  The rank decoding problem has been the subject of much attention in this last
decade. This problem, which is at the base of the security of public-key
cryptosystems based on rank metric codes, is traditionally studied over finite
fields. But the recent generalizations of certain classes of rank-metric codes
from finite fields to finite rings have naturally created the interest to
tackle the rank decoding problem in the case of finite rings. In this paper, we
show that solving the rank decoding problem over finite principal ideal rings
is at least as hard as the rank decoding problem over finite fields. We also
show that computing the minimum rank distance for linear codes over finite
principal ideal rings is equivalent to the same problem for linear codes over
finite fields. Finally, we provide combinatorial type algorithms for solving
the rank decoding problem over finite chain rings together with their average
complexities.
