Recent years have witnessed the trend of increasingly relying on remote and distributed infrastructures. This increases the complexity of access control to data, where access control policies should be flexible and distinguishable among users with different privileges. In this paper, we present EMA-LAB, a novel Multi Authorisation Level Attribute Based Access Control with short ciphertexts size. It relies on the usage of a constant-size threshold attribute based encryption scheme. The EMA-LAB scheme is multifold. First, it ensures a selective access to encrypted data with respect to different security levels. Second, the proposed construction protects the secrecy of enciphered contents against malicious adversaries, even in case of colluding users. Third, EMA-LAB relies on low computation and communication processes, mainly for resource-constrained devices, compared to most closely related schemes.