  An important open problem in supersingular isogeny-based cryptography is to
produce, without a trusted authority, concrete examples of "hard supersingular
curves" that is, equations for supersingular curves for which computing the
endomorphism ring is as difficult as it is for random supersingular curves. A
related open problem is to produce a hash function to the vertices of the
supersingular $\ell$-isogeny graph which does not reveal the endomorphism ring,
or a path to a curve of known endomorphism ring. Such a hash function would
open up interesting cryptographic applications. In this paper, we document a
number of (thus far) failed attempts to solve this problem, in the hope that we
may spur further research, and shed light on the challenges and obstacles to
this endeavour. The mathematical approaches contained in this article include:
(i) iterative root-finding for the supersingular polynomial; (ii) gcd's of
specialized modular polynomials; (iii) using division polynomials to create
small systems of equations; (iv) taking random walks in the isogeny graph of
abelian surfaces; and (v) using quantum random walks.
