Encrypt the message ATTACK using the RSA system with n = 43x61 and e =17

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.

Discover how to encrypt the message ATTACK using RSA encryption with n = 43x61 and e = 17. This detailed guide includes calculations, concepts of modular arithmetic, and the RSA encryption theorem. Suitable for high school students (Grades 10-12).

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