Let $\backslash$Pi $=($Gen$,$ MAC, Verify$)$ be a secure MAC scheme with Gen$(1^$n$)$ in $\{0,1\}^$n$/2$ and MAC : $\{0,1\}^$n$/2\backslash$times $\{0,1\}^$n$/2->\{0,1\}^$n $($i$.$e$.,$ the MAC scheme works with keys and messages of size n$/2$ and returns a tag represented by a bit$-$string of size n$).$

Let F : $\{0,1\}^$n$\backslash$times $\{0,1\}^$n $->\{0,1\}^$n be a length$-$preserving pseudorandom function.

State whether each of the following PRF candidates is or is not a pseudorandom function. If yes, prove it; if not, show a distinguisher that succeeds with nonnegligible probability. Hint: to prove that some of these candidates are not PRFs$,$ it may be useful to assume the existence of MACs or PRFs with input and output of arbitrary size. Feel free to assume that such PRFs and MACs do exist, and pick the parameter sizes that are the most suitable to prove your result.

$($a$)$ F$($k$,$ x$)=$ MAC$([$k$]^$n$/2,[$x$]^$n$/2).$

$($b$)$ F$($k$,$ x$)=$ F$($k$,$ x$)||$AND$($x$).$

$($c$)$ F$($k$,$ x$)=$ MAC$([$F$($k$,1$n$)]^$n$/2,[$x$]^$n$/2).$