Mapping messages or user's identity into a point on elliptic curves is required in many pairing-based cryptographic schemes. In most of these pairing-based schemes, this requirement is realized by a special hash function called MapToPoint function. However, the efficiency of the MapToPoint function is much lower than the general hash functions. In this paper, we propose a new identity-based signature (IBS) scheme without MapToPoint function, which speeds up extracting the secret key and verifying the signatures. The security of the proposed scheme depends on a complex assumption similar to k-CAA. Another benefit of the proposed scheme is that it supports batch verifications such that multiple signatures of distinct messages for distinct users are verified simultaneously. The results show that batch verifications on the proposed IBS scheme is much faster than other IBS schemes. Furthermore, the proposed scheme is used to construct an efficient chameleon signature scheme by cooperating with an identity-based chameleon hash function.

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