$6\mathrm{.}8.$ Suppose $F$ is a secure $PRF$ with input length in$,$ but we want to use it to construct a PRF

with longer input length. Below are some approaches that don't work. For each one,

describe a successful distinguishing attack and compute its advantage:

$($a$)F^{\text{'}}(k,x||x^{\text{'}})=F(k,x)||F(k,x^{\text{'}}),$ where $x$ and $x^{\text{'}}$ are each in bits long.

$($b$)F^{\text{'}}(k,x||x^{\text{'}})=F(k,x)o+F(k,x^{\text{'}}),$ where $x$ and $x^{\text{'}}$ are each in bits long.

$($c$)F^{\text{'}}(k,x||x^{\text{'}})=F(k,x)o+F(k,xo+x^{\text{'}}),$ where $x$ and $x^{\text{'}}$ are each in bits long.

$($d$)F^{\text{'}}(k,x||x^{\text{'}})=F(k,0||x)o+F(k,1||x^{\text{'}}),$ where $x$ and $x^{\text{'}}$ are each in $-1$ bits long.