RSA encryption is a widely used metodad for securing digital commulation. It endives generating a pair of keys and using them tem to encrypt and decrypt messages. Understanding thee practial calculations behind RSA helps in grasping how data security is maintained.

Key Generation Process

Te firtt step in RSA is selecting two large prime numbers, typically denoted as p and q. These primes are used to compute thee modulus n, which is part of the public and private keys.

Calculate n by multiplying p and q: n = p × q. Then, compute Euler 's totient function, ņ( n) = (p -1) × (q -1). Choosing an encryption exponent e that is coprime with (n) is essential. Comnon choices for e include3 or65537.

Te private key exponent d is calculated as the modular inverse of e modulo ņ( n). This means solving for d in thee equation: d × e zanium 1 (modd doposud (n)).

Message Encryption and Decryption

To encrypt a message, convert it into a numical format m, where 0 ≤ m credimp; lt; n. Te ciphertext c is then computed using thee public key (n, e): c = m ^ e modn.

Decryption impeves using the private key d to recover the original message: m = c ^ d modd n. This process ensures that only someone with the e private key can dešifrt the message.

Practical Calculation Example

Podpora p =61 and q =53. Kalkulace n =61 ×53 =3233. Then, ∞ (n) = (61 -1) × (53 -1) =60 ×52 =3120. Choose e =17, which is coprime with3120.

Find d such that d × 17 zaniklý 1 (modd 3120). Te value of d is 2753. Te public key is (n = 3233, e = 17), and the private key is (n = 3233, d = 2753).

To encrypt a message m =65, compute c =65 ^17 modd3233, resulting in c =2790. To dekrypt, compute m =2790 ^2753 modd3233, which yields the original message65.