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In 2009, Tsai proposed an efficient convertible multi-authenticated encryption (CMAE) scheme. However, the author shows that his scheme is distinguishable under adaptive chosen-message attack, and that the designated verifier can generate the signature of the same message for other verifiers. Since no formal model of CMAE has been presented in the literature, the author presents the first complete formal model of CMAE. Then, a new scheme is proposed. The proposed scheme is provably secure in the random oracle model.