Diffie$-$Hellman$(2-2-2-6)$ Consider a Diffie$-$Hellman key exchange with $p=13$ and $g=7.$

$($a$)$ Alice picks $x=5,$ what is the public key A will send to Bob?

$($b$)$ Bob picks $y=11,$ what is the public $B$ he will send to Alice?

$($c$)$ What is the shared key $K$ resulting from the exchange?

$($d$)$ One weakness of $DH$ is that it is vulnerable to Man$-$in$-$the$-$Middle attack. How could

you modify the exchanged messages in DH to prevent this attack? Clearly state all your assumptions $($including any additional cryptographic algorithm or material needed$)$ and the notation you used.