Logarithmic Density of Rank-2 Jacobians Achieves Lower Bounds for Hyperelliptic Curve Cryptography

Scientists are tackling a fundamental problem in number theory: understanding the distribution of rank within genus-2 Jacobians, a crucial area for advancements in cryptography. Razvan Barbulescu, Mugurel Barcau, and Vicentiu Pasol, alongside George C. Turcas et al., demonstrate compelling results regarding the logarithmic density of these ranks, specifically focusing on curves with ranks of 0 or 1. Their research establishes a lower bound for the proportion of curves possessing rank 1, and importantly, identifies explicit families achieving even higher ranks , a significant step forward as these findings directly impact the security and efficiency of hyperelliptic curve cryptography, potentially influencing algorithms like Regev’s.

The research establishes quantitative existence results for these curves, specifically focusing on those whose Jacobians exhibit ranks of at least 1 or 2, ordered by the naive height of their integral Weierstrass models. Employing geometric techniques, the team achieved a lower bound of logarithmic density 13/14 for the subset of genus-2 curves with rank r ≥1, meaning that asymptotically, a substantial proportion of these curves possess a Jacobian with rank at least one. This calculation is based on the existence of approximately X 13 /2 such models among X 7 curves of height less than or equal to X.

The study further unveils a large explicit subfamily where Jacobians have ranks r ≥2, yielding an unconditional logarithmic density of at least 5/7. This discovery represents a considerable step forward in characterizing the prevalence of higher-rank Jacobians within this family of curves. Independently, researchers constructed a genus-2 curve family with split Jacobian and rank 2, achieving a logarithmic density of at least 2/21. This construction provides a concrete example and strengthens the understanding of how to generate curves with specific rank properties. These findings are particularly valuable as counting by conductor is notoriously difficult, prompting the use of height functions on coefficients of integral models for enumeration.

Experiments show that the work builds upon the foundation laid by the Mordell-Weil theorem, which posits that the group of rational points on the Jacobian of any smooth projective curve admits a decomposition into a finite group and a torsion-free part of rank r. The research addresses a fundamental problem in arithmetic geometry: understanding how these ranks are distributed among hyperelliptic curves of a fixed genus. By ordering curves by height, the scientists were able to count curves with height less than or equal to X as X approaches infinity, providing a rigorous framework for quantifying the distribution of ranks. The team’s analysis extends to quadratic and biquadratic twist families in the split-Jacobian setting, revealing a positive proportion of rank-2 twists.

This result has direct implications for Regev’s quantum algorithm in hyperelliptic curve cryptography, potentially influencing the security and efficiency of cryptographic systems. The logarithmic density of rank ≥1 genus-2 Jacobians is proven to be at least 13/14, while the density of rank ≥2 Jacobians is at least 5/7, offering valuable insights for algorithmic complexity and brute-force search strategies. A naive search for curves of rank r ≥1 could succeed in time O(X 1/2 + o(1 ), and finding rank r ≥2 curves requires at most O(X 2 + o(1)) trials.

Genus-2 Curves and Jacobian Mordell-Weil Rank

Scientists investigated the quantitative existence of genus-2 curves over the rational numbers whose Jacobians exhibit a Mordell-Weil rank of at least 2 or 3, classifying these curves by the naive height of their integral Weierstrass models. The study employed geometric techniques to demonstrate that, asymptotically, the Jacobians of nearly all integral models possessing two rational points at infinity attain a rank of 2. Given the existence of such models within curves of height 2, this establishes a logarithmic density lower bound of 1 for the subset of curves with rank 2. Researchers further identified a substantial explicit subfamily where Jacobians achieve ranks of 2 and 3, yielding an unconditional logarithmic density of at least 1.

Independently, the team constructed genus-2 curves with split Jacobians and rank 2, generating a subfamily with a logarithmic density of at least 1. To achieve this, scientists analysed quadratic and biquadratic twist families within the split-Jacobian framework, revealing a positive proportion of rank-1 twists. Experiments utilized the LMFDB database, accessing 66158 Q-isomorphic classes of curves with absolute discriminant less than 106, and identified a curve, y2 + (x3 + x + 1)y = x5 −x4 −5×3 + 9x + 6, with a Jacobian of rank 4 generated by specific Mumford representation elements: (1, x2), (x + 1, 0), (x + 2, −x3 −4), and (x2 −2, 1). The research pioneered a method for constructing genus-2 curves with high rank by gluing together elliptic curves.

Specifically, the team considered the congruent number curve E: y2 = x3−x and the genus-2 curve y2 = 6×6 −9×4 −9×2 + 6, demonstrating that its Jacobian is isomorphic to E × E. This approach enabled the creation of a genus-2 curve defined over Q(t) with a rank of at least 20, achieved through a curve (2, 2)-isogenous to E1 × E2, where E1 is defined over Q and E2 is defined over Q(t) with coefficients of degree 756 and binary size 1761. This innovative technique directly addresses the requirements of Regev’s algorithm in hyperelliptic curve cryptography by providing curves with Jacobians suitable for efficient scalar product computation. . The work establishes that, asymptotically, the Jacobians of almost all integral models of these curves, possessing two rational points at infinity, exhibit rank 0.

This finding yields a logarithmic density lower bound for the subset of curves with rank 0, and further, a large explicit subfamily is identified where Jacobians attain ranks 0, 1, and 2, resulting in an unconditional logarithmic density of at least 1. Furthermore, the study constructs genus-2 curves with split Jacobians and rank 1, establishing a subfamily with a logarithmic density of at least 1. Analysis of quadratic and biquadratic twist families within the split-Jacobian setting reveals a positive proportion of rank-1 twists, which has implications for Regev’s algorithm in hyperelliptic curve cryptography. The authors acknowledge a limitation in their approach, noting the use of a naive height for rational functions, which is comparable to the logarithmic height of the curve.

Future research could focus on refining these height comparisons or extending the analysis to curves of higher genus. These results collectively contribute to a more refined understanding of the Mordell-Weil rank distribution of genus-2 Jacobians, a central problem in arithmetic geometry. The identification of explicit subfamilies with guaranteed rank behaviour is particularly noteworthy, offering concrete examples for further investigation. While the logarithmic density results provide asymptotic information, the explicit construction of curves with specific rank properties offers a more direct path for applications in cryptography and number theory. The authors’ acknowledgement of height comparison limitations suggests a potential avenue for improving the precision of these estimates in future work.