Abstract: The single sign on (SSO) scheme can avoid the waste of resources and information leakage caused by the redundancy of authentication module, and the anonymous single sign on can realize anonymous authentication and authorization under the condition of protecting personal privacy. However, the existing anonymous single sign on schemes do not consider the accountability of fraud caused by the anonymity of users. For this problem, a traceable anonymous single sign on scheme on lattice is proposed. The proposed scheme uses the identity-based cryptosystem on lattice to alleviate the problem of public key certificate management, and realizes the anonymous authentication of the user through the authorized authentication tag and pseudonym. Then, the strong designated verifier technology is used to realize the directional verification of user service requests. And the trusted organization is introduced to recover the user's identity and pursue responsibility through the public key. The proposed scheme is proved to have unlinkability, unforgeability and traceability under the security model. The security and performance analysis results show that under PARMS II and PARMS III, our scheme can generate the access service tickets for 4 service requests by running for about 75 ms and 108 ms respectively. And it can reach the quantum security strength of 230 b and 292 b.