from Crypto.Util.number import inverse, getPrime, bytes_to_long, long_to_bytes
from sympy import nextprime
import os
import math


flag = "h4tum{...}"
m = bytes_to_long(flag.encode())

p : int = getPrime(512)
diff = int.from_bytes(os.urandom(16), 'big')
q : int = nextprime(p - diff)
n = p * q
phi = (p - 1) * (q - 1)
e = 65537
d = inverse(e, phi)

ciphertext = pow(m, e, n)

print(f"Here is the public key:\nn = {n}\ne = {e}")
print(f"\nHere is the encrypted flag:\n{ciphertext}")

t = math.isqrt(n)

for i in range((n+1)//2 - t + 1):
    val = (t + i)**2 - n
    if val > 0:
        root = math.isqrt(val)
        if root * root == val:
            r = math.isqrt((t+i)**2 - n)
            rec_p = (t+i) - r 
            rec_q = (t+i) + r
            rec_phi = (rec_p-1) * (rec_q-1)
            rec_d = inverse(e, rec_phi)
            plain_text = pow(ciphertext, rec_d, n)
            print("We recovered d yay")
            print(f"Cracked flag: {long_to_bytes(plain_text)}")
            break




# Here is the public key:
# Here is the encrypted flag:
# 13158607848927768767396122915610010628021182023103983078656746030071489695567187651581550866461152226282740548654306454938594539749024910072164328096542525661951926739718955208177780816184581302682602611753650161772357698132248994074563798067850345731371433620586869504864647385937063506440611160969011596274