New bounds for distributed storage systems with secure repair

In this paper, we consider exact-repair distributed storage systems. Characterizing the optimal storage-vs-repair bandwidth tradeoff for such systems remains an open problem, in general with results available in the literature for very specific instances. We characterize the optimal tradeoff between storage and repair bandwidth if in addition to exact-repair requirements, an additional requirement of pair-wise symmetry on the repair process is imposed. The optimal tradeoff surprisingly consists of only one efficient point, namely the minimum bandwidth regenerating (MBR) point. These results are also extended to the case in which the stored data (or repair data) must be secure from an external wiretapper, and the correspondingly optimal secure tradeoffs are also characterized. The main technical tool used in the converse proofs is a use of Han's inequality for sub-sets of random variables. Finally, we also present results in which the pair-wise symmetry constraint is relaxed and a new converse bound is obtained for the case of exact repair in which an external adversary can access the repair data of any one node. This bound improves upon the existing best known results for the secure and exact repair problem.