RSA Cryptosystem

The RSA Cryptosystem is a cornerstone of public-key cryptography, widely utilized for secure data transmission. Named after its inventors, Ron Rivest, Adi Shamir, and Leonard Adleman, the RSA cryptosystem revolutionized the field of cryptography by introducing an asymmetric key cryptography framework.

Key Principles

RSA relies on the mathematical properties of prime numbers and involves three primary steps: key generation, encryption, and decryption.

Key Generation

Select two large prime numbers: The security of RSA is predicated on large prime numbers, usually denoted as ( p ) and ( q ).
Compute the modulus ( n ): Multiply the primes to get ( n = p \times q ). The modulus ( n ) is used in both the public and private keys.
Calculate Euler's Totient Function: Compute (\phi(n) = (p-1)(q-1)). This computes the order of the multiplicative group of integers modulo ( n ).
Choose the public exponent ( e ): Select an integer ( e ) such that ( 1 < e < \phi(n) ) and ( gcd(e, \phi(n)) = 1). Common choices for ( e ) are 3, 17 or 65537 as they balance security and efficiency.
Determine the private exponent ( d ): Compute ( d ) as the modular multiplicative inverse of ( e ) modulo (\phi(n)). Hence, ( d \times e \equiv 1 \pmod{\phi(n)} ).

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

Encryption and Decryption

Encryption: To encrypt a message ( m ), compute the ciphertext ( c ) using the public key: ( c \equiv m^e \pmod{n} ).
Decryption: To decrypt the ciphertext ( c ), recover the message ( m ) using the private key: ( m \equiv c^d \pmod{n} ).

Due to the difficulty of factoring large integers, especially the product of two primes, RSA remains a robust method for secure communications.

Applications and Security

RSA is employed in various applications including secure email, digital signatures, and key exchange protocols, such as the Diffie–Hellman key exchange. It is also integral to the SSL/TLS protocols that secure internet communications.

The security of RSA is chiefly based on the computational complexity associated with integer factorization. However, advancements in quantum computing could pose a threat, as quantum algorithms like Shor's algorithm could potentially break RSA by efficiently factoring large numbers.

Limitations and Alternatives

Despite its robustness, RSA has limitations such as slower encryption speeds compared to symmetric-key algorithms like the Advanced Encryption Standard. For enhanced security and efficiency, RSA is often used in conjunction with symmetric-key cryptosystems in a hybrid model.

Alternatives to RSA include Elliptic Curve Cryptography, which offers equivalent security with smaller key sizes, and homomorphic encryption, which allows computations on encrypted data without decryption.