Answer:
The Diffie-Hellman key exchange protocol involves the following steps:
1. **Agree on Parameters**: Alice and Bob agree on a prime number \( p \) and a primitive root \( a \) modulo \( p \). In this case, they choose \( p = 101 \) and \( a = 2 \).
2. **Select Private Keys**: Alice selects a private key \( k_1 \) and Bob selects a private key \( k_2 \). Here, Alice selects \( k_1 = 7 \) and Bob selects \( k_2 = 9 \).
3. **Compute Public Keys**: Alice computes her public key \( A = a^{k_1} \mod p \) and Bob computes his public key \( B = a^{k_2} \mod p \). Using the given values:
- For Alice: \( A = 2^7 \mod 101 = 128 \mod 101 = 27 \)
- For Bob: \( B = 2^9 \mod 101 = 512 \mod 101 = 11 \)
4. **Exchange Public Keys**: Alice and Bob exchange their public keys.
5. **Compute Shared Secret**: Alice computes the shared secret \( s = B^{k_1} \mod p \) and Bob computes the shared secret \( s = A^{k_2} \mod p \).
- For Alice: \( s = 11^7 \mod 101 = 19487171 \mod 101 = 98 \)
- For Bob: \( s = 27^9 \mod 101 = 7625597484987 \mod 101 = 98 \)
Both Alice and Bob arrive at the same shared secret \( s = 98 \).
6. **Encrypt and Decrypt a Message**: To encrypt and decrypt a message using an affine cipher, we need a shared key \( sk \). Since we have obtained \( sk = 98 \), we can now use this key to encrypt and decrypt messages using the affine cipher function \( f(p) = (p \cdot sk) \mod 26 \).
For example, if we want to encrypt the letter "A" (which corresponds to 0 in the English alphabet), we apply the affine cipher:
- Encryption: \( f(0) = (0 \cdot 98) \mod 26 = 0 \)
- Decryption: \( f(0) = (0 \cdot 98) \mod 26 = 0 \)
So, "A" remains "A" after encryption and decryption. Similarly, we can encrypt and decrypt other letters using this shared key.
Step-by-step explanation:
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