We present scalable and parallel versions of Lipmaa's computationally-private information retrieval (CPIR) scheme [20], which provides log-squared communication complexity. In the proposed schemes, instead of binary decision diagrams utilized in the original CPIR, we employ an octal tree based approach, in which non-sink nodes have eight child nodes. Using octal trees offers two advantages: i) a serial implementation of the proposed scheme in software is faster than the original scheme and ii) its bandwidth usage becomes less than the original scheme when the number of items in the data set is moderately high (e.g., 4,096 for 80-bit security level using Damgard-Jurik cryptosystem). In addition, we present a highly-optimized parallel algorithm for shared-memory multi-core/processor architectures, which minimizes the number of synchronization points between the cores. We show that the parallel implementation is about 50 times faster than the serial implementation for a data set with 4,096 items on an eight-core machine. Finally, we propose a hybrid algorithm that scales the CPIR scheme to larger data sets with small overhead in bandwidth complexity. We demonstrate that the hybrid scheme based on octal trees can lead to more than two orders of magnitude faster parallel implementations than serial implementations based on binary trees. Comparison with the original as well as the other schemes in the literature reveals that our scheme is the best in terms of bandwidth requirement.

We present improved and parallel versions of Lipmaa's computationally-private information retrieval (CPIR) protocol based on a additively-homomorphic cryptosystem. Lipmaa's original CPIR utilizes binary decision diagrams, in which non-sink nodes have two children nodes and the data items to be retrieved are placed in the sink nodes. In our scheme, we employ, instead, quadratic and octal trees, where non-sink nodes have four and eight child nodes, respectively. Using other tree forms, which does not change the asymptotic complexity, results in shallow trees by which we can obtain an implementation that is an order of magnitude faster than the original scheme. We also present a non-trivial parallel algorithm that takes advantage of shared-memory multi-core architectures. Finally, our scheme proves to be highly efficient in terms of bandwidth requirement, the amount of data being exchanged in a run of the CPIR protocol.