Multi-keyword rank searchable encryption (MRSE) returns the top- $k$ k results in response to a data user's request of multi-keyword search over encrypted data, and hence provides an efficient way for preserving data privacy in cloud storage systems while without loss of data usability. Many existing MRSE systems are constructed based on an algorithm which we term as $k$ k -nearest neighbor for searchable encryption (KNN-SE). Unfortunately, KNN-SE has a number of shortcomings, which limit its practical applications. In this paper, we propose a new MRSE system which overcomes almost all the defects of the KNN-SE based MRSE systems. Specifically, our new system does not require a predefined keyword set and supports keywords in arbitrary languages, is a multi-user system which supports flexible search authorization and time-controlled revocation, and it achieves better data privacy protection since even the cloud server is not able to tell which documents are the top- $k$ k results returned to a data user. We also conduct extensive experiments to demonstrate the efficiency of the new system.