Authenticated Encryption: Consider a standard authenticated encryption scheme where the message msg is encrypted to produce C $=$ ENC$\_$K$1($msg$),$ and a separate MAC key K$2$ is used to compute an authentication tag T $=$ MAC$\_$K$2($C$).$ The final output of this scheme is the pair $($C$,$ T$).$

A smart colleague suggests an "improved" approach to reduce the number of shared keys. In this new approach, instead of using two separate keys, the MAC key KM is derived deterministically from the message. The modified scheme works as follows: first, the MAC key is computed as KM $=$ H$($msg$),$ where H is a cryptographic hash function. Then, the ciphertext C is generated as C $=$ ENC$\_$K$($msg$).$ Finally, the authentication tag T is computed as T $=$ MAC$\_$KM$($C$).$

In this modified scheme, K is the shared encryption key, and KM is derived from the message msg using a hash function. The receiver decrypts C using K to retrieve msg$,$ derives KM $=$ H$($msg$),$ and then uses KM to verify the MAC tag T$.$

Analyze what could go wrong with this approach. What vulnerabilities or security issues might arise from this modification?

$($Note: ChatGPT provides incorrect$/$imprecise answers in this case.$)$