Algebraic approaches for solving isogeny problems of prime power degrees

Abstract Recently, supersingular isogeny cryptosystems have received attention as a candidate of post-quantum cryptography (PQC). Their security relies on the hardness of solving isogeny problems over supersingular elliptic curves. The meet-in-the-middle approach seems the most practical to solve isogeny problems with classical computers. In this paper, we propose two algebraic approaches for isogeny problems of prime power degrees. Our strategy is to reduce isogeny problems to a system of algebraic equations, and to solve it by Gröbner basis computation. The first one uses modular polynomials, and the second one uses kernel polynomials of isogenies. We report running times for solving isogeny problems of 3-power degrees on supersingular elliptic curves over 𝔽p2 with 503-bit prime p, extracted from the NIST PQC candidate SIKE. Our experiments show that our first approach is faster than the meet-in-the-middle approach for isogeny degrees up to 310.