Exact Abelian Group Structures of Genus 2 Hyperelliptic Jacobians over GF2^k (k = 1 . . . 6): A Verified Database

In the design of modern cryptographic systems, professional standards require us to understandexactly when and where our mathematical models break down. When mapping hyperellipticcurves over finite fields, practitioners typically rely on theoretical boundaries—specifically, theHasse-Weil limits—to estimate the size and shape of the available state space. If mathematicswere a perfectly continuous landscape, every coordinate within these theoretical limits would yielda valid, operational curve.However, computer simulations of complex algebraic state spaces become computationallyprohibitive as the field size increases. Furthermore, we must be careful in how we think about anduse these continuous theoretical models. At a fundamental level, asymptotic formulas are neither correct nor assumed to be accurate when applied to discrete, low-characteristic binary fields (F2through F64 ).In these specific binary environments, theoretical models over-predict reality. Overlappinggeometric obstructions irregularly truncate the available grid. Certain coordinate pairs generatedecomposable surfaces—essentially two weak curves masquerading as a strong one—while othersflatly violate the strict, structural rules unique to characteristic 2 arithmetic.The benefit of using the computer to explicitly map this space is that we obtain an empiricalunderstanding of the different structural anomalies to which characteristic 2 systems are exposed.Once we have this insight, we can consider methodologies to safely navigate these topologicalrisks. Consequently, we have programmatically mapped, independently validated, and catalogedevery valid genus 2 curve into a holistic baseline database.