29 August 2017-Bucharest, Romania. The movie under the stars event, created by the city hall in the park Moghiorosh. You can see a projection screen and people waiting for it to start

The mathematics of RSA encryption

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RSA (Rivest-Shamir-Adleman) is a widely used public-key encryption algorithm that is based on the mathematical concept of prime factorization. It uses two large prime numbers, along with a mathematical function, to generate a pair of keys that can be used to encrypt and decrypt messages.

The process of generating the keys works as follows:

  1. Choose two large prime numbers, p and q.
  2. Calculate n = p * q. This value is used as part of both the public and private keys.
  3. Calculate λ(n) = lcm(λ(p), λ(q)). Here, λ is the Carmichael function and lcm stands for least common multiple. This value is used as part of the private key.
  4. Choose an integer e such that 1 < e < λ(n) and gcd(e, λ(n)) = 1. This value is used as part of the public key.
  5. Calculate d such that d * e ≡ 1 (mod λ(n)). This value is used as part of the private key.

The resulting public key consists of the values of n and e, and the private key consists of the values of n and d. To encrypt a message, the sender uses the recipient’s public key to transform the plaintext into ciphertext. To decrypt the message, the recipient uses their private key to reverse the process and recover the original plaintext.

The security of RSA relies on the difficulty of factoring large composite numbers, such as the value of n. As long as the prime factors of n are kept secret, it is computationally infeasible for an attacker to determine the private key and decrypt the message.

Cracking RSA

It is generally considered to be very difficult, if not impossible, to crack RSA encryption using classical methods. This is because the security of RSA relies on the difficulty of factoring large composite numbers, such as the value of n that is used as part of the key. As long as the prime factors of n are kept secret, it is computationally infeasible for an attacker to determine the private key and decrypt the message.

However, it is important to note that no encryption method is completely foolproof, and new methods and techniques for breaking encryption are being developed all the time. As such, it is always a good idea to use strong keys and to keep your encryption software and protocols up to date in order to help protect against potential attacks.

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