Fundamental limits of covert communication over MIMO AWGN channel

Abstract

Fundamental limits of covert communication have been studied for different models of scalar channels. It was shown that, over n independent channel uses, O( √ n) bits can be transmitted reliably over a public channel while achieving an arbitrarily low probability of detection (LPD) by other stations. This result is well known as the square-root law and even to achieve this diminishing rate of covert communication, all existing studies utilized some form of secret shared between the transmitter and the receiver. In this paper, we establish the limits of LPD communication over the MIMO AWGN channel. In particular, using relative entropy as our LPD metric, we study the maximum codebook size for which the transmitter can guarantee reliability and LPD conditions are met. We first show that, the optimal codebook generating input distribution under δ-PD constraint is the zero-mean Gaussian distribution. Then, assuming channel state information (CSI) on only the main channel at the transmitter, we derive the optimal input covariance matrix, hence, establishing scaling laws of the codebook size. We evaluate the codebook scaling rates in the limiting regimes for the number of channel uses (asymptotic block length) and the number of antennas (massive MIMO). We show that, in the asymptotic block-length regime, square-root law still holds for the MIMO AWGN. Meanwhile, in massive MIMO limit, the codebook size, while it scales linearly with √ n, it scales exponentially with the number of transmitting antennas. Further, we derive equivalent results when no shared secret is present. For that scenario, in the massive MIMO limit, higher covert rate up to the non-LPD constrained capacity still can be achieved, yet, with much slower scaling compared to the scenario with shared secret. The practical implication of our result is that, MIMO has the potential to provide a substantial increase in the file sizes that can be covertly communicated subject to a reasonably low delay. Index TermsLPD communication, Covert MIMO Communication, MIMO physical layer security, LPD Capacity. I. INTRODUCTIONConditions for secure communication under a passive eavesdropping attack fall in two broad categories: 1) low probability of intercept (LPI). 2) low probability of detection (LPD). Communication with LPI requires the message exchanged by two legitimate parties to be kept secret from an illegitimate adversary. Meanwhile, LPD constrained communication is more restrictive as it requires the adversary to be unable to decide whether communication between legitimate parties has taken place. Fundamental limits of LPD constrained communication over scalar AWGN has been established in [1] where the square-root law for LPD communication was established. Assuming a shared secret of sufficient length between transmitter and receiver, square-root law states that, over n independent channel uses of an AWGN channel, transmitter can send O( √ n) bits reliably to the receiver while keeping arbitrary low probability of detection at the adversary. In this paper, we stu...

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