Let's go through the Diffie-Hellman (D-H) key exchange calculations step by step.

1. Given:

   • $p = 73$
   • $\alpha = 5$
   • $A = 44$
   • $B = 33$

2. We calculate $a$ and $b$ using:

   • $a \equiv \alpha^A \, (\text{mod} \, p)$
   • $b \equiv \alpha^B \, (\text{mod} \, p)$

3. Then we calculate the shared key $K$ using:

   • $K \equiv b^A \equiv a^B \, (\text{mod} \, p)$

Step 1: Calculate $a$

$a \equiv 5^{44} \, (\text{mod} \, 73)$ Using modular exponentiation, $5^{44} \mod 73 = 9$.

Thus, $a = 9$.

Step 2: Calculate $b$

$b \equiv 5^{33} \, (\text{mod} \, 73)$ Using modular exponentiation, $5^{33} \mod 73 = 46$.

Thus, $b = 46$.

Step 3: Calculate the Key $K$

Now we compute $K \equiv b^A \equiv a^B \, (\text{mod} \, 73)$. So, we can calculate $K = 46^{44} \mod 73$ or equivalently $K = 9^{33} \mod 73$.

Using modular exponentiation: $K = 9^{33} \mod 73 = 9$

Answer:

The shared key $K$ is 9.

Final Answer: 9

Question 1 Answer:

The correct answer is 9.

Would you like further details on the calculations or have any questions? Here are five related questions that build on this concept:

1. How does the choice of prime $p$ affect the security of the Diffie-Hellman key exchange?
2. Why is modular exponentiation used in Diffie-Hellman rather than straightforward exponentiation?
3. What would happen if a non-prime number was chosen for $p$ in the Diffie-Hellman protocol?
4. How does Diffie-Hellman handle the issue of man-in-the-middle attacks?
5. What are some common cryptographic systems that use Diffie-Hellman for secure key exchange?

Tip: In modular exponentiation, breaking down the power calculation into smaller steps (using techniques like "square-and-multiply") optimizes computations and is essential for large exponents, especially in cryptographic contexts.