Problem
We now investigate how Alice and Bob can perform the key agreement in a way that is not vulnerable to either of the above attacks.
1) Calculate what the Diffie-Hellman key agreement would look like if Alice and Bob kept the same secrets (123 and 456) and modulus (2027) as before, but chose it instead of g, to generate a subgroup of large prime order, rather than the whole group, in the multiplicative group 123027. Concretely, assume that the value of It you found earlier indeed generates a subgroup of prime order (can you verify that it has prime order?), and redo all the calculations for the basic Diffie-Hellman key exchange when Alice and Bob use this It as generator, instead of the whole-group generator g.
2) Brie?y explain why the ?rst of the two attacks above - the attack in which a passive attacker was able to glean some information about Alice's and Bob's secrets 7 is now defeated. Hint: you can focus on the subgroup of order 2: look for and discuss any difference with what you saw earlier.
3) Now explain why the second attack - where BadBob, by choosing a crafty secret key 3, was able to force the key agreement to misbehave in the way you explained earlier - is also defeated. Hint: you can focus your explanation using the same concrete value of 3 as before and see if the attack would be easier to defeat, or at least to detect and mitigate.