$4.$ This problem deals with the protocol in Figure $9\mathrm{.}17.$

$4$a$)$ Show that this protocol is insecure. Hint: Let Trudy be the man in$-$the$-$middle, and show that Trudy can convince Alice that she is Bob, and Trudy can determine Alice$$s session key K$.$

$4$b$)$ Slightly modify the protocol so that Trudy cannot obtain the session key.