# Noisy Channel

Co-authors: @JOHNKRAM

Attachment:

```
#!/usr/local/bin/python
import os
import random

try:
    flag = open("/flag.txt", "rb").read()
except FileNotFoundError:
    flag = b"FLAG{******** REDACTED ********}"

def xor(a, b):
    return bytes(x ^ y for x, y in zip(a, b))

message = input("msg> ").encode()
used = False
for _ in range(20000):
    noise = random.randbytes(len(message))
    # print("noise:", noise.hex()) # You don't need to hear the noise :)
    if input("use? ") == "y":
        message = xor(message, noise)
        used = True
if used and message == b"give me the flag!":
    print(flag)
else:
    print("What I heard:", message.hex())
```

The server generates 20000 random numbers, and we can select some of them so that they xor to zero. Python random uses MT19937, so every bit generated by it, is a linear combination of `32 * 624`-bit internal state in `GF(2)` arithmetic. Since we cannot control the internal state, we hope that, give a linear combination of some `noise`, they sum to zero regardless of the value of the internal state. That is, we need to find a subset of the `noise` that, always xor to zero, no matter what is the internal state.

So, how to find it? First, we need to encode the `20000*136`-bit generated data as linear combinations of `32 * 624`-bit internal state. We regard each generated bit as a vector, of the coefficients for the `32 * 624`-bit internal state. We want to find a linear combination of these vectors, so that all elements become zero. This is essentially constructing a matrix space and finding its kernel (null space). Let's define the process in math:

Let the unknown `32 * 624 = 19968`-bit internal state of MT19937 named \\(s\_1, ..., s\_{19968}\\), so every generated bit \\(g\_i\\) is a linear combination of these unknown bits in `GF(2)` arithmetic:

\\\[ g\_i = a\_{i,1}s\_1 + a\_{i,2}s\_2 + \\cdots + a\_{i,19968}s\_{19968} = \\Sigma\_{j=1}^{19968}a\_{i,j}s\_j \\\]

Now, consider the LSB bit of 20000 generated 136-bit numbers. So we have 20000 \\(g\_i\\). We need to find a linear combination of these \\(g\_i\\) so that they always equal to zero, regardless of \\(s\_i\\):

\\\[ b\_1g\_1 + b\_2g\_2 + \\cdots + b\_{20000}g\_{20000} = \\Sigma\_{i=1}^{20000} b\_ig\_i\\\\ = \\Sigma\_{i=1}^{20000} b\_i(\\Sigma\_{j=1}^{19968}a\_{i,j}s\_j) \\\\ = 0 \\\]

All coefficients of `s_j` should equal to zero. So, we can express the equation into matrix:

\\\[ M = \\begin{pmatrix} a\_{1,1} & a\_{2,1} & \\cdots & a\_{20000,1} \\\\ a\_{1,2} & a\_{2,2} & \\cdots & a\_{20000,2} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ a\_{1,19968} & a\_{2,19968} & \\cdots & a\_{20000,19968} \\end{pmatrix} \\\]

Vector \\(b=(b\_1, b\_2, \\cdots, b\_{20000})^T\\), so:

\\\[ Mb = (0, 0, \\cdots, 0)^T \\\]

To solve \\(b\\), we only need to find the kernel (null space) of matrix \\(M\\), and use its basis. If we find such \\(b\\), we can find a linear combination of the `noise` that, the LSB bit of XOR result should always be zero.

Script:

```
from gf2bv import LinearSystem
from gf2bv.crypto.mt import MT19937
from sage.all import *
import tqdm

ls = LinearSystem([32] * 624)
gen = ls.gens()
rng = MT19937(gen)

msg = b"give me the flag!"

values = []
for _ in range(20000):
    values.append(rng.getrandbits(len(msg) * 8))

states = 32 * 624
count = 20000
nbits = 1
M = matrix(GF(2), states * nbits, count)
for i in tqdm.trange(count):
    for j in range(nbits):
        bits = values[i]._bits[j]
        temp = []
        for k in range(states):
            M[k + states * j, i] = bits & 1
            bits >>= 1

space = M.right_kernel()
print(space)
basis = space.basis()[0]
# save it to file later
print(basis)
```

We found a valid basis that no only makes the LSB bit zero, but also accidentally (actually we can add all 136 bits to the equation, but they turned out to have the same solution) makes the whole XOR result zero:

```
# pip3 install git+https://github.com/maple3142/gf2bv pwntools
from gf2bv import LinearSystem
from gf2bv.crypto.mt import MT19937
from sage.all import *
import random

ls = LinearSystem([32] * 624)
gen = ls.gens()
rng = MT19937(gen)

# the computed vector is saved here
with open("vector", "r") as f:
    for line in f:
        if line.startswith("("):
            vector = [int(i) for i in line.strip()[1:-1].split(", ")]
            break
    print("Got vector", len(vector))

msg = b"give me the flag!"

values = []
for _ in range(20000):
    values.append(rng.getrandbits(len(msg) * 8))

for test in range(5):
    # randomize internal state
    random.seed()
    test_state = list(random.getstate()[1][:-1])

    # linear combination
    res = 0
    for i in range(20000):
        if vector[i]:
            res ^= ls.evaluate(values[i], test_state)
    print(bin(res)) # always zero
```

At last, we can attack simply:

```
from pwn import *

with open("vector", "r") as f:
    for line in f:
        if line.startswith("("):
            vector = [int(i) for i in line.strip()[1:-1].split(", ")]
            break
    print("Got vector", len(vector))

    io = process(["python3", "server.py"])
    io.recvuntil(b"msg> ")
    io.sendline(b"give me the flag!")
    for i in range(20000):
        if vector[i] == 1:
            io.sendline(b"y")
        else:
            io.sendline(b"n")
    io.interactive()
```

The full basis vector \\(b\\) is listed below:

```
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