UIUCTF 2024 Writeup

Jul 1, 2024

Key in a Haystack

Finding a prime factor via small subgroups using Pollard's p-1 algorithm

Problem

We were given a flag encrypted with AES-ECB using a 40-bit prime key. The key is then multiplied with 300 1024-bit primes, and the result is given to us. The following source code is given:

from Crypto.Util.number import getPrime
from Crypto.Util.Padding import pad
from Crypto.Cipher import AES

from hashlib import md5
from math import prod
import sys

from secret import flag

key = getPrime(40)
haystack = [ getPrime(1024) for _ in range(300) ]
key_in_haystack = key * prod(haystack)

enc_flag = AES.new(
	key = md5(b"%d" % key).digest(),
	mode = AES.MODE_ECB
).encrypt(pad(flag, 16))

sys.set_int_max_str_digits(0)

print(f"enc_flag: {enc_flag.hex()}")
print(f"haystack: {key_in_haystack}")

exit(0)

Approach

We need to realize that, because 40-bit prime key is still relatively small, factorizing key-1 is fast. There is a significant probability that key-1 is B-smooth for a reasonably small B. In other words, given a random 40-bit prime key, there is a high probability that all of the prime factors of key-1 are less than or equal to B, where B is small enough to bruteforce.

Pollard’s p-1 algorithm to find a non-trivial factor of a composite number key-1. It is particularly effective if one of the prime factors is B-smooth. The basic concept of Pollard’s p-1 is demonstrated below for factorizing an RSA key.

source: https://www.math.columbia.edu/~goldfeld/PollardAttack.pdf

All that is left to do is determining the appropriate B value. If B is too small, we might not find p because the biggest prime factors of most p-1 is bigger than B. If B is too big, it will take the algorithm too long to complete.

We can approximate the probability of encountering B-smooth key-1, based on how big the B is, using the script below:

from Crypto.Util.number import getPrime
from sympy.ntheory import factorint
import math

# Generate a 40-bit prime
count=0
for i in range(10000):
    p = getPrime(40)
    p_minus_1 = p - 1


# Factorize p-1
    factors = factorint(p_minus_1)

# Determine smoothness
    def is_smooth(factors, B):
        return all(factor <= B for factor in factors)

# Choose a bound B, adjust as needed
    B = 2**13

# Check if p-1 is B-smooth
    smooth = is_smooth(factors, B)
    if smooth:
        count+=1
print(count/10000)

It was found that

For 8-bit B, there is less than 1% chance
For 16-bit B, there is ~16% chance
For 13-bit B, there is ~5% chance

Using an M1 Macbook Air, it takes

~8 seconds to complete the algorithm for 8-bit B
~16 seconds to complete the algorithm for 9-bit B
~32 seconds to complete the algorithm for 10-bit B
~64 seconds to complete the algorithm for 11-bit B
~128 seconds to complete the algorithm for 12-bit B
~256 seconds to complete the algorithm for 13-bit B

Based on the trade-off, we will choose 13-bit B. With, ~5% chance of getting 13-bit-smooth key-1, we need approximately 14 attempts to have at least a 50% chance of getting a smooth p-1 at least once. So it will probably take around 1 hour to complete the bruteforce, which is reasonable.

Solution

Pollard’s p-1 implementation in Sagemath to retrieve the 40-bit key, using 13-bit B

from sage.all import gcd, factorial, ZZ

# 13-bit B
def pollards_p_minus_1_sage(n, a=2, B=2**13):

    # Step 1: Compute b = B!
    b = factorial(B)

    # Step 2: Compute gcd(tmp1, tmp2) where tmp2 = pow(a, b, n) - 1
    tmp1 = n
    tmp2 = power_mod(a, b, n) - 1
    gcd_value = gcd(tmp1, tmp2)

    return gcd_value

# enc_flag: bcb0ed5c0b1624929c58597737cb83f031ec0ad196614b13a4063a7389ab366d9e3a20b11613c3f8b5a6008eb7fa06a7

n=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p_factor = pollards_p_minus_1_sage(n)

print(f"p factor : {p_factor}")

It took about 4 minutes to complete.

For the enc_flag: bcb0ed5c0b1624929c58597737cb83f031ec0ad196614b13a4063a7389ab366d9e3a20b11613c3f8b5a6008eb7fa06a7
the key was found to be 904863707201

Decrypting the flag

from Crypto.Cipher import AES
from Crypto.Util.Padding import unpad
from hashlib import md5

# Provided key
key = 904863707201

# Compute the AES key using MD5 hash of the key
aes_key = md5(b"%d" % key).digest()

# Encrypted flag (provided in hex)
enc_flag_hex = "bcb0ed5c0b1624929c58597737cb83f031ec0ad196614b13a4063a7389ab366d9e3a20b11613c3f8b5a6008eb7fa06a7"
enc_flag = bytes.fromhex(enc_flag_hex)

# Initialize AES cipher in ECB mode
cipher = AES.new(aes_key, AES.MODE_ECB)

# Decrypt the flag
decrypted_flag = unpad(cipher.decrypt(enc_flag), 16)

# Print the decrypted flag
print(f"Decrypted flag: {decrypted_flag.decode()}")

FLAG: uiuctf{Finding_Key_Via_Small_Subgroups}

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Key in a Haystack

Problem

Approach

Solution

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