  A Riemannian manifold is said to be uniformly secure if there is a finite
number $s$ such that all geodesics connecting an arbitrary pair of points in
the manifold can be blocked by $s$ point obstacles. We prove that the number of
geodesics with length $\leq T$ between every pair of points in a uniformly
secure manifold grows polynomially as $T \to \infty$. We derive from this that
a compact Riemannian manifold with no conjugate points whose geodesic flow has
positive topological entropy is totally insecure: the geodesics between any
pair of points cannot be blocked by a finite number of point obstacles.
