Bloom Bloom

6 minutes to read

We are given the Python source code that encrypts the flag:

from random import randint, shuffle
from Crypto.Util.number import getPrime
from Crypto.Cipher import AES
from Crypto.Util.Padding import pad
from hashlib import sha256
from secret import *
import os

assert sha256(KEY).hexdigest().startswith('709149eb5baf8f8cb617226854a7b4f3')

class BBS:
    def __init__(self, bits, length):
        self.bits = bits
        self.out_length = length

    def reset_params(self):
        self.state = randint(2, 2 ** self.bits - 2)
        self.m = getPrime(self.bits//2) * getPrime(self.bits//2) * randint(1, 2)  
    
    def extract_bit(self):
        self.state = pow(self.state, 2, self.m)
        return str(self.state % 2)

    def gen_output(self):
        self.reset_params()
        out = ''
        for _ in range(self.out_length):
            out += self.extract_bit()
        return out

    def encrypt(self, msg):
        out = self.gen_output()
        key = sha256(out.encode()).digest()
        iv = os.urandom(16)
        cipher = AES.new(key, AES.MODE_CBC, iv)
        return (iv.hex(), cipher.encrypt(pad(msg.encode(), 16)).hex())

encryptor = BBS(512, 256)

enc_messages = []
for msg in MESSAGES:
    enc_messages.append([encryptor.encrypt(msg) for _ in range(10)])

enc_flag = AES.new(KEY, AES.MODE_ECB).encrypt(pad(FLAG, 16))

with open('output.txt', 'w') as f:
    f.write(f'{enc_messages}\n')
    f.write(f'{enc_flag.hex()}\n')

And we also have the result in output.txt:

[[('e32c20f22521b1a170968639eb1cf83f', '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'), ('82c4327fc565535adc352336ebce4f8a', '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'), ('9aba0b98c8d542b18f5d86f618f0166f', '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'), ('9fb39e686ba315d74b009a662786fbe1', '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'), ('3cef6a7170546a0a0903a630b9e44032', '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'), ('bcbbb075c9a4496a6d99b0bb75323e06', '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'), ('65caf189809df66fa7689c7f1dd20cfa', '214ce7daa745cf6e18b70078603eb3639624825c64a03cb9650a23c5bbac5118cc7bf2607f747118636e19d6805885a284d50630c1fe4fa0ed59e59e3f9dfa4cb1fc3b58740ed60b2e330f387c154e52ebddd589f01180929fcd006e047472b7bb68bd95aba7eecbb177e1bdcf81b1f62a48dd6ee3d13f5f60ef39fc03f863d5d75e073f452e6741a51d279b81b3129dc7289eff38e72a358eec98913ed4a07791abe3a43fb08aad0862d319407a63360f94a9dbb73ec005fa1c5c1817e6f11bfef9bca6909bf7c0d5ba6a7b4a82b45c68a3a8ec7efa6de2767f2e2f4bab9878fdc20d3b24d1b861222c9aca941999b8798e8c6aa941775b6f9760d450b6d46037c708b021b1967c283736f6b147657f4061a00afd601ecfa3940ed5de5d7865d044d3a03215ee519d1113aedef4586e1af37bed1ae490966d8efaf9f786e820e08f6a2ae4fa1a6c6e917721ff677753e1c666517d2ef9bb49fbe81d5a383bb3fb1c0ed6f766a1b4f53fbcf7251716a80fb2ea92ebb8895880964e77c181e8945a6a6b76158a3e801fdd7cf6c8575a3966178c00dec97094767d591e499d41e07d95ae64569e502c9227a8532c6779bd977c07d4af370e3e96dcc53d3a73d906da425dba2d705ebeba6496bc9173d19757bbb9ebbc4cf6d3a255c867954f0978fbe515351460e54edef9632a58b6e721875c086cb4828e09dbf413a4b8a80e6c'), ('aaed8c3f61ff0aa076add18e2c1ff95c', '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'), ('bb17b9c41a5746f61aa24b35cf7e067f', '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'), ('3396dfd2cbc91d8c990b04852e7291ae', '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')], [('de73200dd21b6c2d61c362362284b159', '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'), ('d8ba2e45a564c3808c64011797746ef8', 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Source code analysis

At first glance, it looks impossible to find the flag, because it depends on KEY, which is not related to the output we have.

So, let’s focus on the encrypted messages (enc_messages):

encryptor = BBS(512, 256)

enc_messages = []
for msg in MESSAGES:
    enc_messages.append([encryptor.encrypt(msg) for _ in range(10)])

Each of the messages is encrypted running encrypt from BBS:

    def encrypt(self, msg):
        out = self.gen_output()
        key = sha256(out.encode()).digest()
        iv = os.urandom(16)
        cipher = AES.new(key, AES.MODE_CBC, iv)
        return (iv.hex(), cipher.encrypt(pad(msg.encode(), 16)).hex())

In this function, the message is encrypted with AES, and the key is the SHA256 hash of gen_output:

    def gen_output(self):
        self.reset_params()
        out = ''
        for _ in range(self.out_length):  
            out += self.extract_bit()
        return out

This function calls reset_params:

    def reset_params(self):
        self.state = randint(2, 2 ** self.bits - 2)
        self.m = getPrime(self.bits//2) * getPrime(self.bits//2) * randint(1, 2)

Which resets the state attibute to a random 512-bit integer, and sets m to the product of two random 256-bit primes times 1 or 2.

Then, it uses extract_bit, which outputs '0' or '1':

    def extract_bit(self):
        self.state = pow(self.state, 2, self.m)  
        return str(self.state % 2)

Solution

Notice that gen_output calls extract_bit a total of 256 times, but the output will always be the same!

Decrypting messages

So, in the end we have encrypted messages with an AES key that is either the SHA256 hash of '0' * 256 or '1' * 256.

We can use the following code to decrypt all messages by trial and error:

#!/usr/bin/env python3

from ast import literal_eval
from hashlib import sha256

from Crypto.Cipher import AES
from Crypto.Util.Padding import unpad

with open('output.txt') as f:
    all_enc_messages = literal_eval(f.readline())
    ct = bytes.fromhex(f.readline())

def try_decrypt(key, iv, ct):
    try:
        pt = unpad(AES.new(key, AES.MODE_CBC, iv=iv).decrypt(ct), AES.block_size)
        return pt
    except ValueError:
        return
    
key0 = sha256(b'0' * 256).digest()
key1 = sha256(b'1' * 256).digest()

messages = []

for enc_messages in all_enc_messages:
    for iv, enc_message in enc_messages:
        if (message := try_decrypt(key0, bytes.fromhex(iv), bytes.fromhex(enc_message))):  
            messages.append(message.decode())
            break
        if (message := try_decrypt(key1, bytes.fromhex(iv), bytes.fromhex(enc_message))):
            messages.append(message.decode())
            break

print('\n\n'.join(messages))

$ python3 solve.py
Welcome! If you see this you have successfully decrypted the first message. To get the symmetric key that decrypts the flag you need to do the following:

1. Collect all 5 shares from these messages
2. Use them to interpolate the polynomial in a finite field that will be revealed in another message
3. Convert the constant term of the polynomial to bytes and use it to decrypt the flag. Here is your first share!

Share#1#: (1, 54374634894026676422403706385209235605654263189334701196866954303113716584451)

Keep up the good work! Offered say visited elderly and. Waited period are played family man formed. He ye body or made on pain part meet. You one delay nor begin our folly abode. By disposed replying mr me unpacked no. As moonlight of my resolving unwilling. Turned it up should no valley cousin he. Speaking numerous ask did horrible packages set. Ashamed herself has distant can studied mrs.

Share#2#: (2, 64946086132218777351978239905100707583094278469454014490074897767952501236845)

Only a few more are left! Of be talent me answer do relied. Mistress in on so laughing throwing endeavor occasion welcomed. Gravity sir brandon calling can. No years do widow house delay stand. Prospect six kindness use steepest new ask. High gone kind calm call as ever is. Introduced melancholy estimating motionless on up as do. Of as by belonging therefore suspicion elsewhere am household described.

Share#3#: (3, 47342035784539865604020206350119236713840941400563052065845192590066845864369)

You are almost there! Not him old music think his found enjoy merry. Listening acuteness dependent at or an. Apartments thoroughly unsatiable terminated sex how themselves. She are ten hours wrong walls stand early. Domestic perceive on an ladyship extended received do. Why jennings our whatever his learning gay perceive. Is against no he without subject. Bed connection unreserved preference partiality not unaffected.  

Share#4#: (4, 3655061496519801785813121517481390175484853044687072580165202783250216005674)

Congratulations!!! Not him old music think his found enjoy merry. Listening acuteness dependent at or an. Apartments thoroughly unsatiable terminated how themselves. She are ten hours wrong walls stand early. Domestic perceive on an ladyship extended received do. You need to interpolate the polynomial in the finite field GF(93932419553446968245028291354625723410141714017280323567892277575234453142991).

Share#5#: (5, 61239598692554281055448838803080244308901202829356284150044638062405311124499)

The above messages show 5 shares $(x_i, y_i)$. They are using a Shamir Secret Sharing scheme to share the KEY.

As shown in the first message, we need to interpolate a polynomial $P$ using the given shares, so that $y_i = P(x_i)$.

We can use Lagrange interpolation to find this polynomial. The finite field is given in the last message.

We can add the following code to the previous script to parse all messages, to get the shares, and also take the prime number for the finite field. Finally, we interpolate the polynomial and take the KEY to decrypt the flag:

from re import search

from Crypto.Util.number import long_to_bytes

from sage.all import GF, PolynomialRing


shares = []

for message in messages:
    if (share_match := search(r'Share#\d+?#: \((\d+?), (\d+?)\)', message)):
        shares.append((int(share_match.group(1)), int(share_match.group(2))))  

    if (gf_match := search(r'GF\((\d+?)\)', message)):
        Fp = GF(int(gf_match.group(1)))

P = PolynomialRing(Fp, 'x')
polynomial = P.lagrange_polynomial(shares)
key = long_to_bytes(int(polynomial(0)))
assert sha256(key).hexdigest().startswith('709149eb5baf8f8cb617226854a7b4f3')

print(unpad(AES.new(key, AES.MODE_ECB).decrypt(ct), AES.block_size).decode())

Flag

If we run the script, we will get the flag:

$ python3 solve.py
HTB{sometimes_the_LSB_can_leak_a_lot_of_crucial_information!}

The full script can be found in here: solve.py.