work

1 files changed, 92 insertions, 48 deletions

diff --git a/nonce-reuse.html b/nonce-reuse.html
index 0795386..21c794d 100644
--- a/nonce-reuse.html
+++ b/nonce-reuse.html
@@ -48,7 +48,10 @@
             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.
+            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.
@@ -65,35 +68,32 @@
             \[
                 mac + mac' = (s + s') + (len + len')h + (c_1 + c'_1)h^2 + (c_0+c'_0)h^3,
             \]
-            Since \(s = s'\) and \(x+x=0 \in \mathbb{F}_{2^{128}}\), we are
+            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.
+            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.
+            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>
-            <input type="button" value="Autofill example">
-            </div>
-
-            <div>
             <label for="key">Key (16 bytes in hex)</label>
-            <input id="key" type="text">
+            <input id="key" type="text" value="59454c4c4f575f5355424d4152494e45">
             </div>
 
             <div>
             <label for="nonce">Nonce (12 bytes in hex)</label>
-            <input id="nonce" type="text">
+            <input id="nonce" type="text" value="4a4f5247454c424f52474553">
             </div>
 
             <div>
@@ -102,27 +102,17 @@
             </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">
+            <label for="mf">Forged message; shorter than the first message (in ASCII)</label>
+            <input id="mf" 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">
+            <input type="button" value="Recover authentication key and forge MAC">
             </div>
         </form>
         <div>
@@ -130,46 +120,100 @@
         <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="#">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."""
+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, the upper and lower floors."
+m2 = b"From any of the hexagons one can see, interminably"
 aad2 = b"Letizia Alvarez de Toledo"
-c1, mac1 = gcm_encrypt(k, nonce, m1, aad1)
-c2, mac2 = gcm_encrypt(k, nonce, m2, aad2)
+
+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
-h, s = nonce_reuse_recover_secrets(nonce, aad1, aad2, c1, c2, mac1, mac2)
+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 was followed by an excessive depression."
-c_forged = xor(c2, xor(m2, m_forged))
+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?"
-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>
+
+# 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>
-            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.
+            <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 &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 = {