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In cloud computing, computationally weak users are always willing to outsource costly computations to a cloud, and at the same time they need to check the correctness of the result provided by the cloud. Such activities motivate the occurrence of verifiable computation (VC). Recently, Parno, Raykova and Vaikuntanathan showed any VC protocol can be constructed from an attribute-based encryption (ABE) scheme for a same class of functions. In this paper, we propose two practical and efficient semi-adaptively secure key-policy attribute-based encryption (KP-ABE) schemes with constant-size ciphertexts. The semi-adaptive security requires that the adversary designates the challenge attribute set after it receives public parameters but before it issues any secret key query, which is stronger than selective security guarantee. Our first construction deals with small universe while the second one supports large universe. Both …