Attribute-based encryption (ABE) is a promising technique for fine-grained access control of encrypted data in a cloud storage, however, decryption involved in the ABEs is usually too expensive for resource-constrained front-end users, which greatly hinders its practical popularity. In order to reduce the decryption overhead for a user to recover the plaintext, Green et al. suggested to outsource the majority of the decryption work without revealing actually data or private keys. To ensure the third-party service honestly computes the outsourced work, Lai et al. provided a requirement of verifiability to the decryption of ABE, but their scheme doubled the size of the underlying ABE ciphertext and the computation costs. Roughly speaking, their main idea is to use a parallel encryption technique, while one of the encryption components is used for the verification purpose. Hence, the bandwidth and the computation cost are doubled. In this paper, we investigate the same problem. In particular, we propose a more efficient and generic construction of ABE with verifiable outsourced decryption based on an attributebased key encapsulation mechanism, a symmetric-key encryption scheme and a commitment scheme. Then, we prove the security and the verification soundness of our constructed ABE scheme in the standard model. Finally, we instantiate our scheme with concrete building blocks. Compared with Lai et al.'s scheme, our scheme reduces the bandwidth and the computation costs almost by half. IndexTerms-Attribute-based encryption, outsourced decryption, verifiability, access control.