  Distributed storage systems in the presence of a wiretapper are considered. A
distributed storage system (DSS) is parameterized by three parameters (n, k,d),
in which a file stored across n distributed nodes, can be recovered from any k
out of n nodes. If a node fails, any d out of (n-1) nodes help in the repair of
the failed node. For such a (n,k,d)-DSS, two types of wiretapping scenarios are
investigated: (a) Type-I (node) adversary which can wiretap the data stored on
any l<k nodes; and a more severe (b) Type-II (repair data) adversary which can
wiretap the contents of the repair data that is used to repair a set of l
failed nodes over time. The focus of this work is on the practically relevant
setting of exact repair regeneration in which the repair process must replace a
failed node by its exact replica. We make new progress on several non-trivial
instances of this problem which prior to this work have been open. The main
contribution of this paper is the optimal characterization of the secure
storage-vs-exact-repair-bandwidth tradeoff region of a (n,k,d)-DSS, with n<=4
and any l<k in the presence of both Type-I and Type-II adversaries. While the
problem remains open for a general (n,k,d)-DSS with n>4, we present extensions
of these results to a (n, n-1,n-1)-DSS, in presence of a Type-II adversary that
can observe the repair data of any l=(n-2) nodes. The key technical
contribution of this work is in developing novel information theoretic converse
proofs for the Type-II adversarial scenario. From our results, we show that in
the presence of Type-II attacks, the only efficient point in the
storage-vs-exact-repair-bandwidth tradeoff is the MBR (minimum bandwidth
regenerating) point. This is in sharp contrast to the case of a Type-I attack
in which the storage-vs-exact-repair-bandwidth tradeoff allows a spectrum of
operating points beyond the MBR point.
