The capacity of private information retrieval with colluding databases

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. T-private PIR is a generalization of PIR to include the requirement that even if any T of the N databases collude, the identity of the retrieved message remains completely unknown to them. The information theoretic capacity of T-private PIR (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 T-private PIR is (1 + T/N + T
²
/N
²
+ ... + T
^K-1
/N
^K-1
)
^-1
.