Study Establishes High Logarithmic Densities for Rank‑1 and Rank‑2 Genus‑2 Jacobians, Impacting Hyperelliptic Cryptography

Global: Logarithmic Density of Rank ≥ 1 and Rank ≥ 2 Genus-2 Jacobians and Applications to Hyperelliptic Curve Cryptography

Researchers Razvan Barbulescu, Mugurel Barcau, Vicentiu Pasol, and George C. Turcăs have presented new quantitative results on the prevalence of genus‑2 curves over (mathbb{Q}) whose Jacobians possess Mordell‑Weil rank at least one or two. The findings, posted to arXiv on January 23, 2026 (arXiv:2601.17142), provide explicit lower bounds on logarithmic densities for these rank conditions and discuss potential ramifications for hyperelliptic curve cryptography.

Background and Motivation

The study addresses a longstanding question in arithmetic geometry: how frequently Jacobians of genus‑2 curves achieve higher Mordell‑Weil rank when curves are ordered by the naive height of their integral Weierstrass models. Understanding this distribution is relevant both to pure number theory and to cryptographic schemes that rely on the hardness of problems defined on such Jacobians.

Methodology and Density Results

Using geometric techniques, the authors show that asymptotically almost all integral models with two rational points at infinity have rank (r ge 1). Since roughly (asymp X^{13/2}) of these models appear among the (X^{7}) curves (y^{2}=f(x)) of height (le X), they derive a logarithmic density lower bound of (13/14) for the subset with rank (r ge 1). In addition, they identify a large explicit subfamily where Jacobians have rank (r ge 2), establishing an unconditional logarithmic density of at least (5/7). A separate construction yields a subfamily with split Jacobian and rank 2, achieving a density of at least (2/21).

Explicit Subfamilies and Rank‑2 Constructions

The paper details a construction of genus‑2 curves whose Jacobians split over (mathbb{Q}) and attain rank 2. This explicit family not only confirms the theoretical density bound of (2/21) but also provides concrete examples that can be examined for cryptographic suitability.

Twist Families and Positive Proportions

Further analysis focuses on quadratic and biquadratic twist families within the split‑Jacobian setting. The authors demonstrate that a positive proportion of these twists also have rank 2, reinforcing the broader claim that higher‑rank Jacobians are more common than previously assumed.

Cryptographic Implications

These density results have direct implications for Regev’s quantum algorithm applied to hyperelliptic curve cryptography. A higher prevalence of rank‑2 Jacobians suggests that certain attack vectors may be more feasible, prompting a reassessment of security parameters for protocols that rely on genus‑2 curves.

Future Directions

The authors note that extending the analysis to higher genus curves and exploring the impact of alternative height orderings could further refine density estimates. Such work may also inform the design of next‑generation cryptographic primitives resistant to quantum attacks.

This report is based on information from arXiv, licensed under Academic Preprint / Open Access. Based on the abstract of the research paper. Full text available via ArXiv.