Abstract: Based on the bilinear Diffie-Hellman assumption, in 2009, Kang et al. proposed an identity-based strong designated verifier signature scheme which only allows the intended verifier to verify the signature. Besides, the designated verifier is not capable of transferring the conviction to any third party. Their scheme was proved secure in the random oracle model. In this paper, however, we will demonstrate that their scheme is still vulnerable to the universal forgery attack for arbitrarily chosen messages. Moreover, an efficient and provably secure improvement to eliminate the security weakness is presented.