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arXiv:2401.15912 · IEEE Transactions on Communications

On PIR and SPIR Over Gaussian MAC

Or Elimelech, Asaf Cohen

Published Aug 11, 2025• 2 citations
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Abstract

This paper revisits the problems of Private Information Retrieval (PIR) and Symmetric PIR (SPIR). In PIR, a user retrieves a desired message fromNreplicated, non-communicating databases, each storing the sameMmessages, while preserving the privacy of the requested message index. SPIR extends this notion further by additionally protecting the privacy of the databases, ensuring that the user learns no information beyond the requested message. In this paper, we assume a block-fading Additive White Gaussian Noise Multiple Access Channel (AWGN MAC) linking the user and the databases. Previous work 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, as well as its scaling laws. Specifically, the achievable rate outperforms the C&F-based approach and scales with the number of databasesNand the powerPsimilarly to the channel capacitywithout the privacy constraint. Furthermore, the analysis demonstrates that the improved rate exhibits only a finite gap from this unconstrained channel capacity –$1~bit/sec/Hz$asNincreases. Finally, we provide two SPIR schemes. The first is a modification for our PIR scheme to attain SPIR with no rate loss, 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 requiring common randomness, at the price of a loss in the achievable rate.

DOI
10.1109/tcomm.2025.3597809

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Private information retrieval (PIR) is the problem of retrieving, as efficiently as possible, one out of K messages from N non-communicating replicated databases (each holds all K messages) while keeping the identity of the desired message index a secret from each individual database. Symmetric PIR (SPIR) is a generalization of PIR to include the requirement that beyond the desired message, the user learns nothing about the other K - 1 messages. The information theoretic capacity of SPIR (equivalently, the reciprocal of minimum download cost) is the maximum number of bits of desired information that can be privately retrieved per bit of downloaded information. We show that the capacity of SPIR is 1-1/N regardless of the number of messages K, if the databases have access to common randomness (not available to the user) that is independent of the messages, in the amount that is at least 1/(N -1) bits per desired message bit. Otherwise, if the amount of common randomness is less than 1/(N -1) bits per message bit, then the capacity of SPIR is zero. Extensions to the capacity region of SPIR and the capacity of finite length SPIR are provided.
Private information retrieval (PIR) is the problem of retrieving as efficiently as possible, one out of K messages from N non-communicating replicated databases (each holds all K messages) while keeping the identity of the desired message index a secret from each individual database. Symmetric PIR (SPIR) is a generalization of PIR to include the requirement that beyond the desired message, the user learns nothing about the other K - 1 messages. The information theoretic capacity of SPIR (equivalently, the reciprocal of minimum download cost) is the maximum number of bits of desired information that can be privately retrieved per bit of downloaded information. We show that the capacity of SPIR is 1-1/N regardless of the number of messages K, if the databases have access to common randomness (not available to the user) that is independent of the messages, in the amount that is at least 1/(N -1) bits per desired message bit, and zero otherwise.
Private information retrieval (PIR) is an important protocol in cryptographic community to ensure privacy of the user when retrieving information from untrusted databases. In a variation to this problem, symmetric private information retrieval (SPIR) is a protocol which ensures the privacy of user as well as databases. A major drawback for the majority of existing works on information theoretic PIR and SPIR is that they focus on scenarios where all databases store the entire data set (i.e. replicated databases). The goal of this thesis is to present PIR and SPIR schemes for storage constrained databases. For both PIR and SPIR, we present a systematic approach to utilize storage constrained databases. The technical contribution involves two main parts: (a) a careful storage scheme to split and store messages across databases; and (b) construction of information theoretic PIR and SPIR schemes from such databases. We also compare our results to the fully replicated scenario, as well as compare the differences between the communication overhead of PIR and SPIR schemes.
In symmetric private information retrieval (SPIR), a user communicates with multiple servers to retrieve from them a message in a database, while not revealing the message index to any individual server (user privacy), and learning no additional information about the database (database privacy). We study the problem of SPIR on graph-replicated database systems, where each node of the graph represents a server and each link represents a message. Each message is replicated at exactly two servers; those at which the link representing the message is incident. To ensure database privacy, the servers share a set of common randomness, independent of the database and the user's desired message index. We study two cases of common randomness distribution to the servers: i) graph-replicated common randomness, and ii) fully-replicated common randomness. Given a graph-replicated database system, in i), we assign one randomness variable independently to every pair of servers sharing a message, while in ii), we assign an identical set of randomness variable to all servers, irrespective of the underlying graph. In both settings, our goal is to characterize the SPIR capacity, i.e., the maximum number of desired message symbols retrieved per downloaded symbol, and quantify the minimum amount of common randomness required to achieve the capacity. To this goal, in setting i), we derive a general lower bound on the SPIR capacity, and show it to be tight for path and regular graphs through a matching converse. Moreover, we establish that the minimum size of common randomness required for SPIR is equal to the message size. In setting ii), the SPIR capacity improves over the first, more restrictive setting. We show this through capacity lower bounds for a class of graphs, by constructing SPIR schemes from PIR schemes.
Private information retrieval (PIR) is a database query protocol that provides user privacy in that the user can learn a particular entry of the database of his interest but his query would be hidden from the data centre. Symmetric private information retrieval (SPIR) takes PIR further by additionally offering database privacy, where the user cannot learn any additional entries of the database. Unconditionally secure SPIR solutions with multiple databases are known classically, but are unrealistic because they require long shared secret keys between the parties for secure communication and shared randomness in the protocol. Here, we propose using quantum key distribution (QKD) instead for a practical implementation, which can realise both the secure communication and shared randomness requirements. We prove that QKD maintains the security of the SPIR protocol and that it is also secure against any external eavesdropper. We also show how such a classical-quantum system could be implemented practically, using the example of a two-database SPIR protocol with keys generated by measurement device-independent QKD. Through key rate calculations, we show that such an implementation is feasible at the metropolitan level with current QKD technology.
Information-theoretically secure Symmetric Private Information Retrieval (SPIR) is known to be infeasible over noiseless channels with a single server. Previous solutions in-volved additional resources such as database replication, shared randomness, or noisy channels. This paper demonstrates that, using a noiseless multiple access channel, SPIR with information-theoretic security guarantees is feasible without shared random-ness, a noisy channel, or data replication. Specifically, we leverage a noiseless binary adder channel and employ two non-colluding servers with independent content. Furthermore, we characterize the optimal file rates, i.e., the file lengths normalized by the number of channel uses, that can be transferred.