The supersingular isogeny path and endomorphism ring problems are equivalent

We prove that the path-finding problem in 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. This is an extended abstract of the full article available at http://arxiv.org/abs/2111.01481. This full article will be referred to as “the full version” throughout the text.