Suppose F is a secure PRF with input length in$,$ but we want to use it to construct another PRF F$$ with a longer input length. Below are some approaches that don$$t work. For each one, describe a successful distinguishing attack and compute its advantage:

$($a$)$ F $($k$,$ x$||$x$)=$ F $($k$,$ x$)||$F $($k$,$ x$),$ where x and x$$ are each in bits long.

$($b$)$ F$($k$||$k$,$x$||$x$)=$ F$($k$,$x $$ x$)$ F$($k$,$x $$ x$),$ where x and x$$ are each in bits long, and k and k$$

belong to the key space of F $.$

$($c$)$ F $($k$,$ x$||$x$)=$ F $($k$,0||$x$)$ F $($k$,1||$x$),$ where x and x$$ are each $($in $1)$ bits lon