$6\mathrm{.}2.$ Let $F$ be a secure PRF$.$

$($a$)$ Let min$\{0,1{\}}^{\text{o}\text{u}\text{t}}$ be a fixed $($public$,$ hard$-$coded, known to the adversary$)$ string. Define:

$F_m(k,x)=F(k,x)o+m.$

Prove that for every $m,F_m$ is a secure PRF$.$

$($b$)$ Define

$F^{\text{'}}(k,x)=F(k,x)o+x.$

Prove that $F^{\text{'}}$ is a secure PRF$.$