  We study the problem of constructing secure regenerating codes that protect
data integrity in distributed storage systems (DSS) in which some nodes may be
compromised by a malicious adversary. The adversary can corrupt the data stored
on and transmitted by the nodes under its control. The "damage" incurred by the
actions of the adversary depends on how much information it knows about the
data in the whole DSS. We focus on the limited-knowledge model in which the
adversary knows only the data on the nodes under its control. The only secure
capacity-achieving codes known in the literature for this model are for the
bandwidth-limited regime and repair degree $d=n-1$, i.e., when a node fails in
a DSS with $n$ nodes all the remaining $n-1$ nodes are contacted for repair. We
extend these results to the more general case of $d\leq n-1$ in the
bandwidth-limited regime. Our capacity-achieving scheme is based on the use of
product-matrix codes with special hashing functions and allow the
identification of the compromised nodes and their elimination from the DSS
while preserving the data integrity.
