Praktyka Obliczenia in Rsa Szyfrowanie: Key Generation andMessage Encryption

RSA szyfruje is a widely used the methode for securing digital communication. It involves generating a pair of keys andd using them tem certipt and decrypt messages. Zrozumiałe, że te praktyczne obliczenia są hind RSA helps in grapping how data security is maintained.

Procesy Key Generation

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

Obliczenie n b y multipliing p and q: n = p × q. Then, compute Euler 's totient function, mbH (n) = (p - 1) × (q - 1). Choosing an critiption excutent e that is coprime with mbH (n) is essential. Common choices for e include 3 or 65537.

Te prywatne key wykładnia d is calculated as te modular inverse of e modulo mbH (n). This means solving for d in thee equation: d × e message 1 (mod mbH (n)).

Message Encryption andd Decryption

Tu szyfruje się message, konwertuje it into a numerical format m, where 0 ≤ m permanent; lt; n. The ciphertext c is then coputed using the public key (n, e): c = m ^ e mod n.

Decryption involves using the private key d to recover the original message: m = c ^ d mod n. This process ensures that only someone with the private key can decrypt the message.

Praktykal Calculation Example

Suppose p = 61 and q = 53. Obliczenia n = 61 × 53 = 3233. Then, Ά( n) = (61 - 1) × (53 - 1) = 60 × 52 = 3120. Choose e = 17, which is coprime witch 3120.

Find d such that d × 17 memod (mod 3120). The value of d is 2753. The public key is (n = 3233, e = 17), ande the private key is (n = 3233, d = 2753).

To code-pt a message m = 65, compute c = 65 ^ 17 mod 3233, resutting in c = 2790. To decrypt, compute m = 2790 ^ 2753 mod 3233, which yields thee original message 65.