Blockchain is a distributed ledger in which a database is distributed across numerous users. Blockchain technologies have recently come to the forefront of the research and industrial communities as they bring potential benefits for many industries. A ring signature is a special type of digital signature which has been widely adopted to protect anonymity and privacy in many cryptocurrencies and blockchain applications. Especially, a unique ring signature offers a special feature enabling the ability to determine whether a signer produces two different ring signatures of the same message with respect to the same ring. The signature size of the previous constructions of unique ring signature is large and grows linearly/sublinearly with the number of ring members. In this paper, we propose a more efficient unique ring signature with logarithmic size. We prove that our scheme is secure under the Decisional Diffie-Hellman and Discrete Logarithm Assumptions, and provide an implementation with a comparison with previous constructions.