  The orbit problem is at the heart of symmetry reduction methods for model
checking concurrent systems. It asks whether two given configurations in a
concurrent system (represented as finite strings over some finite alphabet) are
in the same orbit with respect to a given finite permutation group (represented
by their generators) acting on this set of configurations by permuting indices.
It is known that the problem is in general as hard as the graph isomorphism
problem, whose precise complexity (whether it is solvable in polynomial-time)
is a long-standing open problem. In this paper, we consider the restriction of
the orbit problem when the permutation group is cyclic (i.e. generated by a
single permutation), an important restriction of the problem. It is known that
this subproblem is solvable in polynomial-time. Our main result is a
linear-time algorithm for this subproblem.
