HECC is a cryptography approach that builds upon classical discrete logarithm schemes, using Jacobians as its underlying group, and provides greater security per unit of key size but with different tradeoffs. This paper defines the discrete logarithm problem (DLP) in hyperelliptic Jacobians, examines computational complexity and compares it with the elliptic curve DLP over generic and index-calculus models. Foundations in mathematics such as finite fields, divisor class groups and Mumford representations are explored in order to define group operations and hardness assumptions. We consider sub-exponential index calculus attacks, which are Lq[1/2] in small genus, and this means that security margins are lower with elliptic curves. The findings show that recovery of keys up to 28.6% faster in genus-3 environments and 22.4% fewer successful security bits with the optimal parameters. The design implications are addressed with suggestions of increased genus constraints, field size, or hybrid post-quantum schemes to ensure the security guarantees remain strong all around the world.