Attribute-based encryption (ABE) is a promising encryption for fine-grained sharing of ciphertext based on users' attributes. One drawback of ABE is that the encryption and decryption computational costs grow with the number of attributes and the complexity of the access policy. In scenarios where mobile devices are required, it will make encryption and decryption a possible bottleneck for these applications. To largely eliminate computational overhead for mobile devices, we propose a novel attribute-based online/offline encryption with outsourcing decryption (ABOOE-OD). First, the encryption process is split into the offline and online phases. Second, we modify the key generation and decryption algorithms to handle the ciphertext, and outsource the majority of decryption to cloud servers without compromising its security. Our first ABOOE-OD scheme is proven to be secure against the chosen-plaintext attack. We present an attribute-based online/off- line KEM with outsourcing decapsulation (ABOOKEM-OD), and construct a generic transformation to achieve CCA security for ABOOE-OD from any ABOOKEM-OD with one-wayness. Finally, we compare our schemes with previous schemes, and provide an implementation of our scheme. The performance measurements indicate ABOOE-OD is very suitable for mobile devices to perform the online encryption and decryption operations without significantly draining the battery.