endomorphism

Given a supersingular elliptic curve $E$ and a nonscalar endomorphism $\alpha$ of $E$, we prove that the endomorphism ring of $E$ can be computed in classical time about $\text{disc}(\mathbb{Z}[\alpha])^{1/4}$ , and in quantum subexponential time, assuming the generalised Riemann hypothesis. Previous results either had higher complexities, or relied on heuristic assumptions.

[Submitted on 21 Sep 2023] Download a PDF of the paper titled The supersingular endomorphism ring problem given one endomorphism, by Arthur Herl\’edan Le Merdy (LIP) and 2 other authors