The Discrete Logarithm Problem in Hyperelliptic Curve Groups (HECDLP) is a fundamental basis to more advanced cryptographic systems, and is more secure per key size than classical techniques using elliptic curves. This paper discusses the computational complexity and security consequences of HECDLP by considering the mathematical structure of it on finite field and divisor class groups. The aim is the evaluation of efficient algorithms, such as index calculus and rho versions of Pollard, and the proposal of a hybrid cryptographic model, taking advantage of the properties of hyperelliptic curves. The algorithm combines algebraic geometry, group theory and algorithmic optimization approaches to measure performance using security and efficiency measures. The preliminary performance comparison reports a security-strength score of 92.6% and a computation time of 118 ms for the proposed HECC framework. The reported computation time is 24.36% lower than that of the ECC-based method and 58.45% lower than that of the classical DLP method. These findings require validation through a reproducible experimental protocol and a formally defined security evaluation metric. The suggested model improves the immunity of classical and quantum-inspired attacks, making HECC an effective and scalable cryptography alternative with high security levels to be used in the next-generation applications.