Given a supersingular elliptic curve E and a non-scalar endomorphism $\alpha$ of E, we prove that the endomorphism ring of E can be computed in classical time about disc(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. Along the way, we prove that the Primitivisation problem can be solved in polynomial time (a problem previously believed to be hard), and we prove that the action of smooth ideals on oriented elliptic curves can be computed in polynomial time (previous results of this form required the ideal to be powersmooth, i.e., not divisible by any large prime power). Following the attacks on SIDH, isogenies in high dimension are a central ingredient of our results.

翻译：给定超奇异椭圆曲线E及其非标量自同态$\alpha$，我们证明：在广义黎曼假设下，可在经典时间复杂度约disc(Z[$\alpha$])^1/4及量子次指数时间内计算E的自同态环。此前结果要么复杂度更高，要么依赖于启发式假设。在此过程中，我们证明了原始化问题可在多项式时间内求解（该问题此前被认为困难），并证明了光滑理想在定向椭圆曲线上的作用可在多项式时间内计算（此前此类结果要求理想为幂光滑形式，即不含大素幂因子）。借鉴SIDH攻击方法，高维同源成为我们结果的核心要素。