Attribute-based encryption is an efficient and flexible fine-grained access control scheme. However, how to realize the attribute privacy concerns in the access policy and optimize the heavy computing overhead have been not adequately addressed. First, in view of the open-access policies formulated by data owners in the cloud environment and the linear growth of bilinear pairing operations with the number of attributes in the decryption process, a verifiable outsourced attribute-based encryption with partial policy hidden scheme is proposed, in which the attribute name of access policy can be sent while attribute value involving sensitive information can be hidden, so nobody can infer information from the access policy. Second, the bilinear pairing operation and modular power operation are outsourced to the cloud server, then users only need to perform constant exponential operation to decrypt. In addition, the proposed scheme is based on the composite order bilinear group and satisfies full secure under the standard model. Finally, compared with other schemes in term of function and performance, it shows that this scheme is more efficient and suitable for resource-constrained mobile devices in outsourcing environment.