On the Secure Degrees of Freedom of 2 x 2 x 2 Multi-Hop Network with Untrusted Relays

We study the impact of untrusted relays on the degrees of freedom of multi-antenna multi-hop networks. In par- ticular, we consider the two user two-hop interference network, where two source nodes want to send independent messages securely to their designated receivers through the help of two untrusted relays. The relays are considered untrusted in terms of eavesdropping the messages sent by the sources. Moreover, we also assume that the messages are confidential, i.e., each receiver must not be able to decode the information meant for the other receiver. We assume that all the terminals (i.e., sources, relays, and the receivers) are equipped with multiple number of antennas. The goal of this work is to understand the secure degrees of freedom (SDoF) region of this multi-hop MIMO network under the two constraints of a) untrusted relays; and b) confidential messages. To cope with the untrusted nature of relays, we present achievable schemes in which both sources mix their information symbols with artificial noises so that the signals at each relay are completely immersed in the artificial noises space. However, this mixing must be done carefully, so as to ensure the feasibility of interference neutralization in the second hop to allow successful decoding at the respective destination. To this end, we devise transmission schemes based on interference alignment and interference neutralization techniques. The main contributions of this work are as follows: a) we present an upper bound on the SDoF region as a function of the number of antennas at the terminals, b) we present two achievable schemes, the first scheme is based on secure interference alignment and neutralization and is shown to be information theoretically optimal when all terminals have the same number of antennas; and a second scheme, based on secure sub-space alignment and neutralization, which is shown to be optimal for another specific antenna configuration. To the best of our knowledge, these are the first results on multi-hop MIMO relay networks with untrusted relays and confidential messages.