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<!DOCTYPE html>
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<head>
<title>Forbidden Salamanders</title>
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<div>
<div class="home">
<a href="/forbidden-salamanders" class="home-title">Forbidden Salamanders</a>
<span> at </span><a href="/">cyfraeviolae.org</a>
</div>
<div class="crumbs">
<a href="/git/forbidden-salamanders">source code</a>
<span class="sep"> · </span>
<a href="/forbidden-salamanders/nonce-reuse"><strong>nonce reuse</strong></a>
<span class="sep"> · </span>
<a href="/forbidden-salamanders/nonce-truncation">nonce truncation</a>
<span class="sep"> · </span>
<a href="/forbidden-salamanders/key-commitment">key commitment</a>
</div>
</div>
<p>
<strong>Nonce reuse.</strong> Due to rising entropy
prices, Roseacrucis has started to reuse nonces. You must perform the
Forbidden Attack in order to recover the authentication key and
forge arbitrary ciphertext.
</p>
<br>
<details>
<summary>
Attack outline.
</summary>
<p>
Recall that the AES-GCM ciphertext is computed as the XOR of the
keystream and the message. One can modify the bits of the
ciphertext arbitrarily to effect the same change in the decrypted
plaintext.
</p>
<p>
Where certain bits of the plaintext are already known, the attacker
can fully determine the same bits of the forged plaintext. If
nonces are reused, the keystream will be identical, allowing us to
recover plaintext via
<a href="https://samwho.dev/blog/toying-with-cryptography-crib-dragging/">
crib dragging</a>, which makes this attack particularly effective.
</p>
<p>
However, we still need to compute a new MAC over the forged ciphertext.
Simplifying for a ciphertext \(c\) of two blocks, the GMAC MAC is computed as
\[
mac = s + (len)h + c_1h^2 + c_0h^3,
\]
where \(s\) is a constant depending on the AES-GCM key and the nonce, and \(h\)
is the authentication key depending only on the AES-GCM key.
</p>
<p>
If we intercept a second ciphertext \(c'\) encrypted under the same key and nonce,
we can compute
\[
mac + mac' = (s + s') + (len + len')h + (c_1 + c'_1)h^2 + (c_0+c'_0)h^3,
\]
Since \(s = s'\) and \(x+x=0 \in \mathbb{F}_{2^{128}}\), we are
left with the polynomial equation
\[
0 = (mac + mac') + (len + len')h + (c_1 + c'_1)h^2 + (c_0+c'_0)h^3
\]
where all variables are known other than \(h\). Thus, recovering \(h\)
is a simple matter of finding the roots by <a href="https://en.wikipedia.org/wiki/Factorization_of_polynomials_over_finite_fields">factoring the
polynomial</a> using a computer algebra system such as SageMath.
</p>
<p>
We plug \(h\) back into the first equation to recover \(s\),
and finally, we can forge the MAC for arbitary ciphertext under the
same nonce.
</p>
</details>
<br>
<form>
<div>
<input type="button" value="Autofill example">
</div>
<div>
<label for="key">Key (16 bytes in hex)</label>
<input id="key" type="text">
</div>
<div>
<label for="nonce">Nonce (12 bytes in hex)</label>
<input id="nonce" type="text">
</div>
<div>
<label for="m1">First message (in ASCII)</label>
<input id="m1" type="text">
</div>
<div>
<label for="aad1">First additional authenticated data (in ASCII)</label>
<input id="aad1" type="text">
</div>
<div>
<label for="m2">Second message (in ASCII)</label>
<input id="m2" type="text">
</div>
<div>
<label for="aad2">Second additional authenticated data (in ASCII)</label>
<input id="aad2" type="text">
</div>
<div>
<label for="c">Forged ciphertext (in hex)</label>
<input id="c" type="text">
</div>
<div>
<input type="button" value="Compute authentication key and forge MAC">
</div>
</form>
<div>
<label for="h">Authentication key</label>
<input id="h" type="text" disabled value="deadbeef">
</div>
<div>
<label for="mac">Forged MAC</label>
<input id="mac" type="text" disabled value="deadbeef">
</div>
<br>
<pre>
from <a href="#">aesgcmanalysis</a> import xor, gcm_encrypt, nonce_reuse_recover_secrets, forge_gmac
k = b"YELLOW_SUBMARINE"
nonce = b"JORGELBORGES"
m1 = b"""The universe (which others call the Library) is composed of an indefinite and
perhaps infinite number of hexagonal galleries, with vast air shafts between,
surrounded by very low railings."""
aad1 = b"The Anatomy of Melancholy"
m2 = b"From any of the hexagons one can see, interminably, the upper and lower floors."
aad2 = b"Letizia Alvarez de Toledo"
c1, mac1 = gcm_encrypt(k, nonce, m1, aad1)
c2, mac2 = gcm_encrypt(k, nonce, m2, aad2)
# Recover the authentication key and blind from public information
h, s = nonce_reuse_recover_secrets(nonce, aad1, aad2, c1, c2, mac1, mac2)
# Forge the ciphertext
m_forged = b"As was natural, this inordinate hope was followed by an excessive depression."
c_forged = xor(c2, xor(m2, m_forged))
aad_forged = b"You who read me, are You sure of understanding my language?"
mac_forged = forge_gmac(nonce, h, s, c_forged, aad_forged)
assert gcm_decrypt(k, nonce, c_forged, aad_forged, mac_forged) == m_forged
</pre>
<details>
<summary>
Show me the code.
</summary>
</details>
<details>
<summary>
Show me the math.
</summary>
see homepage
The forged ciphertext can be computed as a xor from original ciphertext,
which should be particularly easy as nonce reuse reveals the XOR and then
crib dragging.
</details>
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