Existing CP-ABE schemes incur very large ciphertext size, which increases linearly with respect to the number of attributes in the access policy. Large ciphertext prevents CPABE from being adopted in the communication constrained environments. In this paper, we proposed a new construction of CP-ABE, named Constant-size CP-ABE (denoted as CCP-ABE) that significantly reduces the ciphertext to a constant size for an AND gate access policy with any given number of attributes. Each ciphertext in CCP-ABE requires only 2 elements on a bilinear group. Based on CCP-ABE, we further proposed an Attribute Based Broadcast Encryption (ABBE) scheme. Compared to existing Broadcast Encryption (BE) schemes, ABBE is more flexible because a broadcasted message can be encrypted by an expressive access policy, either with or without explicit specifying the receivers. Moreover, ABBE significantly reduces the storage and communication overhead to the order of O(logN), where N is the system size.