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 understand
exactly when and where our mathematical models break down. When mapping hyperelliptic
curves over finite fields, practitioners typically rely on theoretical boundaries—specifically, the
Hasse-Weil limits—to estimate the size and shape of the available state space. If mathematics
were a perfectly continuous landscape, every coordinate within these theoretical limits would yield
a valid, operational curve.
However, computer simulations of complex algebraic state spaces become computationally
prohibitive as the field size increases. Furthermore, we must be careful in how we think about and
use 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 (F2
through F64 ).
In these specific binary environments, theoretical models over-predict reality. Overlapping
geometric obstructions irregularly truncate the available grid. Certain coordinate pairs generate
decomposable surfaces—essentially two weak curves masquerading as a strong one—while others
flatly 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 empirical
understanding 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 topological
risks. Consequently, we have programmatically mapped, independently validated, and cataloged
every valid genus 2 curve into a holistic baseline database.