Symbolic Plithogenic Numbers in RSA Cryptography: A Path to Post-Quantum Security

Abstract

The RSA algorithm, with its strong mathematical foundation rooted in number theory and its wide range of applications in secure communication, is one of the most widely used public-key encryption methods. However, advancements in classical computing and the emergence of quantum computers have introduced significant risks that threaten the security of traditional RSA-based systems. Therefore, new mathematical structures, referred to as extended number systems, are needed to increase algebraic complexity and provide more secure encryption. In this context, alternative number systems such as neutrosophic numbers and plithogenic numbers, which possess multidimensional and partially ordered structures, show promising potential for offering stronger cryptographic solutions against both classical and quantum-based attacks. This study presents a reconstruction of the RSA algorithm using symbolic 2-plithogenic and 3-plithogenic numbers, which represent an extended version of the classical and neutrosophic number systems. These numbers differ from traditional integers by incorporating two symbolic components, thereby forming a multidimensional structure. Consequently, the algebraic structure obtained in the processes of encryption and key generation becomes significantly more complex compared to classical methods. In particular, the increased difficulty of integer factorization in such multidimensional structures substantially strengthens cryptographic security.