> For the complete documentation index, see [llms.txt](https://kunalwalavalkar.gitbook.io/write-ups/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://kunalwalavalkar.gitbook.io/write-ups/cryptohack/general/xor/xor-properties.md).

# 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

{% hint style="info" %}
Commutative: A ⊕ B = B ⊕ A\
Associative: A ⊕ (B ⊕ C) = (A ⊕ B) ⊕ C\
Identity: A ⊕ 0 = A\
Self-Inverse: A ⊕ A = 0
{% endhint %}

* The conversion of hexadecimal to bytes is the same as the [Hex](/write-ups/cryptohack/general/encoding/hex.md) 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

```python
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}")
```