Wireless Body Area Network (WBAN) is a highly promising technology enabling health providers to remotely monitor vital parameters of patients via tiny wearable and implantable sensors. In a WBAN, medical data is collected by several tiny sensors and usually transmitted to a server-side (e.g., a cloud service provider) for long-term storage and online/offline processing. However, as the health data includes several sensitive information, providing confidentiality and fine-grained access control is necessary to preserve the privacy of patients. In this paper, we design an attribute-based encryption (ABE) scheme with lightweight encryption and decryption mechanisms. Our scheme enables tiny sensors to encrypt the collected data under an access control policy by performing very few computational operations. Also, the computational overhead on the users in the decryption phase is lightweight, and most of the operations are performed by the cloud server. In comparison with some excellent ABE schemes, our encryption mechanism is more than 100 times faster, and the communication overhead in our scheme decreases significantly. We provide the security definition for the new primitive and prove its security in the standard model and under the hardness assumption of the decisional bilinear Diffie-Hellman (DBDH) problem.