Interference Alignment for the Multi-Antenna Compound Wiretap Channel

We study a wiretap channel model where the sender has $M$ transmit antennas and there are two groups consisting of $J_1$ and $J_2$ receivers respectively. Each receiver has a single antenna. We consider two scenarios. First we consider the compound wiretap model – group 1 constitutes the set of legitimate receivers, all interested in a common message, whereas group 2 is the set of eavesdroppers. We establish new lower and upper bounds on the secure degrees of freedom. Our lower bound is based on the recently proposed \emph{real interference alignment} scheme. The upper bound provides the first known example which illustrates that the \emph{pairwise upper bound} used in earlier works is not tight. The second scenario we study is the compound private broadcast channel. Each group is interested in a message that must be protected from the other group. Upper and lower bounds on the degrees of freedom are developed by extending the results on the compound wiretap channel.

💡 Research Summary

The paper investigates a multi‑antenna compound wiretap channel in which a transmitter equipped with (M) antennas communicates with two groups of single‑antenna receivers: (J_{1}) legitimate users (group 1) and (J_{2}) eavesdroppers (group 2). The authors consider two distinct scenarios. In the first, all legitimate users must decode a common confidential message while the eavesdroppers must obtain no information about it. In the second, a compound private broadcast setting is studied where each group wishes to receive its own confidential message that must be kept secret from the other group.

System model and definitions. The received signal at user (i) is (y_i = \mathbf{h}_i^{\top}\mathbf{x}+z_i), where (\mathbf{h}_i\in\mathbb{R}^M) is the channel vector, (\mathbf{x}) the transmitted vector subject to a power constraint, and (z_i) Gaussian noise. Secure degrees of freedom (sDoF) are defined as the pre‑log factor of the secrecy rate as the transmit power (P\to\infty).

Upper bound – breaking the pairwise bound. Earlier works used a “pairwise upper bound” that treats each legitimate‑eavesdropper pair independently, yielding (\text{sDoF}\le \min{J_{1},M-J_{2}}). The authors show this bound is not tight when the total number of users exceeds the number of transmit antennas. By analyzing the rank of the concatenated channel matrix (