To encrypt the message "ATTACK" using the RSA system with $n = 43 \times 61$ and $e = 17$, we follow these steps:

Step 1: Calculate $n$

First, we calculate $n$: $n = 43 \times 61 = 2623$

Step 2: Convert the message to numbers

Next, we convert each letter in "ATTACK" to a number using the following scheme:

• A = 0, B = 1, C = 2, ..., Z = 25.

Thus, "ATTACK" translates to:

• A = 0
• T = 19
• T = 19
• A = 0
• C = 2
• K = 10

This gives us the numerical representation: $[0, 19, 19, 0, 2, 10]$.

Step 3: Encrypt each number

The encryption formula is: $c = m^e \mod n$ where $m$ is the plaintext number.

Now, we compute the ciphertext for each letter:

1. For $m = 0$: $c = 0^{17} \mod 2623 = 0$

2. For $m = 19$: $c = 19^{17} \mod 2623$ Calculating $19^{17} \mod 2623$ (using modular exponentiation):

   • $19^2 = 361$
   • $19^4 = 361^2 \mod 2623 = 1377$
   • $19^8 = 1377^2 \mod 2623 = 1340$
   • $19^{16} = 1340^2 \mod 2623 = 1723$
   • $19^{17} = 19^{16} \times 19 \mod 2623 = 1723 \times 19 \mod 2623 = 1470$

3. For $m = 0$ (again): $c = 0$

4. For $m = 2$: $c = 2^{17} \mod 2623$

   • $2^{17} = 131072$
   • $131072 \mod 2623 = 1845$

5. For $m = 10$: $c = 10^{17} \mod 2623$

   • $10^{17} = 100000000000000000$
   • $100000000000000000 \mod 2623 = 1870$

Step 4: Compile the ciphertext

The resulting ciphertext for "ATTACK" is:

• $[0, 1470, 1470, 0, 1845, 1870]$

Final Result

Thus, the encrypted message "ATTACK" using RSA with $n = 2623$ and $e = 17$ is: $\text{Ciphertext: } [0, 1470, 1470, 0, 1845, 1870]$

Do you want more details on any part of the RSA encryption process, or do you have any questions? Here are some related questions:

1. How is the decryption process performed in RSA?
2. What are the steps to calculate the totient function $\phi(n)$?
3. How do you choose $e$ in the RSA algorithm?
4. What are the advantages of using RSA encryption?
5. Can you explain how modular exponentiation works?

Tip: Always check that your $e$ and $\phi(n)$ are coprime for RSA to work effectively.