  A passive adversary can eavesdrop stored content or downloaded content of
some storage nodes, in order to learn illegally about the file stored across a
distributed storage system (DSS). Previous work in the literature focuses on
code constructions that trade storage capacity for perfect security. In other
words, by decreasing the amount of original data that it can store, the system
can guarantee that the adversary, which eavesdrops up to a certain number of
storage nodes, obtains no information (in Shannon's sense) about the original
data. In this work we introduce the concept of block security for DSS and
investigate minimum bandwidth regenerating (MBR) codes that are block secure
against adversaries of varied eavesdropping strengths. Such MBR codes guarantee
that no information about any group of original data units up to a certain size
is revealed, without sacrificing the storage capacity of the system. The size
of such secure groups varies according to the number of nodes that the
adversary can eavesdrop. We show that code constructions based on Cauchy
matrices provide block security. The opposite conclusion is drawn for codes
based on Vandermonde matrices.
