  Fix an ordinary abelian variety defined over a finite field. The ideal class
group of its endomorphism ring acts freely on the set of isogenous varieties
with same endomorphism ring, by complex multiplication. Any subgroup of the
class group, and generating set thereof, induces an isogeny graph on the orbit
of the variety for this subgroup. We compute (under the Generalized Riemann
Hypothesis) some bounds on the norms of prime ideals generating it, such that
the associated graph has good expansion properties.
  We use these graphs, together with a recent algorithm of Dudeanu, Jetchev and
Robert for computing explicit isogenies in genus 2, to prove random
self-reducibility of the discrete logarithm problem within the subclasses of
principally polarizable ordinary abelian surfaces with fixed endomorphism ring.
In addition, we remove the heuristics in the complexity analysis of an
algorithm of Galbraith for explicitly computing isogenies between two elliptic
curves in the same isogeny class, and extend it to a more general setting
including genus 2.
