Topological analysis of number theory and its applications in quantum cryptography

This study investigates the application of algebraic topology, in particular homology theory and persistent homology. It includes the analysis of elliptic curves and their use in cryptographic systems and cybersecurity. The study was designed by establishing a topological framework for understanding elliptic curves as tori, highlighting the role of their homology groups in classifying their structure. A detailed case study illustrated how the first homology group H_1 (E)≅Z⨁Z oplus \mathbb{Z} provides the foundational loop structure for elliptic curve cryptography (ECC). Scalar multiplication and key generation were interpreted through topological cycles, reinforcing the complexity that underpins the hardness of the elliptic curve discrete logarithm problem (ECDLP). Persistent homology was then applied to cybersecurity, where topological features such as Betti numbers were used to detect anomalies in network traffic. A practical simulation demonstrated how transient spikes in 𝛽 1 could indicate coordinated attacks, supporting the role of topology in behavioral-based threat detection. Overall, the research bridged abstract topological theory with concrete cryptographic and security applications. It provides a new lens for understanding the shape and complexity of secure mathematical systems.