Exercise $2.$ Insecure Hash Functions $[30$ points$]$ Show that the following hash functions are insecure. To do that explain how to find a specific collision pair.

Consider a function $H$:$\{0,1{\}}^{**}\{0,1{\}}^n.$ On input a message $m$ and two

shares of it $x,w$ such that $m=xo+w,$ the function outputs $y=H(m)=$

$H(x)o+H(w).$ Show that this is NOT a collision resistance hash function $($i$.$e$.,$

show two different messages $m_1,m_2$ such that they map to the same $y).$

Let $H_s^a(x_1||x_2)=H_s(x_1)o+H_s(x_2)$ where $H$ is a collision resistance hash function

and $x=x_1||x_2,$ where $|x_1|=|x_2|$ and the size of $x$ is an even number $($i$.$e$.,x_1,x_2$

are the $2$ parts of $x).$ Show that $H^a$ is NOT collision resistance.