Private information retrieval for k≥1 databases (denoted by (k,l)-PIR) 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 k 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 Ω({sup left}(l+1) the square root of n) (Theorem 3.1); (2) the lower bound for the communication complexity of any linear type (k,l)-PIR is Ω(Corollary 3.2); (3) the lower bound for the communication complexity of any affine type (k,l)-PIR is Ω({sup left}(l+1) the square root of n) (Theorem 4.2).
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