A designated verifier signature (DVS) scheme is a special type of digital signature scheme without the property of non-repudiation. Such schemes only allow the designated verifier to validate the signer's signature. Meanwhile, the designated verifier is not able to convince any third party of the signature’s actual signer, since he can also generate a computationally indistinguishable one compared to the received signature. A strong designated verifier signature (SDVS) scheme further prohibits anyone except for the intended verifier from validating the signature as it requires the designated verifier's private key to finish the verification process. In this paper, we propose an efficient short SDVS scheme with low computation costs. Unlike many SDVS schemes primarily implemented on pairingbased cryptosystems, we focus on the discrete logarithms (DL) based system. Compared with those DL based SDVS schemes, ours provides better functionalities and efficiency and hence benefits the practical applications. In addition, the security requirement of unforgeability against existential forgery on adaptive chosen-message attacks (EU-CMA) is proved in the random oracle model.