On lower bounds for the communication complexity of private information retrieval
Private information retrieval for k greater than or equal to 1 databases (denoted by (k,l)-PIR for short) is a protocol that (1) a user sends an l tuple query to each of k noncommunicating replicated databases; (2) each database responds the user with an answer corresponding to the l tuple query; (3) the user privately retrieve any single bit out of the n bits of data stored in ii databases. In this model, "privacy" implies that the user retrieves the bit he is interested in but releases to each database nothing about which bit he wishes to get. In general, the efficiency of (k, l)-PIR is measured by the total amount of bits exchanged between the user and the k databases, but few about its lower bounds are known except for restricted cases. In this paper, we classify (k, l)-PIR into a linear type, a multilinear type, and an affine type with respect to the relationship between queries to each database (made by the user) and answers to the user (made by each database), and show that (1) the lower bound for the communication complexity of any multilinear type (k, l)-PIR is Omega( (l+1)rootn) (Theorem 3.1); (2) the lower bound for the communication complexity of any linear type (k, l)-PIR is Ohm(rootn) (Corollary 3.2); (3) the lower bound for the communication complexity of any affine type (k, l)-PIR is Omega( (l+1)rootn) (Theorem 4.2).
communication complexity private information retrieval linear type multilinear type affine type