We propose an efficient Key-policy Attribute-based Encryption (KP-ABE)
scheme for general (monotone) Boolean circuits based on secret sharing and on a very particular and simple form of leveled multilinear maps,
called chained multilinear maps. The number of decryption key components is substantially reduced in comparison with the current scheme based on leveled multilinear maps, and the size of the multilinear map (in terms of bilinear map components) is less than the Boolean circuit depth, while it is quadratic in the Boolean circuit depth for the current scheme based on leveled multilinear map. Moreover, it is much easier to find chained multilinear maps than leveled multilinear maps. Selective security of the proposed schemes in the standard model is proved, under the decisional multilinear Diffie-Hellman assumption.