In IT outsourcing, a user may delegate the data storage and query processing functions to a third-party server that is not completely trusted. This gives rise to the need to safeguard the privacy of the database as well as the user queries over it. In this article, we address the problem of running ad hoc equi-join queries directly on encrypted data in such a setting. Our contribution is the first solution that achieves constant complexity per pair of records that are evaluated for the join. After formalizing the privacy requirements pertaining to the database and user queries, we introduce a cryptographic construct for securely joining records across relations. The construct protects the database with a strong encryption scheme. Moreover, information disclosure after executing an equi-join is kept to the minimum—that two input records combine to form an output record if and only if they share common join attribute values. There is no disclosure on records that are not part of the join result. Building on this construct, we then present join algorithms that optimize the join execution by eliminating the need to match every record pair from the input relations. We provide a detailed analysis of the cost of the algorithms and confirm the analysis through extensive experiments with both synthetic and benchmark workloads. Through this evaluation, we tease out useful insights on how to configure the join algorithms to deliver acceptable execution time in practice.