### Some results on query processes and reconstruction functions for unconditionally secure 2-server 1-round binary private information retrieval protocols

D. R. Stinson, R. Wei
2007 Journal of Mathematical Cryptology
In this paper, we investigate query processes and reconstruction functions for unconditionally secure 2-server 1-round binary private information retrieval (PIR) schemes. We begin by formulating a simplified model for PIR schemes which is equivalent to the usual model. We show that a query is equivalent to a boolean function of two variables, and we give a precise characterization of the boolean functions that can be used as "query pairs" to the two servers. We also consider several notions of
more » ... several notions of "privacy" and we make a few remarks about the communication complexity of PIR schemes. deterministic function of Q i and X. 3. Given two responses R 1 and R 2 , U attempts to infer the value of x j . More precisely, U has a reconstruction function rec which, when given an index j, two suitable queries, and their responses, computes the value of x j . Thus we desire that 4. The scheme is private, which means that the query Q 1 should not reveal the value of j to S 1 , and the query Q 2 should not reveal the value of j to S 2 . (Various types of privacy will be defined formally in Section 4.) Example 1.1. We describe a PIR scheme from . On input j, U chooses a random subset Q 1 ⊆ {1, . . . , n} such that |Q 1 | is even. Then U defines Q 2 = Q 1 ∆{j}, where ∆ denotes the symmetric difference of two sets. The responses are defined to be R i = j∈Qi x j mod 2, for i = 1, 2. Finally, U computes x j = R 1 + R 2 mod 2. Remark 1.2. There is one slight difference between our model and the model defined in  . In  , the reconstruction function is also given as input the random number rand . In the schemes we study, rand is used to select a query pair, but once a query pair is selected, the value of rand is no longer relevant.