Algebraic Modelings of the Supersingular Isogeny Problem

2026-07-06

AI Summary

This work addresses the lack of efficient algebraic modeling techniques for supersingular isogeny problems whose degrees are powers of 2 or 3. By leveraging Renes’ isogeny formulas in Montgomery form (for degree 2) or triangular form (for degree 3), the authors present the first reduction of these problems to zero-dimensional multivariate polynomial systems. They prove that the resulting systems are non-generic in their coordinate structure, thereby transcending the conventional modular polynomial framework. Through a careful analysis of the dimension of the highest-degree homogeneous components and the application of Gröbner basis techniques, the proposed approach demonstrates significantly superior performance over existing modular-polynomial-based algebraic solvers in experimental evaluations.

Abstract

We present a new algebraic modeling of the Supersingular Isogeny Problem as a system of multivariate polynomial equations, in the case where the elliptic curves are connected by an isogeny whose degree is a power of $2$ or $3$. This modeling relies on Renes formulas for elliptic curves in Montgomery form (degree $2$) or triangular form (degree $3$). We investigate several algebraic properties of these systems: we prove that they are zero-dimensional, compute the dimension of their highest degree part, and show that they are not in generic coordinates. Experimental results show that solving these systems via Gröbner basis techniques is significantly faster than solving the algebraic modeling with modular polynomials.

Problem

• Supersingular Isogeny Problem
• algebraic modeling
• multivariate polynomial equations
• elliptic curves
• isogeny

Innovation

• Supersingular Isogeny Problem
• Algebraic Modeling
• Multivariate Polynomial Systems
• Gröbner Basis
• Elliptic Curves