Consider the following variations on a Diffie$-$Hellman exchange, where Alice and Bob establish a shared key and carry on a conversation. Which security

vulnerabilities does each of the following protocols have? Assume in the variants using public keys $($protocols $2-6),$ that Alice and Bob each have public$/$private

key pairs and they know each other's public keys. In the variants using a pre$-$shared secret $($protocols $7-8),$ assume there is some secret $K_{AB}$ that both Alice and

Bob know. Assume Trudy can be an active MITM $($for instance, she is malware on a router on the path between Alice and Bob$).$

Security vulnerabilities:

A$.$ Trudy can impersonate Alice to Bob

B$.$ Trudy can impersonate Bob to Alice

C$.$ Trudy can be an active Meddler$-$In$-$The$-$Middle and see what Alice and Bob are saying to one another

D$.$ Trudy can passively eavesdrop on the conversation and see what Alice and Bob are saying to one another

Protocols:

Alice and Bob each send their Diffie$-$Hellman numbers to one another $($Alice sends $g^A$ mod $p,$ and Bob sends $g^B$ mod $p)$ and then they encrypt their

conversation with the Diffie$-$Hellman shared key.

Like $1,$ but Al$$ce and Bob each s$$gn their Diffie$-$Hellman numbers with their private keys and they each verify the other's signature knowing the other's

public key. In other words, Alice sends $[$gA mod p$]$Alice$,$ and Bob sends $[$gB mod p$]$Bob$.$

Like $1,$ but Alice and Bob each encrypt their Diffie$-$Hellman numbers using the other's public key. In other words, Alice sends $\{\begin{array}{c}{g}^A\end{array}$ mod p$\}$Bob$,$ and Bob sends

$\{$g ${?}^B$ mod p$\}$Alice$.$

Like $1,$ but Alice encrypts her Diffie$-$Hellman number with Bob's public key. In other words, Alice sends $\{\begin{array}{c}{g}^A\end{array}$ mod p