A ring-lwe-based hybrid signature approach enhanced with pell-lucas sequences and vernam encryption

In this thesis, the design, security, performance, and potential application domains of advanced digital signature algorithms grounded in number theory are examined comprehensively. Digital signatures are underscored as critical for ensuring data integrity, authentication, and non-repudiation, while the prospective inadequacy of classical schemes against future threats posed by quantum computers is discussed. Accordingly, the development of a novel digital signature algorithm that addresses the security and performance shortcomings of existing schemes while incorporating post-quantum security is targeted. In addition, the necessity of designing effective algorithms for NP-class problems—such as the traveling salesman and knapsack problems—and conducting in depth performance analyses is articulated; the rapid advances in information technologies over the last decade are noted to necessitate the systematic classification of methods in finance, e-commerce, and information security. The security of the proposed algorithms is validated through mathematical analyses and theoretical tests, and significant improvements over classical approaches are demonstrated via measurements of runtime, verification procedures, and resource utilization. Within the scope of the study, widely used symmetric and asymmetric cryptographic algorithms are compared in terms of time, computational, and memory complexity, and the resulting findings are employed to develop strategies that enhance application effectiveness. In conclusion, the selection of digital signature algorithms is emphasized as requiring careful alignment with stated objectives and security requirements; through the integration of information security and algorithm design, innovative solutions are provided that meet contemporary needs and remain resilient to both classical and quantum threats. Key Words: Encryption algorithms, Performance analysis, Cryptology, Signature algorithm