Hirschfeld, JamesAltimani, Nuof2024-11-042024-08https://hdl.handle.net/20.500.14154/73455This dissertation investigates elliptic curve theory and its foundational role in algebraic geometry, number theory, and cryptography. At its core is the Weierstrass equation, which defines the group structure of elliptic curves—a critical property for cryptographic systems that rely on the discrete logarithm problem for secure data exchange. Additionally, this work examines the geometric and algebraic properties of elliptic curves, emphasizing their applications in cryptography. The dissertation presents core concepts, the Weierstrass equation, methods for solving discrete logarithm problems, and cryptographic applications.75enMathematicscubiccurvecrytograpgyThesis