ORAMs in a quantum world

Abstract

We study the security of Oblivious Random Access Machines (ORAM) in the quantum world. First we introduce a new formal treatment of ORAMs, which is at the same time elegant and simpler than the known formalization by Goldreich and Ostrovsky. Then we define a new security model for ORAMs, based on a strong, adaptive, game-based security definition, which we show to be at least as strong as other existing notions in the literature. We extend such security notion to the post-quantum setting in the natural way, i.e., by considering classical ORAMs resistant against quantum adversaries. We show a standard quantum attack against an insecure instantiation of PathORAM, one of the most efficient general ORAM constructions to date, introduced by Stefanov et al. On the other hand, we show that PathORAM is post-quantum secure if instantiated using post-quantum underlying primitives. Furthermore, we initiate the study of quantum ORAMs (QORAMs), that is, ORAM constructions meant to be executed between quantum parties acting on arbitrary quantum data. We address many problems arising when formalizing QORAM security through a novel technique of independent interest (which we call safe extractor), modeling a quantum adversary able to extract information from a quantum system in a computationally undetectable way. Finally, we provide a secure QORAM construction (based on PathORAM and a quantum encryption scheme introduced by Alagic et al.) which has the interesting property of making read and write operations inherently equivalent.