RSA Algorithm - The Mathematics Behind Public-Key Cryptography

Cryptography has always played a crucial role in protecting sensitive information from prying eyes. As technology evolved, the need for more robust and secure encryption methods became apparent, leading to the development of public-key cryptography. In the 1970s, Ron Rivest, Adi Shamir, and Leonard Adleman devised the RSA algorithm, which remains one of the most widely-used public-key cryptosystems today. In this blog post, we’ll explore the mathematics behind the RSA algorithm and how it forms the backbone of secure communication in the digital age.

Understanding Public-Key Cryptography

Before delving into the RSA algorithm, it’s essential to grasp the concept of public-key cryptography. Unlike symmetric-key cryptography, where a single key is used for both encryption and decryption, public-key cryptography employs two separate keys:

  • A public key, which can be shared openly and is used to encrypt data.
  • A private key, which must remain secret and is used to decrypt data.

The beauty of public-key cryptography lies in the fact that, even if an attacker intercepts the encrypted data and knows the public key, they cannot decrypt the information without the private key. This asymmetry makes public-key cryptography much more secure than symmetric-key cryptography for large-scale communication.

The Foundation of RSA

At the core of the RSA algorithm are prime numbers and modular arithmetic. Prime numbers are integers greater than 1 with only two distinct positive divisors: 1 and the number itself. For example, 2, 3, 5, and 7 are prime numbers. The RSA algorithm relies on the difficulty of factoring large composite numbers (i.e., numbers that can be divided by at least one other number besides 1 and itself) into their prime factors.

Modular arithmetic, also known as “clock arithmetic,” involves calculating the remainder when dividing one number by another. In the context of RSA, modular arithmetic ensures that encrypted and decrypted messages stay within a specific range of numbers.

The RSA Algorithm

The RSA algorithm involves three main steps: key generation, encryption, and decryption.

Key Generation

  • Select two large prime numbers, p and q. These numbers should be chosen randomly and kept secret.
  • Calculate the modulus n by multiplying p and q: n = p * q. This value will be used as the modulus for both the public and private keys.
  • Calculate Euler’s totient function, φ(n), which is given by: φ(n) = (p-1)*(q-1).
  • Choose an integer e such that 1 < e < φ(n) and e is coprime to φ(n), meaning e and φ(n) share no common factors other than 1. The number e is the public key exponent.
  • Calculate the private key exponent d, which satisfies the equation: (d * e) % φ(n) = 1. In other words, find the multiplicative inverse of e modulo φ(n).

The public key consists of (n, e), and the private key is (n, d).

Encryption

To encrypt a message M, the sender computes the ciphertext C using the recipient’s public key (n, e) as follows: C = M^e % n.

Decryption

The recipient, who has the private key (n, d), can decrypt the ciphertext C to obtain the original message M by performing the following calculation: M = C^d % n.

Security and Applications of RSA

The security of the RSA algorithm relies on the fact that it is computationally infeasible to factor large composite numbers into their prime factors. If an attacker could easily do this, they could determine the private key (n, d) using the public key (n, e) and decrypt messages. However, no efficient factoring algorithms are currently known, and as a result, RSA remains secure for practical purposes.

RSA is widely used in various applications, such as:

  • Secure communication

    RSA is often used to exchange symmetric encryption keys securely, which are then used to encrypt and decrypt large volumes of data.

  • Digital signatures

    RSA can be used to verify the authenticity and integrity of a message or document by signing it with the sender’s private key and verifying the signature with their public key.

  • Secure online transactions

    In e-commerce and online banking, RSA plays a critical role in ensuring secure communication between clients and servers.