Optimality of the product-matrix construction for secure MSR regenerating codes

In this paper, we consider the security of exact-repair regenerating codes operating at the minimum-storage-regenerating (MSR) point. The security requirement (introduced in Shah et. al.) is that no information about the stored data file must be leaked in the presence of an eavesdropper who has access to the contents of ℓ₁ nodes as well as all the repair traffic entering a second disjoint set of ℓ₂ nodes. We derive an upper bound on the size of a data file that can be securely stored that holds whenever ℓ₂ ≤ d - k + 1. This upper bound proves the optimality of the product-matrix-based construction of secure MSR regenerating codes by Shah et. al.