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：给定一个超奇异椭圆曲线E及其一个非标量自同态α，我们证明了在假设广义黎曼猜想成立的前提下，可以在经典时间约disc(Z[α])^1/4内以及量子次指数时间内计算E的自同态环。此前的结果要么具有更高的复杂度，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以在多项式时间内求解（该问题先前被认为难以解决），并证明了光滑理想在定向椭圆曲线上的作用可以在多项式时间内计算（此前此类结果要求理想为幂光滑的，即不被任何大素数的幂整除）。借鉴对SIDH的攻击，高维同源是我们结果的核心要素。