Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Machine learning for moduli space of genus two curves and an application to isogeny based cryptography

Elira Shaska, Tony Shaska

TL;DR

This work investigates the moduli space of genus $2$ curves $\mathcal M_2$ through a data-driven lens, constructing a labeled database of curves over $\mathbb{Q}$ using Igusa invariants in the weighted projective space $\mathbb{P}_{(2,4,6,10)}$ and studying the distribution of fine versus coarse moduli points via weighted heights. It further demonstrates that machine learning can reliably detect whether a genus $2$ curve has a $(n,n)$-split Jacobian (for $n=2,3,5$) using only the invariants $J_2,J_4,J_6,J_{10}$, achieving up to 99.9% accuracy with supervised models and high ARI with unsupervised clustering on both latent and original features. The paper provides explicit constructions for the $(3,3)$- and $(5,5)$-split loci and discusses the computational challenges of generating and normalizing data in weighted projective space, highlighting a promising intersection of arithmetic geometry and data science with potential cryptographic applications. It also emphasizes the sparsity of fine moduli points at small heights and proposes that ML can guide mathematical proofs by identifying patterns in the moduli landscape, with extensions to higher $n$, positive characteristic, and graded-space neural nets as future directions.

Abstract

We use machine learning to study the moduli space of genus two curves, specifically focusing on detecting whether a genus two curve has $(n, n)$-split Jacobian. Based on such techniques, we observe that there are very few rational moduli points with small weighted moduli height and $(n, n)$-split Jacobian for $n=2, 3, 5$. We computational prove that there are only 34 genus two curves (resp. 44 curves) with (2,2)-split Jacobians (resp. (3,3)-split Jacobians) and weighted moduli height $\leq 3$. We discuss different machine learning models for such applications and demonstrate the ability to detect splitting with high accuracy using only the Igusa invariants of the curve. This shows that artificial neural networks and machine learning techniques can be highly reliable for arithmetic questions in the moduli space of genus two curves and may have potential applications in isogeny-based cryptography.

Machine learning for moduli space of genus two curves and an application to isogeny based cryptography

TL;DR

This work investigates the moduli space of genus curves through a data-driven lens, constructing a labeled database of curves over using Igusa invariants in the weighted projective space and studying the distribution of fine versus coarse moduli points via weighted heights. It further demonstrates that machine learning can reliably detect whether a genus curve has a -split Jacobian (for ) using only the invariants , achieving up to 99.9% accuracy with supervised models and high ARI with unsupervised clustering on both latent and original features. The paper provides explicit constructions for the - and -split loci and discusses the computational challenges of generating and normalizing data in weighted projective space, highlighting a promising intersection of arithmetic geometry and data science with potential cryptographic applications. It also emphasizes the sparsity of fine moduli points at small heights and proposes that ML can guide mathematical proofs by identifying patterns in the moduli landscape, with extensions to higher , positive characteristic, and graded-space neural nets as future directions.

Abstract

We use machine learning to study the moduli space of genus two curves, specifically focusing on detecting whether a genus two curve has -split Jacobian. Based on such techniques, we observe that there are very few rational moduli points with small weighted moduli height and -split Jacobian for . We computational prove that there are only 34 genus two curves (resp. 44 curves) with (2,2)-split Jacobians (resp. (3,3)-split Jacobians) and weighted moduli height . We discuss different machine learning models for such applications and demonstrate the ability to detect splitting with high accuracy using only the Igusa invariants of the curve. This shows that artificial neural networks and machine learning techniques can be highly reliable for arithmetic questions in the moduli space of genus two curves and may have potential applications in isogeny-based cryptography.
Paper Structure (44 sections, 12 theorems, 67 equations, 7 figures, 5 tables, 1 algorithm)

This paper contains 44 sections, 12 theorems, 67 equations, 7 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Neural networks
  4. Weighted greatest common divisors
  5. Weighted projective spaces and moduli space of binary forms
  6. Heights on weighted projective spaces
  7. Abelian surfaces and their isogenies
  8. Computing isogenies between Abelian varieties
  9. Torsion points and dual isogenies
  10. Supersingular Isogeny Diffie-Helman (SIDH)
  11. Key exchange:
  12. Moduli space ${\mathcal{M}}_2$ of genus 2 curves
  13. Genus two curves with extra automorphisms
  14. Genus two curves with $(n, n)$-split Jacobians
  15. $(3,3)$-split:
  16. ...and 29 more sections

Key Result

Theorem 1

Let ${\mathcal{A}}$ be an Abelian variety. Then ${\mathcal{A}}$ is isogenous to where (up to permutation of the factors) ${\mathcal{A}}_i$ , for $i=1, \dots , r$ are simple, non-isogenous, Abelian varieties. Moreover, up to permutations, the factors ${\mathcal{A}}_i^{n_i}$ are uniquely determined up to isogenies.

Figures (7)

  • Figure 1: ${\mathcal{L}}_2$ surface and 1-dimensional subloci of ${\mathcal{L}}_2$: curves with automorphism group $D_4$ or $D_6$
  • Figure 2: Ramification type and Weierstrass points
  • Figure 3: Distribution of points with extra automorphisms
  • Figure 4: Graph of rational points of weighted height $\mathfrak{H}_k \leq 2$
  • Figure 5: Learning curves for Neural Network model
  • ...and 2 more figures

Theorems & Definitions (15)

  • Theorem 1: Poincare-Weil
  • Theorem 2
  • Lemma 1
  • Remark 1
  • Theorem 3
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Corollary 1
  • Example 1
  • ...and 5 more