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<title>Forbidden Salamanders</title>
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<div class="home">
<a href="/forbidden-salamanders" class="home-title">Forbidden Salamanders</a>
<span> at </span><a class="site" href="/">cyfraeviolae.org</a>
<a class="source" href="/git/forbidden-salamanders">[src]</a>
</div>
<div class="crumbs">
<a href="/forbidden-salamanders/key-commitment">key commitment</a>
<span class="sep"> · </span>
<a href="/forbidden-salamanders/nonce-reuse">nonce reuse</a>
<span class="sep"> · </span>
<a href="/forbidden-salamanders/mac-truncation">mac truncation</a>
</div>
</div>
<p>
The FIPS-compliant sorcerer Roseacrucis uses the <a href="https://en.wikipedia.org/wiki/Galois/Counter_Mode">Advanced Encryption Standard in Galois/Counter Mode</a>
to correspond with his retinue. The Library’s cryptanalysts
have intercepted the communication channel, but we need your
help to exploit their broken protocols.
</p>
<p>
Choose one of the following missions.
</p>
<p>
<strong><a href="/forbidden-salamanders/key-commitment">Key commitment</a>.</strong> One of
our agents has infiltrated Roseacrucis’ inner circle, but all
secret keys are required to be surrendered to the
counterintelligence authority. Help her send AES-GCM ciphertexts back to
the Library that decrypt to confidential information under one key,
but innocuous banter under another.
</p>
<p>
<strong><a href="/forbidden-salamanders/nonce-reuse">Nonce
reuse</a>.</strong> Due to rising entropy prices, Roseacrucis has
started to reuse AES-GCM nonces. You must perform the Forbidden Attack in order to
recover the authentication key and forge arbitrary ciphertext.
</p>
<p>
<strong><a href="/forbidden-salamanders/mac-truncation">MAC
truncation</a>.</strong> The sorcerer aims to conserve
bandwidth by truncating AES-GCM MACs. Use the enemy as a decryption
oracle to once again, recover the authentication key and forge
arbitrary ciphertext.
</p>
<br>
<details>
<summary>
Though it is not required to complete your missions, we now
review the construction of AES-GCM.
</summary>
<p>
AES-GCM is a block cipher that accepts a key of 16 bytes,
a nonce of 12 bytes, plaintext, and additional authenticated data.
It returns ciphertext and a message authentication code (MAC).
The construction is <a href="https://csrc.nist.gov/publications/detail/sp/800-38d/final">specified by NIST</a>.
</p>
<p>
The ciphertext is computed as in <a href="https://en.wikipedia.org/wiki/Block_cipher_mode_of_operation#Counter_(CTR)">counter mode</a>, whereas the MAC is computed using the algorithm GMAC.
</p>
<p>
Let
\[
m = \alpha^{128}+\alpha^7 + \alpha^2 + \alpha + 1
\]
\[
\mathbb{K} = \mathbb{F}_{2^{128}}/m.
\]
</p>
<p>
The finite field \(\mathbb{K}\) can be
interpreted as the set of polynomials with coefficients in \(\mathbb{F}_2\)
of degree less than \(128\). Multiplication
is performed modulo \(m\). This field is of characteristic 2;
e.g., \(2=0\), \(a=-a\), \((\alpha^5 + 1)+(\alpha^5 + 1) = 0\).
</p>
<p>
We interpret 16-byte blocks as elements in \(\mathbb{K}\)
in little-endian bit order:
\[
b_0b_1b_2\ldots{}b_{127} \mapsto
b_0 + b_1\alpha + b_2\alpha^2 + \ldots + b_{127}\alpha^{127},
\]
where \(b_0\) is the least significant bit of the first byte of
the block.
</p>
<p>
12-byte nonces are interpreted as 96-bit integers in big-endian
byte order. Let \(\operatorname{Byte} = [0, 2^8-1]\) and
\(x_i\) refer to the \(i\)th 16-byte chunk of the bytestring
\(x\).
</p>
<p>
\(\operatorname{encode_{big}}(x, n)\) encodes an integer \(x\) into \(n\) bytes in big-endian
byte order. \(\operatorname{pad_n}(x, p)\) pads the length of
the bytestring \(x\) to the nearest multiple of \(n\) with the
byte \(p\). \(\operatorname{AES}(k, x)\) refers to
the <a href="https://en.wikipedia.org/wiki/Advanced_Encryption_Standard">128-bit AES block cipher</a>.
</p>
<br>
<div class="algorithm">
<p>\(\operatorname{GMAC}(h\in \mathbb{K}, s\in \mathbb{K}, aad\in \operatorname{Byte}^{y}, c\in \operatorname{Byte}^{z})\)</p>
<ol class="algorithm-code">
<li>\( len = \operatorname{encode_{big}}(8y, 8) \mathbin\Vert \operatorname{encode_{big}}(8z, 8) \)</li>
<li>\( blocks = \operatorname{pad}_{16}(aad, 0) \mathbin\Vert \operatorname{pad}_{16}(c, 0) \mathbin\Vert len \mathbin\Vert s \)</li>
<li>\( N = \frac{\vert blocks \vert}{16} \)</li>
<li>\( \operatorname{return} \sum\limits_{i=1}^{N} blocks_{N-i} h^{i-1}\)</li>
</ol>
</div>
<br>
<br>
<div class="algorithm">
<p>\(\operatorname{AES-GCM}(k\in \operatorname{Byte}^{16}, n\in \operatorname{Byte}^{12}, aad\in \operatorname{Byte}^{y}, m\in \operatorname{Byte}^{z})\)</p>
<ol class="algorithm-code">
<li> \( r = \mathop{\Vert}\limits_{n'=2^{32}n+2}^{2^{32}n+2^{32}-1} \operatorname{AES}(k, n') \)</li>
<li> \( c = r \oplus m \) </li>
<li> \( h = \operatorname{AES}(k, 0) \) </li>
<li> \( s = \operatorname{AES}(k, 2^{32}n + 1) \) </li>
<li> \( \operatorname{return} c, \operatorname{GMAC}(h, s, aad, c) \)</li>
</ol>
</div>
</details>
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