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diff --git a/nonce-reuse.html b/nonce-reuse.html deleted file mode 100644 index 21c794d..0000000 --- a/nonce-reuse.html +++ /dev/null @@ -1,229 +0,0 @@ -<!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> |