Summary: Recently Lewko and Waters proposed the first fully secure multi-authority ciphertext-policy attribute-based encryption (CP-ABE) system in the random oracle model, and leave the construction of a fully secure multi-authority CP-ABE in the standard model as an open problem. Also, there is no CP-ABE system which can completely prevent individual authorities from decrypting ciphertexts. In this paper, we propose a new multi-authority CP-ABE system which addresses these two problems positively. In this new system, there are multiple Central Authorities (CAs) and Attribute Authorities (AAs), the CAs issue identity-related keys to users and are not involved in any attribute related operations, AAs issue attribute-related keys to users and each AA manages a different domain of attributes. The AAs operate independently from each other and do not need to know the existence of other AAs. Messages can be encrypted under any monotone access structure over the entire attribute universe. The system is adaptively secure in the standard model with adaptive authority corruption, and can support large attribute universe.