  The supersingular Endomorphism Ring problem is the following: given a
supersingular elliptic curve, compute all of its endomorphisms. The presumed
hardness of this problem is foundational for isogeny-based cryptography. The
One Endomorphism problem only asks to find a single non-scalar endomorphism. We
prove that these two problems are equivalent, under probabilistic polynomial
time reductions. We prove a number of consequences. First, assuming the
hardness of the endomorphism ring problem, the Charles--Goren--Lauter hash
function is collision resistant, and the SQIsign identification protocol is
sound. Second, the endomorphism ring problem is equivalent to the problem of
computing arbitrary isogenies between supersingular elliptic curves, a result
previously known only for isogenies of smooth degree. Third, there exists an
unconditional probabilistic algorithm to solve the endomorphism ring problem in
time O~(sqrt(p)), a result that previously required to assume the generalized
Riemann hypothesis. To prove our main result, we introduce a flexible framework
for the study of isogeny graphs with additional information. We prove a general
and easy-to-use rapid mixing theorem. The proof of this result goes via an
augmented Deuring correspondence and the Jacquet-Langlands correspondence.
