Nowadays, many customers and enterprises backup their data to cloud storage that performs deduplication to save storage space and network bandwidth. Hence, how to perform secure deduplication becomes a critical challenge for cloud storage. According to our analysis, the state-of-the-art secure deduplication methods are not suitable for cross-user finegrained data deduplication. They either suffer brute-force attacks that can recover files falling into a known set, or incur large computation (time) overheads. Moreover, existing approaches of convergent key management incur large space overheads because of the huge number of chunks shared among users.Our observation that cross-user redundant data are mainly from the duplicate files, motivates us to propose an efficient secure deduplication scheme SecDep. SecDep employs User-Aware Convergent Encryption (UACE) and Multi-Level Key management (MLK) approaches. (1) UACE combines cross-user file-level and inside-user chunk-level deduplication, and exploits different secure policies among and inside users to minimize the computation overheads. Specifically, both of file-level and chunklevel deduplication use variants of Convergent Encryption (CE) to resist brute-force attacks. The major difference is that the file-level CE keys are generated by using a server-aided method to ensure security of cross-user deduplication, while the chunklevel keys are generated by using a user-aided method with lower computation overheads. (2) To reduce key space overheads, MLK uses file-level key to encrypt chunk-level keys so that the key space will not increase with the number of sharing users. Furthermore, MLK splits the file-level keys into share-level keys and distributes them to multiple key servers to ensure security and reliability of file-level keys.Our security analysis demonstrates that SecDep ensures data confidentiality and key security. Our experiment results based on several large real-world datasets show that SecDep is more timeefficient and key-space-efficient than the state-of-the-art secure deduplication approaches.