Algebraic Modelings of the Supersingular Isogeny Problem
2026-07-06 • Symbolic Computation
Symbolic ComputationCryptography and Security
AI summaryⓘ
The authors created a new way to represent a tricky math problem about certain special types of curves, called supersingular elliptic curves, using polynomial equations. Their method works when the connection between these curves involves steps that are powers of 2 or 3. They studied the math properties of these equations and found that their approach makes solving them with computer algebra techniques faster than older methods. The research focuses on a specific formula setup for these curves and shows some important theoretical results about the equations.
Supersingular Isogeny ProblemElliptic CurvesIsogenyMontgomery FormTriangular FormMultivariate Polynomial EquationsGröbner BasisZero-Dimensional IdealsModular Polynomials
Authors
Alessio Caminata, Andrea Sanguineti, Silvia Sconza
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.