backdoor CTF 2015: RSANNE

,

https://github.com/ctfs/write-ups-2015/tree/master/backdoor-ctf-2015/crypto/rsanne

http://factordb.com で一発だったが、本番で通用しなさそうなのでだめ。

solution

id_rsa.pubを見てみるとすごく奇妙なみためをしている。

$ openssl rsa -in id_rsa.pub -pubin -text -noout
Public-Key: (4484 bit)
Modulus:
    0f:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    ff:ff:ff:ff:ff:fd:ff:ff:ff:ff:ff:ff:ff:ff:ff:
    f8:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:00:00:00:00:00:00:00:00:00:00:
    00:00:00:00:00:01
Exponent: 65537 (0x10001)

$2$進数にすると以下で、文字列処理で約数が出せそう。実際出せる。 $2^k-1 = {\overbrace{111 \dots 1}^k}_2$なのを使って、$n = (2^{4484}-1) - 2^{2281} - (2^{2203}-1) + 1$と書ける。 これより$n = 2^{4484} - 2^{2281} - 2^{2203} + 1 = (2^{2281} - 1)(2^{2203} - 1)$となり、Mersenne素数の積であったと分かる。

0b111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111111011111111111111111111111111111111111
11111111111111111111111111111111111111111110000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000000
000001
#!/usr/bin/env python3
import base64
import gmpy2
from Crypto.PublicKey import RSA
from Crypto.Cipher import PKCS1_OAEP

with open('id_rsa.pub') as fh:
    key = RSA.importKey(fh.read())

s = bin(key.n)[2:]
a = s.find('0') + 1
b = len(s) - a
assert a < b
assert s == ('1' * (a-1) + '0' + '1' * (b-a) + '0' * (a-1) + '1')
p = 2**a-1
q = 2**b-1
assert p * q == key.n

d = lambda p, q, e: int(gmpy2.invert(e, (p-1)*(q-1)))
key = RSA.construct((p*q, key.e, d(p, q, key.e)))
key = PKCS1_OAEP.new(key)
with open('flag.enc') as fh:
    c = base64.b64decode(fh.read())
print(key.decrypt(c).decode())