Cloud computing is becoming the preferred solution for efficiently dealing with the increasing amount of genomic data. Yet, outsourcing storage and processing of sensitive data, such as genomic data, comes with important concerns related to privacy and security. This calls for new sophisticated techniques that ensure data protection from untrusted cloud providers and still enables researchers to obtain useful information. We present a novel privacy-preserving algorithm for fully outsourcing the storage of large genomic data files to a public cloud and enable researchers to efficiently search for variants of interest. To preserve data and query confidentiality from possible leakage, our solution exploits optimal encoding for genomic variants and combines it with homomorphic encryption and private information retrieval. The proposed algorithm is implemented in C++ and evaluated on real data as part of the 2016 iDash genome privacy-protection challenge. Results show that our solution outperforms the state-of-the-art and enables researchers to search over millions of encrypted variants in a few seconds. As opposed to prior beliefs that sophisticated privacy-enhancing technologies (PETs) are unpractical for real operational settings, our solution demonstrates that, in the case of genomic data, PETs can represent very efficient enablers.