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Tseng et al. proposed a new signature scheme with message recovery by the concept of self-certified public keys in 2003. Their scheme provides two merits: one is that the signer's public key can simultaneously be authenticated in verifying the signature and the other one is that only the specified verifier can recover the message. Recently, Xie and Yu found that the Tseng et al's scheme is insecure against forgery attack such that they made a small modification for the Tseng et al.'s scheme and claimed that the modified scheme can satisfy the security requirement. Unfortunately, in this paper, we will show that their attack cannot work. Because some researchers have pointed the insecurity of Tseng et al.'s scheme, we also propose a securely authenticated encryption scheme to overcome the weakness.