With the growth of third-party cloud storage services, people often store a large amount of data on the cloud servers. However, to ensure the privacy, the sensitive data need to be encrypted before being uploaded to the cloud. Public-key encryption with keyword search (PEKS) is a promising technique that enables users to search target encrypted data by keywords. Unfortunately, most PEKS schemes are vulnerable to inside keyword guessing attacks (KGA). To solve this issue, Huang and Li introduced the concept of public-key authenticated encryption with keyword search (PAEKS), where the sender not only encrypts a keyword, but also authenticates it. However, our cryptanalysis indicated that the efficiency of most existing PAEKS schemes could be improved. Then, we present a novel pairing-free PAEKS scheme based on elliptic curves in this paper. The security proofs show that our scheme can achieve higher computational efficiency and enhanced security models, namely the multi-ciphertext indistinguishability (MCI) and trapdoor privacy (TP). Furthermore, the comparative results demonstrate the efficiency of our proposed scheme is three to thirteen times higher than that of other PAEKS schemes.