IEEE In a cloud storage system, to allow searching over encrypted data by the cloud server, searchable symmetric encryption (SSE) has been proposed in the literature. The basic security requirement of SSE is that the cloud server learns no information about the files or keywords during the searching process. Some SSE schemes also offer additional functionalities such as detecting cheating behaviour of a malicious server (i.e., verifiability) and allowing update (e.g., modifying, deleting and adding) of documents on the server. However, the previous (verifiable) SSE schemes were mainly designed for single users, which means the searching can only be done by the data owner, whereas in reality people often use cloud storage to share files with other users. In this paper we present a multi-user verifiable searchable symmetric encryption (MVSSE) scheme that achieves all the desirable features of a verifiable SSE and allows multiple users to perform searching. We then define an ideal functionality for MVSSE under the Universally Composable (UC-) security framework and prove that our ideal functionality implies the security requirements of a secure MVSSE, and our multi-user verifiable SSE scheme is UC-secure. We also implement our scheme to verify its high performance based on some real dataset.