Strong designated verifier signature scheme could make it possible for a signer to convince only the designated verifier that the signature is made by the signer. In 2007, Lee et al. proposed a designated verifier signature scheme with message recovery, attaching target message with signature, to prevent any attack by eavesdropping. In this paper, we propose an efficient strong designated verifier signature scheme, using key distribution mechanism where both sender and designated receiver share encryption/decryption key to fulfill encryption/decryption algorithm with low cost of communication and computation: 30% and 50% of Lee et al. scheme, respectively. We would prove its security based on public problem-Gap Diffie-Hellman (GDH) assumption.