Signal Space Alignment for an Encryption Message and Successive Network Code Decoding on the MIMO K-way Relay Channel

This paper investigates a network information flow problem for a multiple-input multiple-output (MIMO) Gaussian wireless network with $K$-users and a single intermediate relay having $M$ antennas. In this network, each user intends to convey a multicast message to all other users while receiving $K-1$ independent messages from the other users via an intermediate relay. This network information flow is termed a MIMO Gaussian $K$-way relay channel. For this channel, we show that $\frac{K}{2}$ degrees of freedom is achievable if $M=K-1$. To demonstrate this, we come up with an encoding and decoding strategy inspired from cryptography theory. The proposed encoding and decoding strategy involves a \textit{signal space alignment for an encryption message} for the multiple access phase (MAC) and \textit{zero forcing with successive network code decoding} for the broadcast (BC) phase. The idea of the \emph{signal space alignment for an encryption message} is that all users cooperatively choose the precoding vectors to transmit the message so that the relay can receive a proper encryption message with a special structure, \textit{network code chain structure}. During the BC phase, \emph{zero forcing combined with successive network code decoding} enables all users to decipher the encryption message from the relay despite the fact that they all have different self-information which they use as a key.