Cryptanalysis of a Lattice-Based PIR Scheme for Arbitrary Database Sizes

Private Information Retrieval (PIR) schemes enable users to securely retrieve files from a server without disclosing the content of their queries, thereby preserving their privacy. In 2008, Melchor and Gaborit proposed a PIR scheme that achieves a balance between communication overhead and server-side computational cost. However, for particularly small databases, Liu and Bi identified a vulnerability in the scheme using lattice-based methods. Nevertheless, the rapid increase in computational cost associated with the attack limited its practical applicability, leaving the scheme's overall security largely intact. In this paper, we present a novel two-stage attack that extends the work of Liu and Bi to databases of arbitrary sizes. To this end, we employ a binary-search-like preprocessing technique, which enables a significant reduction in the number of lattice problems that need to be considered. Specifically, we demonstrate how to compromise the scheme in a matter of minutes using an ordinary laptop. Our findings are substantiated through both rigorous analytical proofs and comprehensive numerical experiments.