Achieving Secrecy Capacity of Minimum Storage Regenerating Codes for all Feasible $(n, k, d)$ Parameter Values

This paper addresses the problem of constructing secure exact-repair regenerating codes at the MSR point for all feasible values of the parameters. The setting involves a passive eavesdropper who is allowed to observe the stored contents of, and the downloads into, an $l$-subset of the $n$ nodes of a distributed storage system (DSS). The objective is to achieve perfect secrecy between the eavesdropped symbols and the file stored on the DSS. Previous secure code constructions (most notably that by Rawat et al.) tackle the problem only for the restricted case wherein the number, $d$, of helper nodes aiding in the recovery of a failed node is equal to $n-1$. This paper builds on Rawat's work, by combining Gabidulin pre-coding and an MSR construction by Ye and Barg to prove the achievability of secrecy capacity at the MSR point for all allowed values of $d$.