Paper 2026/1486

The supersingular isogeny problem in time and memory $p^{1/3+o(1)}$

Abstract

We prove that under a plausible heuristic assumption (on the smoothness of certain random integers), the supersingular isogeny problem can be solved in time and memory $p^{1/3 + o(1)}$. This improves upon the previous best complexity of $p^{1/2} \cdot(\log p)^{O(1)}$. This problem is arguably the central hard problem underlying isogeny-based cryptography, and the cost of its resolution is a major (and often the only) factor in the choice of secure parameters. The impact on concrete parameter sets remains to be clarified, as the asymptotic advantage of the new algorithm is mitigated by a superpolynomial overhead hiding in the $o(1)$ exponent, and by its high memory requirement.