This is a simple implimentation of RSA algorithm in python. It also exposes a few rsa vulnerabilities.

• Choose two large prime numbers p and q.

  • p shouldn't equal q
  • p and q should be a secret

• n = p*q

  • n is the modulus for both public key and private key

• using Euler's totient function phi = (p-1)(q-1)

• choose an interger e such that 2 < e < λ(n) and gcd(e, λ(n)) = 1; that is, e and λ(n) are coprime

  • Generally e is 65537

• Compute d = e (mod λ(n)) using Eucelidean extended algorithm d can be computed efficiently

  • d should be a secret

Public key = (e,n)
Private key = (d,n)

let m be the message

ciphertext = $m^e*mod(n)$

decrypted = $c^d*mod(n)$

Given the performance of this implementation, bit length shouldn't exceed 1024

from pureRsa import Rsa

bits = 1024
bob = Rsa(bits)
alice = Rsa(bits)

#Bob encrypted message with alice's public key
cipherText = bob.encrypt('Euler is in your closet',alice.getPublicKey())

#Alice decrypts with her private key
decrypted = alice.decrypt(cipherText)

print(decrypted)

In order for the brute force to work in a reasonable amount of time, bit length shouldn't exceed 30

from pureRsa import Rsa,Exploits

bits = 27
bob = Rsa(bits)
alice = Rsa(bits)

cipherText = bob.encrypt('Hello, World!',alice.getPublicKey())

e,n = alice.getPublicKey()

x = Exploits(bits)
print('brute forcing...')
x.bruteForce(e,n)
print('Successfully found private key')


print(f'Brute Forced decryption = {x.decrypt(cipherText)}')