(HW1) List some of the attacks on the Diffie-Hellman key exchange protocol we discussed in the lecture. Present your solution for avoiding such attacks. (HW2a) In the Diffie-Helman protocol, g=11, p=29, x=5, and y=7. Variations of data (HW2b) In the Diffie-Helman protocol, what happens is x and y have the same value, that is, Alice and Bob accidentally chosen the same number? Are R1 and R2 same? Do the session key calculated by Alice and Bob have the same value? Explain what would adversary observe? Could she guess Alice’s and Bob’s private key? Use an example to prove your claims. (HW3a) Using RSA scheme, let p=23, q=31, d=457, calculate the public key e. Provide detailed description of all steps, explain what information will be published and what destroyed. Optionally: Encrypt and decrypt simple message M1=100. Variation of data (HW3b) Suppose Fred sees your RSA signature on m1 and m2, (i.e., he sees (m1d mod n) and (m2d mod n)). How does he compute the signature on each of m1j mod n (for positive integer j), m1-1 mod n, m1 x m2 mod n, and in general m1j m2k mod n (for arbitrary j and k)?

(HW1) The Diffie-Hellman key exchange protocol is widely used for establishing a secure communication channel between two parties over an insecure network. However, there are several known attacks on this protocol that can compromise the security of the exchanged keys.

One such attack is the man-in-the-middle (MITM) attack, where an attacker intercepts the communication between Alice and Bob and impersonates both parties to establish separate keys with each of them. This allows the attacker to decrypt and read all the messages exchanged between Alice and Bob.

Another attack is the small-subgroup attack, where the attacker performs calculations in a small subgroup of the larger group used in the protocol. This can lead to the attacker obtaining the private keys of Alice and Bob.

To avoid these attacks, various countermeasures can be employed:

1. Certificate-based authentication: Alice and Bob can obtain digital certificates from trusted authorities to verify each other’s identity. This ensures that there is no MITM attack and that they are indeed communicating with each other.

2. Use of secure channels: It is important to ensure that the communication channel used for the key exchange is secure. This can be achieved through the use of secure socket layer (SSL)/transport layer security (TLS) protocols to encrypt the communication.

3. Diffie-Hellman key exchange with strong parameters: Using large prime numbers as the modulus and generator in the Diffie-Hellman protocol can protect against small-subgroup attacks. It is also important to periodically update these parameters to ensure the security of the protocol.

4. Key validation: After the key exchange, both parties should validate the received keys to ensure they have been generated correctly. This can be done using hash functions or digital signatures.

(HW2a) In the given scenario, the Diffie-Hellman parameters are g = 11, p = 29, x = 5, and y = 7. The shared secret key can be calculated as follows:

Alice’s calculation:
Alice computes R1 = g^x mod p = 11^5 mod 29 = 13
Bob’s calculation:
Bob computes R2 = g^y mod p = 11^7 mod 29 = 24

Yes, in the case where x and y have the same value, Alice and Bob accidentally chosen the same number, the calculated values of R1 and R2 will be the same. However, the session keys calculated by Alice and Bob will still have different values. The session key calculated by Alice is S = R2^x mod p = 24^5 mod 29 = 3, while the session key calculated by Bob is S = R1^y mod p = 13^7 mod 29 = 3.

Explain what an adversary would observe:
If an adversary observes the values of R1 and R2, they would see that both values are the same. However, they would not be able to guess the private keys of Alice and Bob solely based on this observation.

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