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A normal security notion for MAC's is that of unforgeability. So given some set of message,tag pairs $(m_0,t_0),\ldots,(m_k,t_k)$ is should be hard to create a tag for a new message not among the $\{m_0,\ldots,m_k\}$, stated informally. In your case, you could just use $E_k(m_0)$ with the secret MAC key $k$ ($H_0$ in your notation); no need for the extra ...

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