We prove that the path-finding problem in $\ell$-isogeny graphs and the endomorphism ring problem for supersingular elliptic curves are equivalent under reductions of polynomial expected time, assuming the generalised Riemann hypothesis. The presumed hardness of these problems is foundational for isogeny-based cryptography. As an essential tool, we develop a rigorous algorithm for the quaternion analog of the path-finding problem, building upon the heuristic method of Kohel, Lauter, Petit and Tignol. This problem, and its (previously heuristic) resolution, are both a powerful cryptanalytic tool and a building-block for cryptosystems.

翻译：我们证明,如果假设通用的里伊曼假设,超单向椭圆曲线的“以美元为单位”图中的“路由调查问题”和“内晶”环状问题,与假定通用的里伊曼假设的多元预期时间的减少相当。 这些问题的假定硬性是基于异源的加密学的基础。 作为基本工具,我们根据Kohel、Lauter、Petit和Tignol的超常方法,为“路由调查问题”的顶部类比制定严格的算法。 这个问题及其(先前的)超常解决方法既是强大的加密工具,也是加密系统的建筑块。