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<!DOCTYPE html>
<html>
<head>
<title>Forbidden Salamanders</title>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="stylesheet" type="text/css" href="/static/styles.css">
<link rel="stylesheet" type="text/css" href="/forbidden-salamanders/static/styles.css">
<link rel="shortcut icon" type="image/x-icon" href="/forbidden-salamanders/static/favicon.ico">
</head>
<body>
<div class="container">
<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>aes-gcm nonce reuse</strong></a>
<span class="sep"> · </span>
<a href="/forbidden-salamanders/nonce-truncation">aes-gcm nonce truncation</a>
<span class="sep"> · </span>
<a href="/forbidden-salamanders/key-commitment">aes-gcm 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>
{% if form.errors %}
<div class="errors">
Errors:
<ul>
{% for name, errors in form.errors.items() %}
{% for error in errors %}
<li> {{name}}: {{ error }} </li>
{% endfor %}
{% endfor %}
</ul>
</div>
{% endif %}
<form action="/forbidden-salamanders/nonce-reuse" method="post">
<div><em>
Roseacrucis chooses a key, a nonce, and two messages. He encrypts both messages under the same nonce.
</em></div><br>
<div>
<label for="key">Key (16 bytes in hex)</label>
<input name="key" id="key" type="text" value="{{ key if key else '746c6f6e6f7262697374657274697573' }}" minlength=32 maxlength=32 required>
</div>
<div>
<label for="nonce">Nonce (12 bytes in hex)</label>
<input name="nonce" id="nonce" type="text" value="{{ nonce if nonce else '4a4f5247454c424f52474553' }}" minlength=24 maxlength=24 required>
</div>
<div>
<label for="m1">First intercepted message</label>
<input name="m1" id="m1" type="text" required maxlength=64 value="{{m1 if m1 else 'The universe (which others call the Library)'}}">
</div>
<div>
<label for="m2">Second intercepted message</label>
<input name="m2" id="m2" type="text" required maxlength=64 value="{{m2 if m2 else 'From any of the hexagons one can see, interminably'}}">
</div>
<br><div><em>
After intercepting the ciphertexts, you choose a new message to forge under the same key and nonce.
</em></div><br>
<div>
<label for="mf">Forged message; shorter than the first message</label>
<input name="mf" id="mf" type="text" required maxlength=64 value="{{mf}}">
</div>
<div>
<button type="submit">Recover authentication key and forge MAC</button>
</div>
</form>
<form action="/forbidden-salamanders/nonce-reuse" method="get">
<div>
<button type="submit">Reset</button>
</div>
</form>
{% if macs %}
<div class="solution">
<p>
Forged ciphertext: <code>{{ c_forged.hex() }}</code>
{% if macs|length == 1 %}
<br>
Forged MAC: <code>{{macs[0][2].hex()}}</code>
<br>
Authentication key: <code>{{macs[0][0].hex()}}</code></li>
{% endif %}
</p>
{% if macs|length != 1 %}
Forged MAC candidates:
<ul>
{% for h, _, mac in macs %}
<li>
MAC: <code>{{mac.hex()}}</code>
<ul class="inner-ul">
<li>Authentication key: <code>{{h.hex()}}</code></li>
</ul>
</li>
{% endfor %}
</ul>
{% endif %}
</div>
{% endif %}
<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:
\[
c' = c \oplus m \oplus m'.
\]
</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 matter of finding the roots by <a href="https://en.wikipedia.org/wiki/Factorization_of_polynomials_over_finite_fields">factoring the
polynomial</a>.
</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. Note that there may be multiple possible monomial roots;
in this case, one can check each possibility online.
</p>
</details>
<details>
<summary>
Show me the code.
</summary>
<pre>
from <a href="/git/forbidden-salamanders">aesgcmanalysis</a> import xor, gmac, gcm_encrypt, gcm_decrypt, nonce_reuse_recover_secrets
k = b"tlonorbistertius"
nonce = b"jorgelborges"
m1, aad1 = b"The universe (which others call the Library)", b""
m2, aad2 = b"From any of the hexagons one can see, interminably", b""
c1, mac1 = gcm_encrypt(k, nonce, aad1, m1)
c2, mac2 = gcm_encrypt(k, nonce, aad2, m2)
# Recover the authentication key and blind from public information
possible_secrets = nonce_reuse_recover_secrets(nonce, aad1, aad2, c1, c2, mac1, mac2)
# Forge the ciphertext
m_forged = b"As was natural, this inordinate hope"
c_forged, aad_forged = xor(c1, xor(m1, m_forged)), b""
for h, s in possible_secrets:
print("MAC candidate": gmac(h, s, aad_forged, c_forged))</pre></details>
<details>
<summary>
Show me the math.
</summary>
<p>
Once the polynomial difference is computed, one can use SageMath
to compute the factors:
</p>
<pre>
K = GF(2**128, name='x', modulus=x^128+x^7+x^2+x+1)
x = K.gen()
S = PolynomialRing(K, 'y')
y = S.gen()
p = (x^127 + x^125 + x^121 + x^118 + x^117 + x^116 + x^113 + x^111 + x^108 + x^105 + x^102 + x^99 + x^97 + x^95 + x^89 + x^87 +
x^85 + x^84 + x^81 + x^78 + x^73 + x^71 + x^69 + x^67 + x^65 + x^63 + x^62 + x^59 + x^57 + x^55 + x^54 + x^53 + x^52 + x^51 +
x^49 + x^47 + x^46 + x^45 + x^43 + x^41 + x^39 + x^38 + x^37 + x^36 + x^33 + x^29 + x^28 + x^25 + x^21 + x^20 + x^17 + x^15 + x
^13 + x^9 + x^7 + x^4 + x^3 + x)*y^5 + (x^127 + x^125 + x^121 + x^118 + x^117 + x^116 + x^113 + x^111 + x^108 + x^105 + x^102 +
x^99 + x^97 + x^95 + x^89 + x^87 + x^85 + x^84 + x^81 + x^78 + x^73 + x^71 + x^69 + x^67 + x^65 + x^63 + x^62 + x^59 + x^57 +
x^55 + x^54 + x^53 + x^52 + x^51 + x^49 + x^47 + x^46 + x^45 + x^43 + x^41 + x^39 + x^38 + x^37 + x^36 + x^33 + x^29 + x^28 + x
^25 + x^21 + x^20 + x^17 + x^15 + x^13 + x^9 + x^7 + x^4 + x^3 + x)*y^4 + (x^127 + x^125 + x^124 + x^123 + x^122 + x^120 + x^11
9 + x^118 + x^115 + x^112 + x^111 + x^110 + x^109 + x^108 + x^106 + x^101 + x^100 + x^99 + x^98 + x^96 + x^93 + x^88 + x^87 + x
^84 + x^83 + x^82 + x^78 + x^77 + x^74 + x^71 + x^70 + x^69 + x^68 + x^65 + x^62 + x^61 + x^60 + x^57 + x^55 + x^53 + x^50 + x^
49 + x^47 + x^46 + x^44 + x^43 + x^41 + x^40 + x^39 + x^37 + x^36 + x^35 + x^34 + x^33 + x^31 + x^30 + x^25 + x^19 + x^16 + x^1
4 + x^13 + x^12 + x^11 + x^10 + x^5 + x^2 + x + 1)*y^3 + (x^124 + x^123 + x^121 + x^120 + x^119 + x^118 + x^117 + x^116 + x^115
+ x^113 + x^112 + x^110 + x^107 + x^106 + x^104 + x^103 + x^101 + x^99 + x^98 + x^95 + x^94 + x^83 + x^82 + x^81 + x^80 + x^79
+ x^77 + x^76 + x^75 + x^73 + x^72 + x^67 + x^64 + x^63 + x^62 + x^61 + x^59 + x^57 + x^56 + x^54 + x^53 + x^51 + x^49 + x^48
+ x^46 + x^45 + x^44 + x^42 + x^36 + x^35 + x^34 + x^33 + x^31 + x^28 + x^22 + x^21 + x^17 + x^12 + x^11 + x^10 + x^8 + x^5 + x
^4 + 1)*y^2 + (x^120 + x^119 + x^56 + x^55)*y + (x^127 + x^126 + x^125 + x^124 + x^123 + x^118 + x^117 + x^116 + x^115 + x^114
+ x^113 + x^105 + x^103 + x^98 + x^96 + x^94 + x^91 + x^90 + x^89 + x^87 + x^86 + x^85 + x^83 + x^81 + x^80 + x^78 + x^77 + x^7
0 + x^66 + x^63 + x^62 + x^59 + x^58 + x^54 + x^53 + x^52 + x^50 + x^47 + x^45 + x^44 + x^42 + x^40 + x^39 + x^37 + x^35 + x^34
+ x^30 + x^29 + x^27 + x^26 + x^25 + x^23 + x^18 + x^16 + x^14 + x^13 + x^12 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3
+ 1)
for factor, _ in p.factor():
if factor.degree() == 1:
print('Authentication key:', factor - y)</pre>
<p>
However, the library powering this demonstration implements <a href="https://en.wikipedia.org/wiki/Factorization_of_polynomials_over_finite_fields">polynomial factoring over finite fields</a> from scratch, which is an edifying exercise.
</p>
<p>
We present advice for those who wish to implement polynomial factorization as well:
</p>
<ul>
<li>The gcd of two polynomials is unique only up to multiplication by a non-zero constant because “greater” is defined for polynomials in terms of degree. When used in algorithms, gcd refers to the <em>monic</em> gcd, which is unique.</li>
<li>The <a href="https://math.stackexchange.com/a/943626/1084004">inverse Frobenius automorphism</a> (i.e., square root) in \(\mathbb{F}_{2^{128}}\) is given by \(\sqrt{x} = x^{2^{127}}\).</li>
</ul>
</details>
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