work

author: cyfraeviolae <cyfraeviolae> 2022-08-24 00:38:04 -0400
committer: cyfraeviolae <cyfraeviolae> 2022-08-24 00:38:04 -0400
commit: d4e149445bda4a73dc3bb987e6ba296c7d6fe84e (patch)
tree: 5edc1b104f45239367a3a380a911de3b97dafa80 /templates/nonce-reuse.html
parent: 32137f612509ee577703d4316dc6a2ec937da709 (diff)
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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>
+		<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>
+        <br>
+        <form action="/forbidden-salamanders/nonce-reuse" method="post">
+            <div>
+            <label for="key">Key (16 bytes in hex)</label>
+			<input name="key" id="key" type="text" value="{{ key.hex() if key else '59454c4c4f575f5355424d4152494e45' }}" minlength=32 maxlength=32>
+            </div>
+
+            <div>
+            <label for="nonce">Nonce (12 bytes in hex)</label>
+			<input name="nonce" id="nonce" type="text" value="{{ nonce.hex() if nonce else '4a4f5247454c424f52474553' }}" minlength=24 maxlength=24>
+            </div>
+
+            <div>
+            <label for="m1">First intercepted message (in ASCII)</label>
+			<input name="m1" id="m1" type="text" required maxlength=100 value="{{m1}}">
+            </div>
+
+            <div>
+            <label for="m2">Second intercepted message (in ASCII)</label>
+			<input name="m2" id="m2" type="text" required maxlength=100 value="{{m2}}">
+            </div>
+
+            <div>
+            <label for="mf">Forged message; shorter than the first message (in ASCII)</label>
+			<input name="mf" id="mf" type="text" required maxlength=100 value="{{mf}}">
+            </div>
+
+            <div>
+				<button type="submit">Recover authentication key and forge MAC</button>
+            </div>
+        </form>
+		{% if macs %}
+        <div>
+			<p>
+				Forged ciphertext: <code>{{ c_forged.hex() }}</code>
+			</p>
+			Forged MAC candidates:
+			<ul>
+				{% for h, _, mac in macs %}
+				<li>
+					MAC: <code>{{mac.hex()}}</code>
+					<ul>
+						<li>Authentication key: <code>{{h.hex()}}</code></li>
+					</ul>
+				</li>
+				{% endfor %}
+			</ul>
+			<form action="/forbidden-salamanders/nonce-reuse" method="get">
+            <div>
+				<button type="submit">Reset</button>
+            </div>
+			</form>
+        </div>
+		{% endif %}
+        <br>
+		<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 = b"The universe (which others call the Library)"
+aad1 = b"The Anatomy of Melancholy"
+m2 = b"From any of the hexagons one can see, interminably"
+aad2 = b"Letizia Alvarez de Toledo"
+
+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"
+assert len(m_forged) <= len(m1)
+c_forged = xor(c1, xor(m1, m_forged))
+aad_forged = b"You who read me, are You sure of understanding my language?"
+
+# Check possible candidates for authentication key
+succeeded = False
+for h, s in possible_secrets:
+    mac_forged = gmac(h, s, aad_forged, c_forged)
+    try:
+        assert gcm_decrypt(k, nonce, aad_forged, c_forged, mac_forged) == m_forged
+        succeeded = True
+        print(c_forged.hex(), mac_forged.hex())
+    except AssertionError:
+        pass
+assert succeeded</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 &ldquo;greater&rdquo; 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>
+<script>
+MathJax = {
+  tex: {
+		extensions: ["AMSmath.js", "AMSsymbols.js"]
+  }
+};
+</script>
+<script id="MathJax-script" async
+  src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js">
+</script>
+  </body>
+</html>
author	cyfraeviolae <cyfraeviolae>	2022-08-24 00:38:04 -0400
committer	cyfraeviolae <cyfraeviolae>	2022-08-24 00:38:04 -0400
commit	d4e149445bda4a73dc3bb987e6ba296c7d6fe84e (patch)
tree	5edc1b104f45239367a3a380a911de3b97dafa80 /templates/nonce-reuse.html
parent	32137f612509ee577703d4316dc6a2ec937da709 (diff)