How many part does a public and private key has in RSA cipher

/ / Published in Cybersecurity, EITC/IS/CCF Classical Cryptography Fundamentals, Introduction to public-key cryptography, The RSA cryptosystem and efficient exponentiation

The RSA cryptosystem, named after its inventors Rivest, Shamir, and Adleman, is one of the most well-known public-key cryptographic systems. It is widely used for secure data transmission. RSA is based on the mathematical properties of large prime numbers and the computational difficulty of factoring the product of two large prime numbers. The system relies on a pair of keys: a public key, which is used for encryption, and a private key, which is used for decryption. Each of these keys comprises specific components that work together to enable secure communication.

In the RSA cryptosystem, the public key consists of two parts:
1. The Modulus (n): This is a large integer obtained by multiplying two distinct prime numbers, p and q. The security of the RSA algorithm relies on the difficulty of factoring this large number back into its prime components. The modulus is used in both the public and private keys.
2. The Public Exponent (e): This is a smaller integer that is typically chosen to be a prime number such as 3, 17, or 65537. The public exponent is used in the encryption process and is part of the public key. It is chosen such that it is relatively prime to (p-1)(q-1), where p and q are the prime factors of the modulus.

The public key is thus represented as (n, e).

The private key also consists of two parts:
1. The Modulus (n): The same modulus used in the public key is also used in the private key.
2. The Private Exponent (d): This is a large integer computed using the public exponent e and the totient of the modulus n (denoted as φ(n)). The private exponent is calculated such that it satisfies the congruence relation:

    \[ e \cdot d \equiv 1 \ (\text{mod} \ \phi(n)) \]

where φ(n) = (p-1)(q-1). The private exponent d is used in the decryption process.

The private key is thus represented as (n, d).

To illustrate the RSA key generation process, let us consider a simplified example with small prime numbers for clarity:

1. Choose two distinct prime numbers, p and q:
Let p = 61 and q = 53.

2. Compute the modulus n:

    \[ n = p \times q = 61 \times 53 = 3233 \]

3. Compute the totient φ(n):

    \[ \phi(n) = (p-1) \times (q-1) = (61-1) \times (53-1) = 60 \times 52 = 3120 \]

4. Choose a public exponent e:
Let e = 17, which is a commonly used public exponent and is relatively prime to 3120.

5. Compute the private exponent d:
The private exponent d is the modular multiplicative inverse of e modulo φ(n). This means we need to find d such that:

    \[ e \cdot d \equiv 1 \ (\text{mod} \ \phi(n)) \]

Using the Extended Euclidean Algorithm, we find that d = 2753.

Thus, the public key is (n, e) = (3233, 17), and the private key is (n, d) = (3233, 2753).

To understand how RSA encryption and decryption work, consider the following example:

1. Encryption:
Suppose Alice wants to send a confidential message to Bob. She knows Bob's public key (n, e) = (3233, 17). Alice converts her plaintext message into an integer m such that 0 ≤ m < n. For simplicity, let m = 65.

Alice computes the ciphertext c using Bob's public key:

    \[ c = m^e \ (\text{mod} \ n) = 65^{17} \ (\text{mod} \ 3233) \]

Calculating the above expression, we get:

    \[ c = 2790 \]

2. Decryption:
Bob receives the ciphertext c = 2790 and uses his private key (n, d) = (3233, 2753) to decrypt it. Bob computes the plaintext message m using his private key:

    \[ m = c^d \ (\text{mod} \ n) = 2790^{2753} \ (\text{mod} \ 3233) \]

Calculating the above expression, we get:

    \[ m = 65 \]

Therefore, Bob successfully recovers the original message m = 65.

It is important to note that in practical implementations, the prime numbers p and q are chosen to be very large (typically hundreds of digits long) to ensure the security of the RSA cryptosystem. The security of RSA relies on the fact that, while it is computationally easy to multiply two large prime numbers to obtain the modulus n, it is extremely difficult to factor n back into its prime components p and q. This difficulty is what makes RSA a secure cryptographic system.

Additionally, the choice of the public exponent e is important. While e = 3 or e = 65537 are commonly used values, e should always be chosen such that it is relatively prime to φ(n). This ensures that the private exponent d can be computed and that the encryption and decryption processes work correctly.

In practical applications, RSA is often used in conjunction with other cryptographic techniques to enhance security and efficiency. For example, RSA is commonly used to encrypt a symmetric key, which is then used to encrypt the actual message using a faster symmetric-key algorithm such as AES (Advanced Encryption Standard). This approach leverages the strengths of both asymmetric and symmetric cryptography.

Furthermore, RSA is widely used in digital signatures, where the private key is used to sign a message, and the public key is used to verify the signature. This provides authentication and integrity, ensuring that the message has not been tampered with and that it comes from a legitimate source.

The RSA cryptosystem is a foundational element of modern public-key cryptography. Understanding the structure and function of the public and private keys in RSA is essential for anyone studying or working in the field of cybersecurity. The public key consists of the modulus (n) and the public exponent (e), while the private key consists of the modulus (n) and the private exponent (d). These keys work together to enable secure encryption and decryption, ensuring the confidentiality and integrity of data in a wide range of applications.

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