Logarithmic Density of Rank $\geq 1$ and Rank $\geq 2$ Genus-2 Jacobians and Applications to Hyperelliptic Curve Cryptography

• URL: http://arxiv.org/abs/2601.17142v1
• Date: Fri, 23 Jan 2026 19:40:04 GMT
• Title: Logarithmic Density of Rank $\geq 1$ and Rank $\geq 2$ Genus-2 Jacobians and Applications to Hyperelliptic Curve Cryptography
• Abstract summary: We study existence results for genus-$2$ curves over $mathbbQ$ whose Jacobians have Mordell-Weil rank at least $1$ or $2$.<n>We present a large explicit subfamily where Jacobians have ranks $r geq 2$, yielding an unconditional logarithmic density of at least $5/7$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this work we study quantitative existence results for genus-$2$ curves over $\mathbb{Q}$ whose Jacobians have Mordell-Weil rank at least $1$ or $2$, ordering the curves by the naive height of their integral Weierstrass models. We use geometric techniques to show that asymptotically the Jacobians of almost all integral models with two rational points at infinity have rank $r \geq 1$. Since there are $\asymp X^{\frac{13}{2}}$ such models among the $X^7$ curves $y^2=f(x)$ of height $\leq X$, this yields a lower bound of logarithmic density $13/14$ for the subset of rank $r \geq 1$. We further present a large explicit subfamily where Jacobians have ranks $r \geq 2$, yielding an unconditional logarithmic density of at least $5/7$. Independently, we give a construction of genus-$2$ curves with split Jacobian and rank $2$, producing a subfamily of logarithmic density at least $ 2/21$. Finally, we analyze quadratic and biquadratic twist families in the split-Jacobian setting, obtaining a positive proportion of rank-$2$ twists. These results have implications for Regev's quantum algorithm in hyperelliptic curve cryptography.

Related papers

• Exponential Convex Calibration Dimension for the Multi-Label Jaccard Measure [12.752262012085419]
  We prove that the Jaccard score, shifted-loss, and ordinary loss matrices are nonsingular and that the loss columns have affine dimension $2s-1$.<n>We also give two-dimensional approximation guarantees with explicit regret transfers.
  arXiv  Detail & Related papers  (2026-08-13T17:59:17Z)
• The Condition-Number Barrier in Sparse Least Squares [77.64108812086542]
  In [AS21], Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in convex optimization cannot be improved by a sparse-time algorithm.<n>We establish their conjectured lower bound for least-squares objectives, conditional on the randomized exact-volume Small-Set Expansion Hypothesis.
  arXiv  Detail & Related papers  (2026-08-03T17:57:01Z)
• On the Power of Adaptivity for $\varepsilon$-Best Arm Identification in Linear Bandits [8.86974002494041]
  We study the minimax sample complexity of $varepsilon$-best arm identification in linear bandits.<n>The goal is to output an arm $widehatxinmathcalX$ such that $langle ge max_xinmathcalX langle x,rangle ge max_xinmathcalX langle x,rangle ge max_xinmathcalX langle x
  arXiv  Detail & Related papers  (2026-05-15T06:35:31Z)
• Detection of local geometry in random graphs: information-theoretic and computational limits [4.67223325681232]
  We study the problem of detecting local geometry in random graphs.<n>We introduce a model $mathcalG(n, p, d, k)$, where a hidden community of average size $k$ has edges drawn as a random geometric graph.<n>All remaining edges follow the Erds--Rényi model $mathcalG(n, p)$.
  arXiv  Detail & Related papers  (2026-03-25T17:20:01Z)
• Optimal Scalar Quantization for Matrix Multiplication: Closed-Form Density and Phase Transition [50.36362492608702]
  We study entrywise scalar quantization of two matrices prior to multiplication.<n>We obtain a closed-form optimal point density [ star(u) propto exp!left(-fracu26right)bigl( (1-2)+2u22bigr), qquad u=fracx_X,, and prove a correlation-driven phase transition.
  arXiv  Detail & Related papers  (2026-03-20T01:53:44Z)
• Permanents of matrix ensembles: computation, distribution, and geometry [0.0]
  We use the GPU to greaatly accelerate the computation of permanents over $mathbbC,$ $mathbbR,$ $mathbbF_p$ and $mathbbQ.$<n>We study the permanent along geodesics on the unitary group.<n>For the geodesic from the identity to the $n$-cycle permutation matrix, we find a universal scaling function $f(t)=frac1nln|perm((t))|$ that is independent of$n$ in
  arXiv  Detail & Related papers  (2026-02-08T22:31:42Z)
• Information-Computation Tradeoffs for Noiseless Linear Regression with Oblivious Contamination [65.37519531362157]
  We show that any efficient Statistical Query algorithm for this task requires VSTAT complexity at least $tildeOmega(d1/2/alpha2)$.
  arXiv  Detail & Related papers  (2025-10-12T15:42:44Z)
• Improved convergence rate of kNN graph Laplacians [11.93971616098517]
  General class of $k$NN graph where the graph affinity is $W_ij = epsilon-d/2 . We prove the point-wise convergence of the $k$NN graph Laplacian to the limiting manifold operator.
  arXiv  Detail & Related papers  (2024-10-30T17:01:00Z)
• Machine learning for moduli space of genus two curves and an application to isogeny based cryptography [0.0]
  We use machine learning to study the moduli space of genus two curves.<n>We show that machine learning can be highly reliable for arithmetic questions in the moduli space of genus two curves.
  arXiv  Detail & Related papers  (2024-03-25T22:52:50Z)
• A Unified Framework for Uniform Signal Recovery in Nonlinear Generative Compressed Sensing [68.80803866919123]
  Under nonlinear measurements, most prior results are non-uniform, i.e., they hold with high probability for a fixed $mathbfx*$ rather than for all $mathbfx*$ simultaneously. Our framework accommodates GCS with 1-bit/uniformly quantized observations and single index models as canonical examples. We also develop a concentration inequality that produces tighter bounds for product processes whose index sets have low metric entropy.
  arXiv  Detail & Related papers  (2023-09-25T17:54:19Z)
• Detection of Dense Subhypergraphs by Low-Degree Polynomials [72.4451045270967]
  Detection of a planted dense subgraph in a random graph is a fundamental statistical and computational problem. We consider detecting the presence of a planted $Gr(ngamma, n-alpha)$ subhypergraph in a $Gr(n, n-beta) hypergraph. Our results are already new in the graph case $r=2$, as we consider the subtle log-density regime where hardness based on average-case reductions is not known.
  arXiv  Detail & Related papers  (2023-04-17T10:38:08Z)
• Low-degree learning and the metric entropy of polynomials [44.99833362998488]
  We prove that any (deterministic or randomized) algorithm which learns $mathscrF_nd$ with $L$-accuracy $varepsilon$ requires at least $Omega(sqrtvarepsilon)2dlog n leq log mathsfM(mathscrF_n,d,|cdot|_L,varepsilon) satisfies the two-sided estimate $$c (1-varepsilon)2dlog
  arXiv  Detail & Related papers  (2022-03-17T23:52:08Z)
• Random Graph Matching in Geometric Models: the Case of Complete Graphs [21.689343447798677]
  This paper studies the problem of matching two complete graphs with edge weights correlated through latent geometries. We derive an approximate maximum likelihood estimator, which provably achieves, with high probability, perfect recovery of $pi*$. As a side discovery, we show that the celebrated spectral algorithm of [Ume88] emerges as a further approximation to the maximum likelihood in the geometric model.
  arXiv  Detail & Related papers  (2022-02-22T04:14:45Z)
• Optimal Robust Linear Regression in Nearly Linear Time [97.11565882347772]
  We study the problem of high-dimensional robust linear regression where a learner is given access to $n$ samples from the generative model $Y = langle X,w* rangle + epsilon$ We propose estimators for this problem under two settings: (i) $X$ is L4-L2 hypercontractive, $mathbbE [XXtop]$ has bounded condition number and $epsilon$ has bounded variance and (ii) $X$ is sub-Gaussian with identity second moment and $epsilon$ is
  arXiv  Detail & Related papers  (2020-07-16T06:44:44Z)
• Near-Optimal SQ Lower Bounds for Agnostically Learning Halfspaces and ReLUs under Gaussian Marginals [49.60752558064027]
  We study the fundamental problems of agnostically learning halfspaces and ReLUs under Gaussian marginals. Our lower bounds provide strong evidence that current upper bounds for these tasks are essentially best possible.
  arXiv  Detail & Related papers  (2020-06-29T17:10:10Z)
• Nearly Optimal Regret for Stochastic Linear Bandits with Heavy-Tailed Payoffs [35.988644745703645]
  We analyze the linear bandits with heavy-tailed payoffs, where the payoffs admit finite $1+epsilon$ moments. We propose two novel algorithms which enjoy a sublinear regret bound of $widetildeO(dfrac12Tfrac11+epsilon)$.
  arXiv  Detail & Related papers  (2020-04-28T13:01:38Z)
• Curse of Dimensionality on Randomized Smoothing for Certifiable Robustness [151.67113334248464]
  We show that extending the smoothing technique to defend against other attack models can be challenging. We present experimental results on CIFAR to validate our theory.
  arXiv  Detail & Related papers  (2020-02-08T22:02:14Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.