## 1 Introduction
[17] introduced a top-down, levelled, multi-replica, MHT-based data auditing scheme for dynamic big data storage in the cloud, which supports fully dynamic data updates and authentication of block indices. Public verifiability is one of the most important properties for PDP schemes, which means that external verifiers are able to check the integrity of cloud data. With this practical property, cloud users can delegate the rights to check data possession to a third-party auditor (TPA) to do the periodical checking job. However, data privacy may be leaked to the TPA during the checking process, which may cause financial loss for the users who have stored confidential or sensitive data on cloud servers. Recently, a number of schemes [18; 19; 20; 21; 22] have been developed that allow a TPA to check the integrity of the stored data. [PERSON] et al. [20] proposed a privacy-preserving public cloud data auditing system by combining the public key-based homomorphic authenticator with random masking. This scheme explained the definition of data privacy against the TPA. [PERSON] et al. [21] performed further study on data privacy preservation and proposed the notion of 'zero-knowledge public auditing' to defeat off-line guessing threat. [PERSON] et al. [22] enhanced the privacy of remote data integrity-checking protocol for secure cloud storage. The aforementioned schemes [18; 19; 20; 21; 22; 16] all work only in the public key infrastructure (PKI) [23]-based system, which may suffer from heavy public key management. [PERSON] et al. [24] further presented an identity-based PDP scheme to eliminate heavy public key management. This scheme leaks no information of the stored data to the TPA, but cannot protect the privacy of the user's identity. If the data is shared by a group of users, such as a company, the TPA will know all the details of the group users' identities. In order to reduce the complexity of the PDP scheme, many researchers [24; 25; 26; 27] have focused on studying identity-based PDP schemes. By utilizing identity-based cryptography (IBC) [28], there is no need for a PKI to perform complex certificate management such as distribution, storage, revocation, and verification. Unfortunately, IBC has an inherent drawback of key escrow. [PERSON] et al. [29] first proposed a certificateless public auditing mechanism for verifying data integrity in the cloud in order to eliminate the problem of key escrow. In this scheme, Key Generation Center will generate only the partial key, so that in any case it will not compromise the user's private key. [PERSON] et al. [30] introduced a certificateless PDP scheme for shared group data, but it lost the privacy preservation of cloud data and the user's identity to the TPA. **Motivation**: In this paper, we mainly focus on preserving the privacy of group shared data against the third-party auditor (TPA) during the integrity checking process. Suppose there is a group of users from one company who share company data on a given cloud server. Each user of the group can upload and share their data on the cloud server. The manager of the group requires a TPA to periodically check the integrity of the outsourced data, but he does not allow the TPA to extract any information related to their data, not even the identity of the group users. Users of the group may leave the company, so the problem of user revocation from the group needs to be considered. Thus, we need a primitive to meet such requirements and to guarantee the integrity of the outsourced data on the cloud server. **Our contribution**: The contributions of this paper are summarized as follows:  
* First, we propose a new PDP protocol (CL-PGSDP) for group shared data by utilizing certificateless cryptography [31], which eliminates the problems of certificate management and key escrow. * Second, by making use of the idea of zero-knowledge proof protocol, the equality of discrete logarithm [32; 33; 34], and randomization method, we construct a privacy-preserving CL-PGSDP protocol.