Since the ciphertext length of attribute-based encryption scheme from multilinear maps is large, the decryption is inefficient and the scheme has key escrow problem, a key-policy attribute-based encryption scheme from multilinear maps was proposed by using outsourcing technology and user's secret value. The proposed scheme supported general polynomial-size circuit and arbitrary fanout, the private key was generated by key generation center and user. The length of the ciphertext is fixed to | G|+| Z|, compared with the known ciphertext scheme with the minimum ciphertext, the storage cost is decreased by 25% after setting reasonable parameters in accordance with the standards elliptic curves. Users only need to compute transformation ciphertext and the ciphertext is verifiable. The decryption multilinear operation count is only 3, which greatly reduces the computional cost. Selective security is proved in standard model under the multilinear decisional Diffie-Hellman problem. Additionally, it also can be applied in small mobile devices with limited computing capability.