Cryptography stands as the frontline defense, safeguarding sensitive information against unauthorized access. Encryption and decryption are two cryptographic techniques that conceal and transmit critical information to authorized individuals without the interference of a third party in a network. Cryptography relies on mathematical concepts such as number theory, graph theory, group theory, probability theory, statistics and algebra. Graph theory enhances cryptographic security by utilizing the complexity of graphs, which pose significant challenges to potential attackers. There are mainly two types of cryptographic algorithms as private key cryptosystem and public key cryptosystem. This research introduces a novel cryptosystem utilizing the Generalized Peterson graph GP (7,2). After analyzing various generalized Peterson graphs GP(7,2) was selected for its properties in graph decomposition. We use graph decomposition theories to develop the cryptosystem. The main advantage of choosing GP (7,2) is that, by decomposing GP (7,2), 𝐶7, 𝑃4, 2 copies of 𝑃2and 3 copies of 𝑆3 can be generated. An encryption and decryption scheme utilizing GP (7,2) is proposed, incorporating these subgraphs to enhance cryptographic operations. Security is further enhanced by employing 10 lists of arithmetic progression as coding tables. Additionally, the binary digit labeling for path 𝑃4 and an encoding table for frequently used symbols are introduced to increase the robustness of the cryptosystem. An algorithm is developed and tested to validate the proposed methodology, confirming its effectiveness and reliability. Results indicate that the proposed methodology effectively encrypts, and decrypts sentences of variable lengths, showcasing its flexibility compared to other graph based cryptosystems in the current literature. In conclusion, goal of this research was to propose a new private key block crypto system which is based on graphs and graph theory concepts and which could potentially address the limitations in the current literature such as complexity of graph decomposition becoming exponentially difficult with the number of words in the sentence getting higher. And this proposed methodology can be further improved as well. An open area for future work involves expanding the symbol set to include more characters, such as upper and lower case letters, special characters and numbers. This enhancement would accommodate a wider range of inputs in real world scenarios. The decomposition of a Generalized Peterson graph has been chosen, and future work will explore different types of graph techniques to make the cryptosystem stronger and more difficult to hack.