Problem $3-$ Basic Concept of ENC and MAC

In our semester, we have emphasized couple times that private$-$key encryption

and message authentication codes are different. In this task, you are going to

show the following:

There exists a private$-$key encryption whose ciphertexts do not provide

data authentication.

Particularly, you need to describe $(1)$ a secure private$-$key encryption

scheme KeyGen$(),$ Enc$(),$Dec$().(2)$ An adversary's strategy in the MAC

game: he can make one challenge of message $m,$ and get Enc$(m).$ Then he

can produce another message $m^{\text{'}}m$ with a valid encryption Enc$(m^{\text{'}}).$

There exists a message authentication code such that its tag does not

provide secrecy$.$

Particularly, you need to describe $(1)$ a secure message authentication

code KeyGen$(),$ Tag$(),$ Verify$().(2)$ An adversary's strategy in the semantic

security game of encryption: he provides $m_0,m_1$ and gets $C=$Tag$(m_b)$

where $b$ is a uniform bit from $\{0,1\}.$ From the ciphertext $C,$ adversary is

going to send another bit $b^{\text{'}}$ and wins if $b=b^{\text{'}}.$ Your adversary should be

able to win with probability better than $0\mathrm{.}5,$ significantly.