With the rapid development and wide application of cloud computing and 5G communication, the number of cloud mobile users has increased rapidly. The privacy protection of cloud data is getting more and more attention. Public key encryption scheme with keyword search (PEKS) and secure channel free public key encryption with keyword search (SCF-PEKS) allow any user in the system to send encrypted files to the server for retrieval by the receiver, which plays a certain role of privacy protection. However, Rhee et al. found that the scheme may result in loss of privacy security of keywords in their work. Meanwhile, many of public-key searchable encryption schemes are calculated based on bilinear pairings, and their computational efficiency is limited on battery-limited devices. To address these issue, we propose a non bilinear pairs secure channel free public key authentication encryption with keyword search scheme (NBP-SCF-PAEKS). The scheme has higher computational efficiency than the bilinear pair scheme, and has access control function in keyword retrieval process. Without random oracle model, we prove that the scheme can resist the online keyword guessing attack and the offline keyword guessing attack, by ensuring the multi-keyword ciphertext indistinguishability under adaptive chosen keyword attack and the keyword trapdoor indistinguishability under adaptive chosen keyword attack through game-hopping method. Compared with other schemes, the simulation results show that the scheme is efficient and secure.