To enhance data users’ search experience, reduce the computation overhead of end-user devices and improve data retrieval efficiency in the Key-Policy Attribute-Based Encryption (KP-ABE) framework, we propose a novel Boolean query searchable KP-ABE scheme (which we call FineDReaM) that supports simultaneously the functionalities: (i) expressive Boolean policy search over encrypted data, (ii) preserving keyword privacy, (iii) authorized search, (iv) outsourced decryption, (v) verification of search results and outsourced decryption. Our FineDReaM is secure against keyword guessing attacks on both search token and ciphertext. Only authorized users are allowed to obtain the data retrieval service offered by the cloud server. Our design makes the decryption cost of end users lightweight and constant. The cloud server cannot deceive data users into accepting wrong search results. FineDReaM achieves a non-interactive verifiability framework. Precisely, a data user can verify the correctness of the search results and outsourced decryption without interacting with any third party. We formalize more general security definitions of FineDReaM by taking into account various possible adversarial capabilities. The presented security analysis exhibits that FineDReaM can realize data confidentiality, keyword privacy and verifiability in the random oracle model.