'''
Simplification of Euler's Totient function knowing
the prime factorisation for the public key N value.  
'''
def _totient(p: int, q: int) -> int:
    return (p - 1) * (q - 1)

'''
Implements RSA encryption as modular exponentiation. 
'''
def encrypt(plaintext: int, e: int, N: int) -> int:
    return pow(plaintext, e, N)
def decrypt(ciphertext: int, d: int, N: int) -> int:
    return pow(ciphertext, d, N)

def gen_private_key(e: int, p: int, q: int) -> int:
    return pow(e, -1, _totient(p, q))