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Direct Anonymous Attestation (DAA) is an anonymous signature scheme, which allows the Trusted Platform Module (TPM), a small chip embedded in a host computer, to attest to the state of the host system, while preserving the privacy of the user. DAA provides two signature modes: fully anonymous signatures and pseudonymous signatures. One main goal of designing DAA schemes is to reduce the TPM signing workload as much as possible, as the TPM has only limited resources. In an optimal DAA scheme, the signing workload on the TPM will be no more than that required for a normal signature like ECSchnorr. To date, no scheme has achieved the optimal signing efficiency for both signature modes.In this paper, we propose the first DAA scheme which achieves the optimal TPM signing efficiency for both signature modes. In this scheme, the TPM takes only a single exponentiation to generate a signature, and this single exponentiation can be precomputed. Our scheme can be implemented using the existing TPM 2.0 commands, and thus is compatible with the TPM 2.0 specification. We benchmarked the TPM 2.0 commands needed for three DAA use cases on an Infineon TPM 2.0 chip, and also implemented the host signing and verification algorithm for our DAA scheme on a laptop with 1.80GHz Intel Core i7-8550U CPU. Our experimental results show that our DAA scheme obtains a total signing time of about 144 ms for either signature mode, while with pre-computation we can obtain a signing time of about 65 ms. Based on our benchmark results for the pseudonymous signature mode, our scheme is roughly 2× (resp., 5×) faster than the existing DAA schemes supported by TPM 2.0 in terms of total (resp., online) signing efficiency.