$(2)$ Let PrivK$-$EAV$2$A$,\backslash$Pi $0($n$)$ be the game in Figure $1$ where A is allowed to choose challenge messages of arbitrary length for breaking encryption scheme $\backslash$Pi $),$ and consider the security notion derived by this modified game: Intuitively, $\backslash$Pi $0$ is EAV$2-$secure$,$ if no efficient A can determine cb better than guessing, even when $|$m$0|6=|$m$1|.$ Show that no EAV$2-$secure scheme $\backslash$Pi $0$ exists.

Hint: First, note that $\backslash$Pi $0$ is defined over message space M $=\{0,1\},$ i$.$e$.,$ messages of arbitrary length. Then, assume that polynomial p$($n$)$ is an upper bound on the time spent by Enc$0$ for encrypting a single bit, and consider what happens when A chooses m$0$ in $\{0,1\}$ and a random m$1$ in $\{0,1\}$p$($n$)+$c$,$ where c $>=1$ a positive integer.