Distributed Storage Codes Meet Multiple-Access Wiretap Channels

We consider {\it i)} the overhead minimization of maximum-distance separable (MDS) storage codes for the repair of a single failed node and {\it ii)} the total secure degrees-of-freedom (S-DoF) maximization in a multiple-access compound wiretap channel. We show that the two problems are connected. Specifically, the overhead minimization for a single node failure of an {\it optimal} MDS code, i.e. one that can achieve the information theoretic overhead minimum, is equivalent to maximizing the S-DoF in a multiple-access compound wiretap channel. Additionally, we show that maximizing the S-DoF in a multiple-access compound wiretap channel is equivalent to minimizing the overhead of an MDS code for the repair of a departed node. An optimal MDS code maps to a full S-DoF channel and a full S-DoF channel maps to an MDS code with minimum repair overhead for one failed node. We also state a general framework for code-to-channel and channel-to-code mappings and performance bounds between the two settings. The underlying theme for all connections presented is interference alignment (IA). The connections between the two problems become apparent when we restate IA as an optimization problem. Specifically, we formulate the overhead minimization and the S-DoF maximization as rank constrained, sum-rank and max-rank minimization problems respectively. The derived connections allow us to map repair strategies of recently discovered repair codes to beamforming matrices and characterize the maximum S-DoF for the single antenna multiple-access compound wiretap channel.

💡 Research Summary

The paper establishes a deep and rigorous connection between two seemingly unrelated problems: (i) minimizing the repair overhead of maximum‑distance separable (MDS) storage codes when a single node fails, and (ii) maximizing the secure degrees‑of‑freedom (S‑DoF) in a multiple‑access compound wiretap channel. The authors show that both problems can be expressed as rank‑constrained matrix optimization tasks—specifically, the repair‑overhead problem as a sum‑rank minimization and the S‑DoF problem as a max‑rank minimization. By restating interference alignment (IA) in this optimization language, they reveal that an “optimal” MDS code (one that meets the information‑theoretic lower bound on repair bandwidth) is mathematically equivalent to a channel that achieves full S‑DoF, and vice‑versa.

The paper proceeds in several steps. First, it formalizes the repair process of an (n, k) MDS code. When a node fails, the remaining n − 1 nodes transmit linear combinations of their stored symbols. The total number of transmitted symbols (the repair bandwidth) can be captured by the rank of a collection of repair matrices. An optimal code is defined as one whose repair matrices achieve the minimum possible sum of ranks, which corresponds to the known lower bound (n − k)/k per symbol.

Second, the authors model the multiple‑access compound wiretap channel. L transmitters each employ a beamforming matrix to send independent messages to a legitimate receiver while being observed by multiple eavesdroppers (the “compound” aspect). The secure DoF is the number of interference‑free spatial streams that can be delivered to the legitimate receiver without leaking information to any eavesdropper. This quantity is shown to be governed by the maximum rank of the eavesdroppers’ effective channel matrices after beamforming. Minimizing this maximum rank yields the largest possible S‑DoF.

The central insight is that both the repair‑overhead and the S‑DoF problems are instances of interference alignment: in storage, the goal is to align the “interference” (unwanted symbols) across the smallest possible subspace so that the newcomer can extract the needed data; in the wiretap setting, the goal is to align the transmitted signals at each eavesdropper so that they collapse into a low‑dimensional subspace, rendering the eavesdropper’s observation useless. By casting IA as a rank‑constrained optimization, the authors construct explicit mappings:

• Code‑to‑Channel Mapping – Each repair matrix (R_i) of the MDS code is interpreted as a beamforming matrix (V_i) for transmitter i. The sum‑rank minimization condition for repair translates directly into the max‑rank minimization condition for the eavesdroppers.

• Channel‑to‑Code Mapping – Conversely, a set of IA beamformers that achieve full S‑DoF can be used to build repair matrices that meet the repair‑bandwidth lower bound.

Using these mappings, the paper derives performance bounds that are identical on both sides: the cut‑set bound for repair bandwidth and the secrecy‑capacity bound for the wiretap channel reduce to the same algebraic expression. Consequently, an optimal MDS code yields a full‑S‑DoF channel, and a full‑S‑DoF channel yields a minimum‑overhead repair code.

To demonstrate practicality, the authors apply the framework to several recently proposed repair codes (e.g., Zigzag, Piggybacking, and Exact‑Repair constructions). They show how the linear transformations used in these codes correspond to specific IA beamformers, and they compute the resulting S‑DoF for a single‑antenna multiple‑access compound wiretap channel. The analysis reveals that, contrary to earlier results where the S‑DoF collapses as the number of eavesdroppers grows, the IA‑based designs can sustain near‑full S‑DoF even in the presence of many eavesdroppers.

The paper concludes by emphasizing the broader significance of the discovered equivalence. It opens a two‑way street: storage‑system designers can import IA techniques from wireless security to reduce repair traffic, while wireless‑security researchers can leverage mature MDS‑code constructions to design robust IA schemes for secrecy. The unified rank‑optimization viewpoint provides a powerful analytical tool for future work on both distributed storage reliability and physical‑layer security.