  We initiate the study of computational problems on elliptic curve isogeny
graphs defined over RSA moduli. We conjecture that several variants of the
neighbor-search problem over these graphs are hard, and provide a comprehensive
list of cryptanalytic attempts on these problems. Moreover, based on the
hardness of these problems, we provide a construction of groups with infeasible
inversion, where the underlying groups are the ideal class groups of imaginary
quadratic orders.
  Recall that in a group with infeasible inversion, computing the inverse of a
group element is required to be hard, while performing the group operation is
easy. Motivated by the potential cryptographic application of building a
directed transitive signature scheme, the search for a group with infeasible
inversion was initiated in the theses of Hohenberger and Molnar (2003). Later
it was also shown to provide a broadcast encryption scheme by Irrer et al.
(2004). However, to date the only case of a group with infeasible inversion is
implied by the much stronger primitive of self-bilinear map constructed by
Yamakawa et al. (2014) based on the hardness of factoring and
indistinguishability obfuscation (iO). Our construction gives a candidate
without using iO.
