In cryptography, we are always looking for hard mathematical problems on which to build secure protocols for exchanging messages. Modern cryptographic systems rely mainly on the hardness of integer factorization and the Discrete Logarithm Problem. However, both of these can be efficiently solved on sufficiently powerful quantum computers using Shor’s algorithm (1994). Hence the need to look for alternative problems that remain difficult even in the quantum setting. One promising candidate is the Supersingular Isogeny Problem, which gave rise to Isogeny-Based Cryptography. We will take a friendly look at the problem and provide an algebraic description of it in the form of a polynomial system.

We construct algebraic models for the Supersingular Isogeny Problem, for isogenies of degree powers of 2 and 3, using modular polynomials and explicit formulae from the works of Burdges, DeFeo, Renes, Costello, and Hisil. These constructions yield multivariate polynomial systems which we study through tools from computational algebra, including Gröbner bases and related techniques in commutative algebra. We further present experimental results that estimate the maximum step degree observed during the solution process, providing insight into the complexity and feasibility of solving these systems in practice. This is an ongoing joint work with Alessio Caminata (Università di Genova) and Silvia Sconza (University of Zurich).