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Strong designated verifier signature (generally abbreviated to SDVS) allows signers to obtain absolute control over who can verify the signature, while only the designated verifier other than anyone else can verify the validity of a SDVS without being able to transfer the conviction. Certificateless PKC has unique advantages comparing with certificate-based cryptosystems and identity-based PKC, without suffering from key escrow. Motivated by these attractive features, we propose a novel efficient CL-SDVS scheme without bilinear pairings or map-to-point hash operations. The proposed scheme achieves all the required security properties including EUF-CMA, non-transferability, strongness and non-delegatability. We also estimate the computational and communication efficiency. The comparison shows that our scheme outperforms all the previous CL-(S)DVS schemes. Furthermore, the crucial security properties of the CL-SDVS scheme are formally proved based on the intractability of SCDH and ECDL assumptions in random oracle model.