  We introduce and study the random "locked" constraint satisfaction problems.
When increasing the density of constraints, they display a broad "clustered"
phase in which the space of solutions is divided into many isolated points.
While the phase diagram can be found easily, these problems, in their clustered
phase, are extremely hard from the algorithmic point of view: the best known
algorithms all fail to find solutions. We thus propose new benchmarks of really
hard optimization problems and provide insight into the origin of their typical
hardness.
