1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
|
<!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>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:
\[
c' = c \oplus m \oplus m'.
\]
</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>
<div>
<label for="key">Key (16 bytes in hex)</label>
<input id="key" type="text" value="59454c4c4f575f5355424d4152494e45">
</div>
<div>
<label for="nonce">Nonce (12 bytes in hex)</label>
<input id="nonce" type="text" value="4a4f5247454c424f52474553">
</div>
<div>
<label for="m1">First message (in ASCII)</label>
<input id="m1" type="text">
</div>
<div>
<label for="m2">Second message (in ASCII)</label>
<input id="m2" type="text">
</div>
<div>
<label for="mf">Forged message; shorter than the first message (in ASCII)</label>
<input id="mf" type="text">
</div>
<div>
<input type="button" value="Recover 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="mf">Forged ciphertext, assuming first message is known</label>
<input id="mf" type="text">
</div>
<div>
<label for="mac">Forged MAC</label>
<input id="mac" type="text" disabled value="deadbeef">
</div>
<br>
<details>
<summary>
Show me the code.
</summary>
<pre>
from <a href="/git/forbidden-salamanders">aesgcmanalysis</a> import xor, 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>
A description of the construction of GMAC can be found at the <a
href="/forbidden-salamanders">mission homepage</a>.
</p>
<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>, 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>
<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>
|
