On PIR and SPIR Over Gaussian MAC

Abstract: This paper revisits the problems of Private Information Retrieval (PIR) and Symmetric PIR (SPIR). In PIR, there are $N$ replicated non-communicating databases containing the same $M$ messages and a user wishing to retrieve one message without revealing the message's index to the databases. SPIR extends this notion further by additionally protecting the privacy of the databases, ensuring that the user learns no information beyond the requested message. However, we assume a block-fading Additive White Gaussian Noise Multiple Access Channel (AWGN MAC) linking the user and the databases.} Previous work \cite{shmuel2021private} presented a joint channel-PIR scheme utilizing the Compute and Forward (C&F) protocol, demonstrating the potential of a joint PIR-channel coding scheme over a separated one, yet still lagging behind the channel capacity and requiring significant computational complexity. We propose an improved scheme that offers reduced computational complexity while improving the achievable rate for finite parameters and its scaling laws. Specifically, the achievable rate outperforms the C&F-based approach and scales with the number of databases $N$ and the power $P$ similarly to the channel capacity scaling laws \textit{without the privacy constraint}. Furthermore, the analysis demonstrates that the improved rate exhibits only a finite gap from this unconstrained channel capacity -- $1$ $bit/sec/Hz$ as $N$ increases. \textcolor{black}{Additionally, we provide two SPIR schemes. The first is a modification for our PIR scheme to attain SPIR, which is accomplished by introducing shared common randomness among databases. The second is a novel joint channel-SPIR scheme that utilizes the channel and lattice codes' characteristics to nontrivially achieve SPIR without using common randomness.