Single Database Private Information Retrieval Implies Oblivious Transfer

A Single-Database Private Information Retrieval (PIR) is a protocol that allows a user to privately retrieve from a database an entry with as small as possible communication complexity. We call a PIR protocol non-trivial if its total communication is strictly less than the size of the database. Non-trivial PIR is an important cryptographic primitive with many applications. Thus, understanding which assumptions are necessary for implementing such a primitive is an important task, although (so far) not a well-understood one. In this paper we show that any non-trivial PIR implies Oblivious Transfer, a far better understood primitive. Our result not only significantly clarifies our understanding of any non-trivial PIR protocol, but also yields the following consequences: Any non-trivial PIR is complete for all two-party and multi-party secure computations. There exists a communication-efficient reduction from any PIR protocol to a 1-out-of-n Oblivious Transfer protocol (also called SPIR). There is strong evidence that the assumption of the existence of a one-way function is necessary but not sufficient for any non-trivial PIR protocol