Towards communication-efficient quantum oblivious key distribution

Symmetrically private information retrieval, a fundamental problem in the field of secure multiparty computation, is defined as follows: A database $D$ of $N$ bits held by Bob is queried by a user Alice who is interested in the bit ${D}_{b}$ in such a way that (1) Alice learns ${D}_{b}$ and only ${D}_{b}$ and (2) Bob does not learn anything about Alice's choice $b$. While solutions to this problem in the classical domain rely largely on unproven computational complexity theoretic assumptions, it is also known that perfect solutions that guarantee both database and user privacy are impossible in the quantum domain. Jakobi et al. [Phys. Rev. A 83, 022301 (2011)] proposed a protocol for oblivious transfer using well-known quantum key device (QKD) techniques to establish an oblivious key to solve this problem. Their solution provided a good degree of database and user privacy (using physical principles like the impossibility of perfectly distinguishing nonorthogonal quantum states and the impossibility of superluminal communication) while being loss-resistant and implementable with commercial QKD devices (due to the use of the Scarani-Acin-Ribordy-Gisin 2004 protocol). However, their quantum oblivious key distribution (QOKD) protocol requires a communication complexity of $O(N\mathrm{log}N)$. Since modern databases can be extremely large, it is important to reduce this communication as much as possible. In this paper, we first suggest a modification of their protocol wherein the number of qubits that need to be exchanged is reduced to $O(N)$. A subsequent generalization reduces the quantum communication complexity even further in such a way that only a few hundred qubits are needed to be transferred even for very large databases.