Verifying the integrity of cloud data shared by a group is one of the most common usage of cloud storage integrity checking. In cloud storage integrity checking, the private key which is used to generate data signatures by user may be unavailable because of the damage or the fault of storage medium. However, currently existing cloud storage integrity checking schemes for shared data do not consider this realistic problem. This paper first explores how to deal with the problem of the private key unavailability in cloud storage integrity checking for shared data. A new scheme that enables cloud storage integrity checking for shared data with private key-recovery ability is proposed. In this scheme, when a group user's private key is unavailable, this user's private key can be recovered with the help of t or more users in the group. At the same time, a random masking technology is designed to guarantee the security of participating members' private keys. The user can also verify the correctness of the recovered private key. Finally, the analysis of security and experimental results are provided to show that the proposed scheme is secure and efficient.