CPA$-$security, CCA$-$security $[19$ Marks$].$

$($a$)(9$ marks$)$ Let $F$:$\{0,1{\}}^n\{0,1{\}}^n\{0,1{\}}^n$ be a length$-$preserving pseudorandom permutation. Suppose

the shared key is a uniform kin$\{0,1{\}}^n.$ Consider an encryption scheme as follows. To encrypt min

$\{0,1{\}}^{2n},$ parse $m$ as $m_1||m_2$ with $|m_1|=|m_2|,$ then choose uniform rin$\{0,1{\}}^n,$ and send $c=En{c}_k(m)$:$=$

$($:$r,m_{1}o+F_k(r),m_{2}o+F_k(r+1)$:$).$ Is this scheme CPA$-$secure$?$ Explain your answer.

$($b$)(10$ marks$)$ Let $F$:$\{0,1{\}}^n\{0,1{\}}^n\{0,1{\}}^n$ be a length$-$preserving pseudorandom permutation. Sup$-$

pose the shared key is a uniform kin$\{0,1{\}}^n.$ Consider an encryption scheme as follows. Given as inputs

a message min$\{0,1{\}}^{\frac{n}{2}}$ and a key kin$\{0,1{\}}^n,$ algorithm Enc chooses a random string $r$ of length $\frac{n}{2},$

i$.$e$.,$ rlarr$\{0,1{\}}^{\frac{n}{2}}$ and computes $c$:$=F_k(r||m).$ Show how to decrypt. Is this scheme CPA$-$secure$?$ Is this

scheme CCA$-$secure$?$ Explain your answer.