  Let p>3 be a prime and let E, E' be supersingular elliptic curves over F_p.
We want to construct an isogeny phi: E --> E'. The currently fastest algorithm
for finding isogenies between supersingular elliptic curves solves this problem
by performing a "meet-in-the-middle" breadth-first search in the full
supersingular 2-isogeny graph over F_{p^2}. In this paper we consider the
structure of the isogeny graph of supersingular elliptic curves over F_p. We
give an algorithm to construct isogenies between such supersingular elliptic
curves that works faster than the usual algorithm. We then discuss how this
results can be used to obtain an improved algorithm for the general
supersingular isogeny problem.
