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

• Published in: Journal of algebraic combinatorics Vol. 61; no. 2; p. 23
• Main Authors: Shaska, Elira, Shaska, Tanush
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2025; Springer Nature B.V
• Subjects: Artificial neural networks; Combinatorics; Computer Science; Convex and Discrete Geometry; Cryptography; Datasets; Experiments; Group Theory and Generalizations; Jacobians; Lattices; Machine learning; Mathematics; Mathematics and Statistics; Neural networks; Order; Ordered Algebraic Structures; Isogenies; Primary: 68T07; Secondary: 68T20; Weighted projective space; Genus two curves; 68Q32
• ISSN: 0925-9899; 1572-9192
• DOI: 10.1007/s10801-025-01393-8

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 ≤ 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.