In clinical genomics, the continuous evolution of bioinformatic algorithms and sequencing platforms makes it beneficial to store patients' complete aligned genomic data in addition to variant calls relative to a reference sequence. Due to the large size of human genome sequence data files (varying from 30 GB to 200 GB depending on coverage), two major challenges facing genomics laboratories are the costs of storage and the efficiency of the initial data processing. In addition, privacy of genomic data is becoming an increasingly serious concern, yet no standard data storage solutions exist that enable compression, encryption, and selective retrieval. Here we present a privacy-preserving solution named SECRAM (elective retrieval on Encrypted and Compressed Reference oriented Alignment Map) for the secure storage of compressed aligned genomic data. Our solution enables selective retrieval of encrypted data and improves the efficiency of downstream analysis (e.g., variant calling). Compared with BAM, the de facto standard for storing aligned genomic data, SECRAM uses 18% less storage. Compared with CRAM, one of the most compressed nonencrypted formats (using 34% less storage than BAM), SECRAM maintains efficient compression and downstream data processing, while allowing for unprecedented levels of security in genomic data storage. Compared with previous work, the distinguishing features of SECRAM are that (1) it is position-based instead of read-based, and (2) it allows random querying of a subregion from a BAM-like file in an encrypted form. Our method thus offers a space-saving, privacy-preserving, and effective solution for the storage of clinical genomic data.