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Users of mobile services wish to retrieve nearby points of interest without disclosing their locations to the services. This article addresses the challenge of optimizing the query performance while satisfying given location privacy and query accuracy requirements. The article's proposal, SpaceTwist, aims to offer location privacy for k nearest neighbor (kNN) queries at low communication cost without requiring a trusted anonymizer. The solution can be used with a conventional DBMS as well as with a server optimized for location-based services. In particular, we believe that this is the first solution that expresses the server-side functionality in a single SQL statement. In its basic form, SpaceTwist utilizes well-known incremental NN query processing on the server. When augmented with a server-side granular search technique, SpaceTwist is capable of exploiting relaxed query accuracy guarantees for obtaining better performance. We extend SpaceTwist with so-called ring ranking, which improves the communication cost, delayed termination, which improves the privacy afforded the user, and the ability to function in spatial networks in addition to Euclidean space. We report on analytical and empirical studies that offer insight into the properties of SpaceTwist and suggest that our proposal is indeed capable of offering privacy with very good performance in realistic settings.