Mobile cloud computing is a revolutionary computing paradigm for mobile applications, which enables storage and computation migration from mobile users to resource-rich and powerful cloud server. This migration causes some privacy issues in providing secure data storage, fine-grained access control and anonymity of users. The existing multi-authority ciphertext policy attribute based encryption (CP-ABE) access control scheme guarantees the confidentiality of sensitive data in the cloud server and provides fine-grained access control using defined policies. However it costs too much computation time on encryption and decryption and consumes enormous power resources, making it unsuitable for the mobile devices which are usually equipped with a limited power support. To cope with these challenging concerns, this paper proposes a new data access control scheme for cloud computing by using a new cryptographic primitive known as online/offline multi-authority ABE and the transform key technique. This scheme implements fine-grained access of data and reduces online computation cost of the encryption and decryption on the user side. The proposed scheme acquires user's secret key received from different authorities. That results in protecting privacy of each user against single authority. At last, the security and performance analysis demonstrate that this scheme has high security in terms of data confidentiality and high efficiency in terms of online computation cost.