Semantic Private Information Retrieval
Sajani Vithana, Karim Banawan, Şennur Ulukuş · IEEE Transactions on Information Theory · 2021
We investigate the problem of semantic private information retrieval (semantic PIR). In semantic PIR, a user retrieves a message out of$K$independent messages stored in$N$replicated and non-colluding databases without revealing the identity of the desired message to any individual database. The messages come withdifferent semantics, i.e., the messages are allowed to havenon-uniform a priori probabilitiesdenoted by$(p_{i}>0,\: i \in [K])$, which are a proxy for their respective popularity of retrieval, andarbitrary message sizes$(L_{i},\: i \in [K])$. This is a generalization of the classical private information retrieval (PIR) problem, where messages are assumed to have equal message sizes. We derive the semantic PIR capacity for general$K$,$N$. The results show that the semantic PIR capacity depends on the number of databases$N$, the number of messages$K$, the a priori probability distribution of messages$p_{i}$, and the message sizes$L_{i}$. We present two achievable semantic PIR schemes: The first one is a deterministic scheme which is based on message asymmetry. This scheme employs non-uniform subpacketization. The second scheme is probabilistic and is based on choosing one query set out of multiple options at random to retrieve the required message without the need for exponential subpacketization. We derive necessary and sufficient conditions for the semantic PIR capacity to exceed the classical PIR capacity with equal priors and sizes. Our results show that the semantic PIR capacity can be larger than the classical PIR capacity when longer messages have higher popularities. However, when messages are equal-length, the non-uniform priors cannot be exploited to improve the retrieval rate over the classical PIR capacity. We provide two extensions of the semantic PIR problem, namely, the semantic PIR from MDS-coded databases and the semantic PIR from colluding databases. For both extensions, we derive the exact PIR capacity in addition to providing a corresponding optimal scheme.