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.