Private Information Retrieval over Gaussian MAC

Consider the problem of Private Information Retrieval (PIR), where a user wishes to retrieve a single message from $N$ non-communicating and non-colluding databases (servers). All servers store the same set of $M$ messages and they respond to the user through a block fading Gaussian Multiple Access Channel (MAC). The goal in this setting is to keep the index of the required message private from the servers while minimizing the communication overhead. This work provides joint privacy and channel coding retrieval schemes for the Gaussian MAC with and without fading. The schemes exploit the linearity of the channel while using the Compute and Forward (CF) coding scheme. Consequently, single-user encoding and decoding are performed to retrieve the required message. In the case of a channel without fading, the achievable retrieval rate is shown to outperform a separation-based scheme, in which the retrieval and the channel coding are designed separately. Moreover, this rate equals the channel capacity without privacy constraints for even $N$ (hence the joint scheme is optimal) and has a negligible gap from the capacity for odd $N$ as $N$ grows. When the channel suffers from fading, the asymmetry between the servers' channels forces a more complicated solution, which involves a hard optimization problem. Nevertheless, we provide coding schemes and lower bounds on the expected achievable retrieval rate which are shown to have the same scaling laws as the channel capacity, both in the number of servers and the SNR.