RSA Cryptography: A Comprehensive Guide

Introduction to RSA

RSA (Rivest-Shamir-Adleman) is one of the most widely used public-key cryptographic systems. It is used to secure sensitive data, ensure confidentiality, and verify authenticity in digital communications. Named after its inventors Ron Rivest, Adi Shamir, and Leonard Adleman, RSA operates on the mathematical principles of number theory and modular arithmetic.

How RSA Works

Key Concepts

1. Public Key: Used for encryption and shared publicly.
2. Private Key: Used for decryption and kept secret.
3. Asymmetric Encryption: Encryption and decryption use different keys, ensuring data security.

Key Generation

The strength of RSA lies in its key generation process, which involves:

1. Selecting Two Large Prime Numbers

   • Choose two distinct prime numbers, p and q.
   • These numbers should be sufficiently large to ensure security.

2. Calculating the Modulus n

   • Compute n = p * q.
   • The modulus n is used as part of the public and private keys.

3. Calculating Euler's Totient Function φ(n)

   • φ(n) = (p - 1) * (q - 1).
   • This value represents the count of integers less than n that are relatively prime to n.

4. Choosing the Public Exponent e

   • Select an integer e such that:
     • 1 < e < φ(n)
     • gcd(e, φ(n)) = 1
   • Common choices for e include 3, 17, and 65537, as they are efficient for encryption.

5. Calculating the Private Key d

   • Compute d such that:
     • d ≡ e^(-1) (mod φ(n))
     • This means d is the modular multiplicative inverse of e modulo φ(n).

6. Public and Private Keys

   • Public Key: (e, n)
   • Private Key: (d, n)

Encryption

To encrypt a message M:

1. Convert M into a numerical representation m where 0 ≤ m < n.

2. Compute the ciphertext c using the formula:

   c = m^e mod n

Decryption

To decrypt the ciphertext c:

1. Compute the original message m using the formula:

   m = c^d mod n

2. Convert m back into the original plaintext message.

Security of RSA

Why RSA is Secure

1. Factoring Large Integers

   • The security of RSA is based on the difficulty of factoring the modulus n into its prime components p and q.

2. Size of Keys

   • Modern RSA implementations use key sizes of at least 2048 bits to ensure security.

3. Cryptanalysis Resistance

   • RSA resists many attacks, such as brute force and chosen ciphertext attacks, when properly implemented.

Vulnerabilities

RSA can be vulnerable if:

• Small prime numbers are used.
• Insecure random number generation is employed.
• Keys are reused improperly.

Applications of RSA

1. Secure Communications

   • Ensures confidentiality in protocols like HTTPS and TLS.

2. Digital Signatures

   • Verifies the authenticity and integrity of messages.

3. Data Encryption

   • Protects sensitive information in transit and storage.