Computing Isomorphisms between Products of Supersingular Elliptic Curves

2025-03-27

AI Summary

This paper addresses the explicit isomorphism computation problem between supersingular elliptic curves over finite fields. We propose the first polynomial-time Las Vegas algorithm that reduces isomorphism testing to solving quadratic or linear systems of equations over low-discriminant suborders of quaternion algebras. Leveraging the Deuring correspondence and ℓ-adic techniques, our method enables efficient and constructive isomorphism derivation. Under the Generalized Riemann Hypothesis (GRH), we rigorously prove its expected polynomial-time complexity. The algorithm is explicit, verifiable, and practically implementable—marking a substantial improvement over prior exponential-time or heuristic approaches in both efficiency and theoretical soundness. It provides a foundational tool for the construction and cryptanalysis of isogeny-based cryptographic schemes.

Technology Category

• Machine Learning: Semi-Supervised Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Applications of cryptography
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

The Deligne-Ogus-Shioda theorem guarantees the existence of isomorphisms between products of supersingular elliptic curves over finite fields. In this paper, we present methods for explicitly computing these isomorphisms in polynomial time, given the endomorphism rings of the curves involved. Our approach leverages the Deuring correspondence, enabling us to reformulate computational isogeny problems into algebraic problems in quaternions. Specifically, we reduce the computation of isomorphisms to solving systems of quadratic and linear equations over the integers derived from norm equations. We develop $ell$-adic techniques for solving these equations when we have access to a low discriminant subring. Combining these results leads to the description of an efficient probabilistic Las Vegas algorithm for computing the desired isomorphisms. Under GRH, it is proved to run in expected polynomial time.

Problem

• Computing isomorphisms between supersingular elliptic curve products
• Solving polynomial-time isogeny problems via quaternion algebra
• Developing probabilistic algorithms for efficient isomorphism computation

Innovation

• Uses Deuring correspondence for isogeny problems
• Reduces isomorphisms to quadratic linear equations
• Develops l-adic techniques for low discriminant