The RSA algorithm is a widely used form of public-key cryptography that facilitates secure data transmission. Named after its inventors, Ron Rivest, Adi Shamir, and Leonard Adleman, who introduced it in 1977, RSA remains one of the most effective methods for securing sensitive data transmitted over the internet[1][5][6][12].
Key Components of RSA
RSA operates on a set of algorithms that manage key generation, encryption, and decryption:
Key Generation:
Select two large prime numbers, $$ p $$ and $$ q $$.
Compute $$ n = p \times q $$. This $$ n $$ serves as the modulus for both the public and private keys.
Calculate the totient, $$ \phi(n) = (p-1) \times (q-1) $$.
Choose an integer $$ e $$ such that $$ 1 < e < \phi(n) $$ and $$ e $$ is coprime to $$ \phi(n) $$, which will be the public key exponent.
Determine $$ d $$ as the modular multiplicative inverse of $$ e $$ modulo $$ \phi(n) $$, which serves as the private key exponent[1][5][6][7][8].
Encryption:
The message $$ M $$, which is less than $$ n $$, is encrypted using the recipient's public key $$ (e, n) $$ to produce the ciphertext $$ C $$ as $$ C = M^e \mod n $$[1][5][6].
Decryption:
The ciphertext $$ C $$ is decrypted using the private key $$ (d, n) $$ to retrieve the original message $$ M $$ as $$ M = C^d \mod n $$[1][5][6].
Security and Practicality
The security of RSA is based on the practical difficulty of the factorization of large integers, a problem known as the RSA problem. The assumption is that while it is feasible to multiply large numbers, the factorization of their product, particularly when the integers are large prime numbers, is computationally intensive and not feasible with current technology, especially as the key size increases (commonly 2048 bits today)[1][5][6].
