5. We now consider a key exchange protocol which is a variant of the Diffie- Hellman key exchange (DHKE). In this protocol, we assume that there is a public prime number p and a public generator of Z: Each participant T in this protocol has a private key sT E Zp and a public key PT = aT mod p. When a participant A wants to exchange a key with B, the following two messages are exchanged. . A generates a random number TA Zp, computes nAaA mod p and sends nA to B B upon receiving the first message from A, generates a random num- ber rB E Zp, computes na mod p and sends the pair nB to A. The secret key shared between A and B is +sr mod p (a) Suppose that U (with private key su and playing the role A) and V (with private key sv and playing the role B) exchange keys using the above variant of DHKE. Let Ko be the key shared between U and V after this session. How do U and V compute Ko using the messages nu and ny exchanged between them? (b) The numbers ru and rv must be kept secret by U and V respectively Why? (c) Continuing from part (a), suppose a malicious entity M (with private key sM) eavesdrops the exchange between U and V and records nu and nv M (playing the role of A) starts a new session with U (playing the role of B) by sending the message ny that it recorded. U responds by generating a fresh random n, computing ni- mod p and sending back n Let Ki be the that is supposed to be shared between U and M. Now, U can compute Ki, but M cannot. Why? 5. We now consider a key exchange protocol which is a variant of the Diffie- Hellman key exchange (DHKE). In this protocol, we assume that there is a public prime number p and a public generator of Z: Each participant T in this protocol has a private key sT E Zp and a public key PT = aT mod p. When a participant A wants to exchange a key with B, the following two messages are exchanged. . A generates a random number TA Zp, computes nAaA mod p and sends nA to B B upon receiving the first message from A, generates a random num- ber rB E Zp, computes na mod p and sends the pair nB to A. The secret key shared between A and B is +sr mod p (a) Suppose that U (with private key su and playing the role A) and V (with private key sv and playing the role B) exchange keys using the above variant of DHKE. Let Ko be the key shared between U and V after this session. How do U and V compute Ko using the messages nu and ny exchanged between them? (b) The numbers ru and rv must be kept secret by U and V respectively Why? (c) Continuing from part (a), suppose a malicious entity M (with private key sM) eavesdrops the exchange between U and V and records nu and nv M (playing the role of A) starts a new session with U (playing the role of B) by sending the message ny that it recorded. U responds by generating a fresh random n, computing ni- mod p and sending back n Let Ki be the that is supposed to be shared between U and M. Now, U can compute Ki, but M cannot. Why
