An Ω(n1/3) Lower Bound for Bilinear Group-Based Private Information Retrieval∗

A two-server private information retrieval (PIR) scheme allows a user U to retrieve the i-th bit of an n-bit string x replicated on two servers while each server individually learns no information about i. The main parameter of interest in a PIR scheme is its communication complexity: the number of bits exchanged by the user and the servers. Substantial effort has been invested by researchers over the last decade in the search for efficient PIR schemes. A number of different schemes (Chor et al., 1998, Beimel et al., 2005, Woodruff and Yekhanin, CCC’05) have been proposed; however, all of them result in the same communication complexity of O(n1/3). The best known lower bound to date is 5 logn by Wehner and de Wolf (ICALP’05). The tremendous gap between upper and lower bounds is the focus of our paper. We show an Ω(n1/3) lower bound in a restricted model that nevertheless captures all known upper bound techniques. ∗A preliminary version of this paper appeared in the proceedings of the 47th IEEE Symposium on Foundations of Computer Science (FOCS’06) [15]. †Supported by the Charles Simonyi Endowment and NSF grant ITR-0324906. ‡Supported by NSF grant CCR 0219218. ACM Classification: H.3.3, F.1.3.e, F.1.2.b AMS Classification: 68P20, 68Q17, 20C20