Abstract: In the existing adaptively secure ABE (attribute-based encryption) schemes, the decryption cost goes linearly with the number of attributes that are used in decryption. An adaptively secure key-policy ABE (FKP-ABE) scheme with fast decryption is proposed, where constant size of computation overhead is implemented in decryption. This scheme is constructed on composite order bilinear groups and supports any monotonic access structure that can be expressed by LSSS (linear secret sharing schemes). The proposed scheme is proved to be adaptively secure in the standard model.