## Maid### Challenge
```python#!/usr/bin/python3
from Crypto.Util.number import *from gmpy2 import *from secret import *from flag import flag
global nbitnbit = 1024
def keygen(nbit):    while True:        p, q = [getStrongPrime(nbit) for _ in '01']        if p % 4 == q % 4 == 3:            return (p**2)*q, p
def encrypt(m, pubkey):    if GCD(m, pubkey) != 1 or m >= 2**(2*nbit - 2):        return None    return pow(m, 2, pubkey)
def flag_encrypt(flag, p, q):    m = bytes_to_long(flag)    assert m < p * q    return pow(m, 65537, p * q)
def die(*args):    pr(*args)    quit()
def pr(*args):    s = " ".join(map(str, args))    sys.stdout.write(s + "\n")    sys.stdout.flush()
def sc():    return sys.stdin.readline().strip()
def main():    border = "+"    pr(border*72)    pr(border, "  hi all, welcome to Rooney Oracle, you can encrypt and decrypt any ", border)    pr(border, "  message in this oracle, but the flag is still encrypted, Rooney   ", border)    pr(border, "  asked me to find the encrypted flag, I'm trying now, please help! ", border)    pr(border*72)
    pubkey, privkey = keygen(nbit)    p, q = privkey, pubkey // (privkey ** 2)
    while True:        pr("| Options: \n|\t[E]ncrypt message \n|\t[D]ecrypt ciphertext \n|\t[S]how encrypted flag \n|\t[Q]uit")        ans = sc().lower()        if ans == 'e':            pr("| Send the message to encrypt: ")            msg = sc()            try:                msg = int(msg)            except:                die("| your message is not integer!!")            pr(f"| encrypt(msg, pubkey) = {encrypt(msg, pubkey)} ")        elif ans == 'd':            pr("| Send the ciphertext to decrypt: ")            enc = sc()            try:                enc = int(enc)            except:                die("| your message is not integer!!")            pr(f"| decrypt(enc, privkey) = {decrypt(enc, privkey)} ")        elif ans == 's':            pr(f'| enc = {flag_encrypt(flag, p, q)}')        elif ans == 'q':            die("Quitting ...")        else:            die("Bye ...")
if __name__ == '__main__':    main()```
We are given access to an oracle which allows us to encrypt and decrypt data with
$$c = m^2 \pmod n, \qquad n = p^2 q$$
and we can request a flag encrypted as
$$c = m^{e} \pmod {pq}, \qquad e = 65537$$
Where $p,q$ are 1024 bit primes. The goal is to use the oracle to factor $n$ and hence obtain the flag.
### Solution
The first step is to obtain $n = p^2 q$, which we can do by computing:
$$n = \gcd(m_1^2 - c_1, m^2_2 - c_2)$$
by using the oracle to obtain $c_i$ from integers $m_i$. 
Note: there may by other factors, and we actually compute $kn$ for some $k \in \mathbb{Z}$. We can ensure $k = 1$ by computing many $m_i,c_i$ and taking the gcd many times.
The second step is to obtain one of the prime factors.
We cannot send very large numbers to encrypt, but can get around this by sending $E(-X)$ for arbitary sized $X$. As the encryption is simply squaring, this makes no difference. **Defund** noticed that if you decrypt $X = D(2)$ and then compute $E(-X)$, you do not obtain $2$, but rather some very large integer.
I'm not sure how **Defund** solved it, but playing with the numbers while writing it up, I noticed that:
$$\gcd(n, D(-1) - 1) = p$$
which allowed me to solve. Looking around online, some other people seem to have solved by doing something like 
$$\gcd(n, D(m)^2 - m ) = p^2$$
but seeing as we have no source, a bit of guess work needed to be done one way or another, which seems like a shame.
#### Implementation
```pythonfrom pwn import *from Crypto.Util.number import *from math import gcdimport random
r = remote('04.cr.yp.toc.tf', 38010)
def encrypt(msg):    r.recvuntil(b"[Q]uit")    r.sendline(b"E")    r.recvuntil(b"encrypt: ")    r.sendline(str(msg))    r.recvuntil(b" = ")    return int(r.recvline().strip())
def decrypt(msg):    r.recvuntil(b"[Q]uit")    r.sendline(b"D")    r.recvuntil(b"decrypt: ")    r.sendline(str(msg))    r.recvuntil(b" = ")    return int(r.recvline().strip())
def get_flag():    r.recvuntil(b"[Q]uit")    r.sendline(b"S")    r.recvuntil(b" = ")    return int(r.recvline().strip())
def recover_n():    # Obtain kn    m = 2**1536 - random.randint(1,2**1000)    c = encrypt(m)    n = m**2 - c    # Remove all factors of two    while n%2 == 0:        n = n // 2    # Compute a few more GCD to remove any other factors.    for _ in range(10):        m = 2**1536 - random.randint(1,2**1000)        c = encrypt(m)        n = gcd(n, m**2 - c)    return n
def dec_flag(p,q):    c = get_flag()    d = pow(0x10001, -1, (p-1)*(q-1))    m = pow(c,d,p*q)    return long_to_bytes(m)
def recover_factors(n):    X = decrypt(-1)    p = gcd(X - 1, n)    assert isPrime(p)    q = n // (p*p)     assert isPrime(q)    return p, q
n = recover_n()p, q = recover_factors(n)flag = dec_flag(p,q)print(flag)# CCTF{___Ra8!N_H_Cryp70_5YsT3M___}```
##### Flag
`CCTF{___Ra8!N_H_Cryp70_5YsT3M___}`
### Decryption
At the end of the CTF, Factoreal shared the decrypt function. Seeing this, the methods of solving make sense, but it seems like a shame that this wasn't included within the challenge
```pythondef decrypt(c, privkey):    m_p = pow(c, (privkey + 1) // 4, privkey)    i = (c - pow(m_p, 2)) // privkey    j = i * inverse(2*m_p, privkey) % privkey    m = m_p + j * privkey    if 2*m < privkey**2:        return m    else:        return privkey**2 - m```