A third party-based data storage audit service is a developing branch where a user may use its mobile devices with limited energy to trigger a third party to audit the user's data in a cloud service provider. A basic tool for a third party to check the integrity of a user's data is the designated verifier signature (DVS), which allows a designated verifier to simulate a signer's signatures. However, the cloud computing is a technique that may be used a long time in the future, the development of quantum computation shows threats to basic building blocks of cloud computing services. It is then desirable to design DVS schemes potentially secure against quantum attacks. We here give a lattice-based DVS scheme to satisfy the requirement. It is based on Lyubashevsky's signature scheme in EUROCRYPT 2012 and has a shorter signature size.