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With the development of mobile communication technology, location-based services (LBS) are booming prosperously. Meanwhile privacy protection has become the main obstacle for the further development of LBS. The k-nearest neighbor (k-NN) search is one of the most common types of LBS. In this paper, we propose an efficient private circular query protocol (EPCQP) with high accuracy rate and low computation and communication cost. We adopt the Moore curve to convert two-dimensional spatial data into one-dimensional sequence and encrypt the points of interest (POIs) information with the Brakerski-Gentry-Vaikuntanathan homomorphic encryption scheme for privacy-preserving. The proposed scheme performs the secret circular shift of the encrypted POIs information to hide the location of the user without a trusted third party. To reduce the computation and communication cost, we dynamically divide the table of the POIs information according to the value of k. Experiments show that the proposed scheme provides high accuracy query results while maintaining low computation and communication cost.