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With the wide application of the mobile smart terminals, the data privacy protection of the mobile smart terminals stored in the cloud is more and more important. Public key encryption with keyword search (PEKS) and secure channel free PEKS (SCF-PEKS) have been proposed for public key searchable encryption previously. However, the security of keyword search is far from enough. In addition, these schemes are mostly based on bilinear pairing and the computational efficiency is relatively low. In this paper, we propose a novel non-bilinear pairs SCF-PEKS schemes for mobile smart terminal that offer a high computational efficiency along with better security assurances than that of the existing alternatives. Without random oracle model, we prove the security and privacy of the scheme's keyword ciphertext and keyword trapdoor through the game hopping method. Therefore, the scheme is capable of resisting outside online keyword guessing attack and inside offline keyword guessing attack. Based on the comparison and experimental results, the scheme turns out to be secure and practicable.