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The provided image outlines the RSA encryption and decryption process, which is a widely used public-key cryptosystem. Let's break down each step:

1. **Selection of Prime Numbers:**
   - \( p \) and \( q \) are chosen as large random prime numbers.

2. **Calculation of \( N \):**
   - \( N = p \cdot q \)
   - \( N \) is the modulus used for both the public and private keys.

3. **Euler's Totient Function \( \phi(N) \):**
   - \( \phi(N) = \phi(p) \cdot \phi(q) = (p - 1) \cdot (q - 1) \)
   - This function is crucial for determining the public and private keys.

4. **Selection of Public Key \( e \):**
   - \( e = 65537 \) (a commonly used value in RSA, as it is a prime number that meets the criteria)
   - \( 1 < e < \phi(N) \) and \( \gcd(e, \phi(N)) = 1 \)
   - These conditions ensure that \( e \) is suitable for the encryption process.
   - The public key is \( K(e, N) \).

5. **Calculation of Private Key \( d \):**
   - \( d \equiv e^{-1} \mod \phi(N) \)
   - \( d \) is the modular multiplicative inverse of \( e \) modulo \( \phi(N) \).
   - The private key is \( k(d, N) \).

6. **Encryption:**
   - \( c = m^e \mod N \)
   - \( m \) is the plaintext message.
   - \( c \) is the encrypted ciphertext.

7. **Decryption:**
   - \( m = c^d \mod N \)
   - The ciphertext \( c \) is decrypted back to the plaintext message \( m \).

### Explanation

- **Step 1:** Two large prime numbers, \( p \) and \( q \), are chosen to ensure the strength of the encryption.
- **Step 2:** The product of these primes, \( N \), forms part of both the public and private keys.
- **Step 3:** Euler's Totient Function \( \phi(N) \) is calculated. This function counts the number of integers up to \( N \) that are relatively prime to \( N \).
- **Step 4:** A public exponent \( e \) is selected. It is a value that is relatively prime to \( \phi(N) \) and typically small (65537 is a common choice for efficiency and security).
- **Step 5:** The private exponent \( d \) is computed. It is the modular inverse of \( e \) modulo \( \phi(N) \), ensuring that \( (e \cdot d) \mod \phi(N) = 1 \).
- **Step 6:** Encryption transforms the plaintext message \( m \) into ciphertext \( c \) using the public key.
- **Step 7:** Decryption uses the private key to transform the ciphertext back into the original message.

Would you like more details or have any questions?

Here are some questions you might consider asking:
1. How are \( p \) and \( q \) typically chosen in practice?
2. What is the significance of choosing \( e \) as 65537?
3. How is the modular multiplicative inverse \( d \) calculated efficiently?
4. Can you explain why RSA is considered secure?
5. What are some potential vulnerabilities of RSA encryption?

**Tip:** When choosing \( p \) and \( q \) for RSA, it's important to use strong random number generators to ensure the primes are large and unpredictable.

Understand the RSA encryption and decryption process through detailed steps and formulas. Learn about prime number selection, Euler's Totient function, public and private key generation, and the security aspects of RSA encryption. Suitable for advanced high school students and beyond.

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