Identity-based encryption (IBE) allows one party to send ciphered messages to another using an arbitrary identity string as an encryption key. Since IBE does not require prior generation and distribution of keys, it greatly simplifies key management in public-key cryptography. According to the Menezes–Okamoto–Vanstone (MOV) reduction theory, the IBE scheme based on bilinear map loses the high efficiency of elliptic curve because of the requirement of large security parameters. Therefore, it is important to build a provably secure IBE scheme without bilinear map. To this end, this paper proposes an improved IBE scheme that is different from the previous schemes because this new scheme does not use symmetric encryption algorithm. Furthermore, it can be proven to be secure against adaptively chosen identity and chosen plaintext attacks in the standard model. Elaborated security and performance analysis demonstrate that this new scheme outperforms the previous ones in terms of the time complexity for encryption and decryption.