Problem $1(0$ Points$).{?}^1$ Suppose $f,g$:$\{0,1{\}}^{2n}\{0,1{\}}^m$ are one$-$way functions. For xin$\{0,1{\}}^{2n},$

let $x_1$ and $x_2$ denote the first and second halves of $x,$ respectively. Furthermore, $(*,*)$ is the

concatenation operator for bit$-$strings and $o+$ is the bit$-$wise exclusive or operator. Prove that in

general:

$($a$)f_a(x_1,x_2)$:$=(f(x_1,x_2),x_1)$ is not a one$-$way function.

Hint: Can we have one$-$way functions which do not depend on the entire input?

$($b$)f_b(x_1,x_2)$:$=(f(x_1,x_2),x_{1}o+x_2)$ is not a one$-$way function.

$($c$)f_c(x_1,x_2)$:$=(f(x_1),g(x_2))$ is a one$-$way function.

$($d$)f_d(x)$:$=(f(x),g(x))$ is not a one$-$way function.