Quantum cryptography and quantum search algorithm are considered as two important research topics in quantum information science.In this paper,we propose a asymmetrical quantum encryption protocol based on the properties of quantum one-way function and quantum search algorithm.Depending on the no-cloning theorem and trapdoor one-way functions of the public-key,the eavesdropper cannot extract any private-information from the public-keys and ciphertexts.Introducing key-generation randomized logarithm to improve security of our proposed protocol,i.e.,one private-key corresponds to an exponential number of public-keys and every quantum public-key is an unknown and non-orthogonal quantum state to the sender.Using unitary operations and the single photon measurement,secret messages can be directly sent from the sender to the receiver.The security of the proposed protocol is proved that it is information-theoretically secure.Furthermore,compared the symmetrical Quantum key distribution,our protocol is not only efficient to reduce additional communication,but also easier to carry out in practice,because no entangled states and complex operations are required.