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Based on the Schnorr signature scheme, a new signature scheme with non-repudiation is proposed. In this scheme, only the signer and the designated receiver can verify the signature signed by the signer, and if necessary, both the signer and the designated receiver can prove and show the validity of the signature signed by the signer. The proof of the validity of the signature is noninteractive and transferable. To verify and prove the validity of the signature, the signer and the nominated receiver neednt store extra information besides the signature. At the same time, neither the signer nor the designated receiver can deny a valid signature signed. Then, there is no repudiation in this new signature scheme. According to the security analysis of this scheme, it is found the proposed scheme is secure against existential forgery on adaptive chosen message attack.