The popularity of mobile social networking services (mSNSs) is propelling more and more businesses, especially those in retailing and marketing, into mobile and location-based forms. To address the trust issue, the service providers are expected to deliver their location-based services in an authenticatable manner, so that the correctness of the service results can be verified by the client. However, existing works on query authentication cannot preserve the privacy of the data being queried, which are sensitive user locations when it comes to location-based services and mSNSs. In this paper, we address this challenging problem by proposing a comprehensive solution that preserves unconditional location privacy when authenticating range queries. Three authentication schemes for $R$-tree and grid-file index, together with two optimization techniques, are developed. Cost models, security analysis, and experimental results consistently show the effectiveness, reliability and robustness of the proposed schemes under various system settings and query workloads.