Cloud storage is a cost-effective platform to accommodate massive data at low cost. However, advances of cloud services propel data generation, which pushes storage servers to its limit. Deduplication is a popular technique enjoyed by most current cloud servers, which detects and deletes redundant data to save storage and bandwidth. For security concerns, proof-of-ownership (PoW) can be used to guarantee ownership of data such that no malicious user could pass deduplication easily or utilize such mechanism for malicious purposes. Generally, PoW is implemented in static data archive where the data file is supposed to be read-only. However, to satisfy users' needs for dynamical manipulation on data and support real-time data services, it is required to devise efficient PoW for dynamic archive. In this work, we propose the notion of bidirectional and malleable proof-of-ownership (BM-PoW) for the above challenge. Our proposed BM-PoW consists of bidirectional PoW (B-PoW), malleable PoW (M-PoW) and dispute arbitration protocol DAP. We provide the security analysis of our proposal, and performance evaluation that suggests our proposed B-PoW is secure and efficient for large file in static data archive. In addition, our proposed M-PoW achieves acceptable performance under dynamic setting where data is supposed to be outsourced first and updated later in dynamic data archive.