Perfect Secrecy $[17$ Marks$].$

Consider a private$-$key encryption scheme with message space $M$:$=\{a,b,c,d\}($with respective probabilities

for $a,b,c,d$ being $0\mathrm{.}25,0\mathrm{.}3,0\mathrm{.}15,0\mathrm{.}3),$ and key space $K$:$=\{k_1,k_2,k_3\}($with respective probabilities for choosing

keys $k_1,k_2,k_3$ being $0\mathrm{.}25,0\mathrm{.}5,0\mathrm{.}25$ independent of any message$).$ Suppose that the ciphertext space is $C$:$=$

$\{1,2,3,4\},$ with the encryption function given by the following table

The table is to be read as saying that $En{c}_{k_1}(a)=3,En{c}_{k_3}(c)=1,$ etc.

$($a$)(7$ marks$)$ Compute the message entropy $H(M),$ the key entropy $H(K)$ and the ciphertext entropy $H(C).$

$($b$)(4$ marks$)$ Use the rules of entropy to show that when the key is chosen independently of the message, it

holds that $H(K,M,C)=H(K)+H(M)$ and also $H(K,M,C)=H(K,C).$

$($c$)(3$ marks$)$ Using the result from $($b$)$ above, compute how many bits of information about the key are

revealed by a single ciphertext on average.

$($d$)(3$ marks$)$ Is this encryption scheme perfectly secret? Explain your answer.