XOR Properties

Below is a series of outputs where three random keys have been XOR'd together and with the flag. Use the above properties to undo the encryption in the final line to obtain the flag.
KEY1 = a6c8b6733c9b22de7bc0253266a3867df55acde8635e19c73313
KEY2 ^ KEY1 = 37dcb292030faa90d07eec17e3b1c6d8daf94c35d4c9191a5e1e
KEY2 ^ KEY3 = c1545756687e7573db23aa1c3452a098b71a7fbf0fddddde5fc1
FLAG ^ KEY1 ^ KEY3 ^ KEY2 = 04ee9855208a2cd59091d04767ae47963170d1660df7f56f5faf

Commutative: A ⊕ B = B ⊕ A
Associative: A ⊕ (B ⊕ C) = (A ⊕ B) ⊕ C
Identity: A ⊕ 0 = A
Self-Inverse: A ⊕ A = 0

The conversion of hexadecimal to bytes is the same as the Hex level.
After that we can find the key using the given XOR properties.

( FLAG ^ KEY1 ^ KEY3 ^ KEY2 ) ^ ( KEY2 ^ KEY3 ) 
= ( FLAG ^ KEY1 ) ^ [( KEY2 ^ KEY3 ) ^ ( KEY2 ^ KEY3 )]   # Associative property
= ( FLAG ^ KEY1 ) ^ 0                                     # Self-inverse property
= ( FLAG ^ KEY1 )                                         # Identity property
= ( FLAG ^ KEY1 ) ^ KEY1
= FLAG ^ ( KEY1 ^ KEY1 )                                  # Associative property
= FLAG ^ 0                                                # Self-inverse property
= FLAG                                                    # Identity property

So we have XOR FLAG ^ KEY1 ^ KEY3 ^ KEY2 with KEY2 ^ KEY3 and then the result should be XOR with KEY1.

Solution

from pwn import xor

hex_1 = 'a6c8b6733c9b22de7bc0253266a3867df55acde8635e19c73313'
hex_21 = '37dcb292030faa90d07eec17e3b1c6d8daf94c35d4c9191a5e1e'
hex_23 = 'c1545756687e7573db23aa1c3452a098b71a7fbf0fddddde5fc1'
hex_flag123 = '04ee9855208a2cd59091d04767ae47963170d1660df7f56f5faf'

def hex_to_bytes(hex_string):
	byte_string = bytes.fromhex(hex_string)
	return byte_string

bytes_1 = hex_to_bytes(hex_1)
bytes_21 = hex_to_bytes(hex_21)
bytes_23 = hex_to_bytes(hex_23)
bytes_flag123 = hex_to_bytes(hex_flag123)

def xOr(arg1, arg2):
	result = xor(arg1, arg2)
	return result

flag1 = xOr(bytes_flag123, bytes_23)
flag = xOr(flag1, bytes_1).decode()
print(f"{flag}")

Solution​

Solution