Cloud storage is an efficient way for data owner (DO) to store data, and cloud service provider (CSP) periodically provides data integrity proof to DO for integrity auditing. Most existing integrity auditing schemes delegate the third party auditor (TPA) to participate in. However, dishonest TPA may conspire with CSP to deceive DO. Besides, most existing deduplication schemes only consider the deduplication of ciphertext, and rarely consider the deduplication of authentication tags. In this paper, we propose a blockchain assisted integrity auditing scheme with secure deduplication for cloud storage. Our scheme achieves unpredictable challenge messages and avoids collusion between malicious entities. We support less integrity proofs generation for duplicate data owned by multiple data owners, which significantly improves the auditing efficiency. Data owners can benefit from unpredictable challenge messages, low-overhead proof-of-ownership and efficient verification process. Security analysis and performance evaluation provide security assurance and practical evidence of our scheme, respectively. The experimental results show that our scheme is secure with moderate performance overhead.