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Diffstat (limited to 'nonce-reuse.html')
| -rw-r--r-- | nonce-reuse.html | 140 |
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 “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 = { |