Merge pull request #61 from jonasnick/tagged-derive

Use a tagged hash in bip-schnorr nonce derivation

bip-schnorr.mediawiki

@@ -50,13 +50,13 @@ encodings and operations.
 
 === Design ===
 
-'''Schnorr signature variant''' Elliptic Curve Schnorr signatures for message ''m'' and public key ''P'' generally involve a point ''R'', integers ''e'' and ''s'' picked by the signer, and generator ''G'' which satisfy ''e = H(R || m)'' and ''sG = R + eP''. Two formulations exist, depending on whether the signer reveals ''e'' or ''R'':
-# Signatures are ''(e,s)'' that satisfy ''e = H(sG - eP || m)''. This avoids minor complexity introduced by the encoding of the point ''R'' in the signature (see paragraphs "Encoding the sign of R" and "Implicit Y coordinate" further below in this subsection).
-# Signatures are ''(R,s)'' that satisfy ''sG = R + H(R || m)P''. This supports batch verification, as there are no elliptic curve operations inside the hashes.
+'''Schnorr signature variant''' Elliptic Curve Schnorr signatures for message ''m'' and public key ''P'' generally involve a point ''R'', integers ''e'' and ''s'' picked by the signer, and generator ''G'' which satisfy ''e = hash(R || m)'' and ''sG = R + eP''. Two formulations exist, depending on whether the signer reveals ''e'' or ''R'':
+# Signatures are ''(e,s)'' that satisfy ''e = hash(sG - eP || m)''. This avoids minor complexity introduced by the encoding of the point ''R'' in the signature (see paragraphs "Encoding the sign of R" and "Implicit Y coordinate" further below in this subsection).
+# Signatures are ''(R,s)'' that satisfy ''sG = R + hash(R || m)P''. This supports batch verification, as there are no elliptic curve operations inside the hashes.
 
 We choose the ''R''-option to support batch verification.
 
-'''Key prefixing''' When using the verification rule above directly, it is possible for a third party to convert a signature ''(R,s)'' for key ''P'' into a signature ''(R,s + aH(R || m))'' for key ''P + aG'' and the same message, for any integer ''a''. This is not a concern for Bitcoin currently, as all signature hashes indirectly commit to the public keys. However, this may change with proposals such as SIGHASH_NOINPUT ([https://github.com/bitcoin/bips/blob/master/bip-0118.mediawiki BIP 118]), or when the signature scheme is used for other purposes&mdash;especially in combination with schemes like [https://github.com/bitcoin/bips/blob/master/bip-0032.mediawiki BIP32]'s unhardened derivation. To combat this, we choose ''key prefixed''<ref>A limitation of committing to the public key (rather than to a short hash of it, or not at all) is that it removes the ability for public key recovery or verifying signatures against a short public key hash. These constructions are generally incompatible with batch verification.</ref> Schnorr signatures; changing the equation to ''sG = R + H(R || P || m)P''.
+'''Key prefixing''' When using the verification rule above directly, it is possible for a third party to convert a signature ''(R,s)'' for key ''P'' into a signature ''(R,s + a⋅hash(R || m))'' for key ''P + aG'' and the same message, for any integer ''a''. This is not a concern for Bitcoin currently, as all signature hashes indirectly commit to the public keys. However, this may change with proposals such as SIGHASH_NOINPUT ([https://github.com/bitcoin/bips/blob/master/bip-0118.mediawiki BIP 118]), or when the signature scheme is used for other purposes&mdash;especially in combination with schemes like [https://github.com/bitcoin/bips/blob/master/bip-0032.mediawiki BIP32]'s unhardened derivation. To combat this, we choose ''key prefixed''<ref>A limitation of committing to the public key (rather than to a short hash of it, or not at all) is that it removes the ability for public key recovery or verifying signatures against a short public key hash. These constructions are generally incompatible with batch verification.</ref> Schnorr signatures; changing the equation to ''sG = R + hash(R || P || m)P''.
 
 '''Encoding R and public key point P''' There exist several possibilities for encoding elliptic curve points:
 # Encoding the full X and Y coordinates of ''P'' and ''R'', resulting in a 64-byte public key and a 96-byte signature.
@@ -81,7 +81,13 @@ It is important to not mix up the 32-byte bip-schnorr public key format and othe
 
 Implicit Y coordinates are not a reduction in security when expressed as the number of elliptic curve operations an attacker is expected to perform to compute the secret key. An attacker can normalize any given public key to a point whose Y coordinate is a quadratic residue by negating the point if necessary. This is just a subtraction of field elements and not an elliptic curve operation.
 
-'''Final scheme''' As a result, our final scheme ends up using public key ''pk'' which is the X coordinate of a point ''P'' on the curve whose Y coordinate is a quadratic residue and signatures ''(r,s)'' where ''r'' is the X coordinate of a point ''R'' whose Y coordinate is a quadratic residue. The signature satisfies ''sG = R + H(r || p || m)P''.
+'''Tagged Hashes''' Cryptographic hash functions are used for multiple purposes in the specification below and in Bitcoin in general. To make sure hashes used in one context can't be reinterpreted in another one, hash functions can be tweaked with a context-dependent tag name, in such a way that collisions across contexts can be assumed to be infeasible. Such collisions obviously can not be ruled out completely, but only for schemes using tagging with a unique name. As for other schemes collisions are at least less likely with tagging than without.
+
+For example, without tagged hashing a bip-schnorr signature could also be valid for a signature scheme where the only difference is that the arguments to the hash function are reordered. Worse, if the bip-schnorr nonce derivation function was copied or independently created, then the nonce could be accidentally reused in the other scheme leaking the private key.
+
+This proposal suggests to include the tag by prefixing the hashed data with ''SHA256(tag) || SHA256(tag)''. Because this is a 64-byte long context-specific constant, optimized implementations are possible (identical to SHA256 itself, but with a modified initial state). Using SHA256 of the tag name itself is reasonably simple and efficient for implementations that don't choose to use the optimization.
+
+'''Final scheme''' As a result, our final scheme ends up using public key ''pk'' which is the X coordinate of a point ''P'' on the curve whose Y coordinate is a quadratic residue and signatures ''(r,s)'' where ''r'' is the X coordinate of a point ''R'' whose Y coordinate is a quadratic residue. The signature satisfies ''sG = R + tagged_hash(r || p || m)P''.
 
 === Specification ===
 
@@ -109,7 +115,7 @@ The following convention is used, with constants as defined for secp256k1:
 *** Fail if ''c &ne; y<sup>2</sup> mod p''.
 *** Return the unique point ''P'' such that ''x(P) = x'' and ''y(P) = y''.
 ** The function ''point(x)'', where ''x'' is a 32-byte array, returns the point ''P = lift_x(int(x))''.
-** The function ''hash(x)'', where ''x'' is a byte array, returns the 32-byte SHA256 hash of ''x''.
+** The function ''hash<sub>tag</sub>(x)'' where ''tag'' is a UTF-8 encoded tag name and ''x'' is a byte array returns the 32-byte hash ''SHA256(SHA256(tag) || SHA256(tag) || x)''.
 ** The function ''jacobi(x)'', where ''x'' is an integer, returns the [https://en.wikipedia.org/wiki/Jacobi_symbol Jacobi symbol] of ''x / p''. It is equal to ''x<sup>(p-1)/2</sup> mod p'' ([https://en.wikipedia.org/wiki/Euler%27s_criterion Euler's criterion])<ref>For points ''P'' on the secp256k1 curve it holds that ''jacobi(y(P)) &ne; 0''.</ref>.
 
 === Public Key Generation ===
@@ -133,7 +139,7 @@ The signature is valid if and only if the algorithm below does not fail.
 * Let ''P = point(pk)''; fail if ''point(pk)'' fails.
 * Let ''r = int(sig[0:32])''; fail if ''r &ge; p''.
 * Let ''s = int(sig[32:64])''; fail if ''s &ge; n''.
-* Let ''e = int(hash(bytes(r) || bytes(P) || m)) mod n''.
+* Let ''e = int(hash<sub>BIPSchnorr</sub>(bytes(r) || bytes(P) || m)) mod n''.
 * Let ''R = sG - eP''.
 * Fail if ''infinite(R)''.
 * Fail if ''jacobi(y(R)) &ne; 1'' or ''x(R) &ne; r''.
@@ -152,7 +158,7 @@ All provided signatures are valid with overwhelming probability if and only if t
 ** Let ''P<sub>i</sub> = point(pk<sub>i</sub>)''; fail if ''point(pk<sub>i</sub>)'' fails.
 ** Let ''r = int(sig<sub>i</sub>[0:32])''; fail if ''r &ge; p''.
 ** Let ''s<sub>i</sub> = int(sig<sub>i</sub>[32:64])''; fail if ''s<sub>i</sub> &ge; n''.
-** Let ''e<sub>i</sub> = int(hash(bytes(r) || bytes(P<sub>i</sub>) || m<sub>i</sub>)) mod n''.
+** Let ''e<sub>i</sub> = int(hash<sub>BIPSchnorr</sub>(bytes(r) || bytes(P<sub>i</sub>) || m<sub>i</sub>)) mod n''.
 ** Let ''R<sub>i</sub> = lift_x(r)''; fail if ''lift_x(r)'' fails.
 * Fail if ''(s<sub>1</sub> + a<sub>2</sub>s<sub>2</sub> + ... + a<sub>u</sub>s<sub>u</sub>)G &ne; R<sub>1</sub> + a<sub>2</sub>R<sub>2</sub> + ... + a<sub>u</sub>R<sub>u</sub> + e<sub>1</sub>P<sub>1</sub> + (a<sub>2</sub>e<sub>2</sub>)P<sub>2</sub> + ... + (a<sub>u</sub>e<sub>u</sub>)P<sub>u</sub>''.
 
@@ -165,11 +171,11 @@ Input:
 To sign ''m'' for public key ''bytes(dG)'':
 * Let ''P = dG''
 * Let ''d = d' '' if ''jacobi(y(P)) = 1'', otherwise let ''d = n - d' ''.
-* Let ''k' = int(hash(bytes(d) || m)) mod n''<ref>Note that in general, taking the output of a hash function modulo the curve order will produce an unacceptably biased result. However, for the secp256k1 curve, the order is sufficiently close to ''2<sup>256</sup>'' that this bias is not observable (''1 - n / 2<sup>256</sup>'' is around ''1.27 * 2<sup>-128</sup>'').</ref>.
+* Let ''k' = int(hash<sub>BIPSchnorrDerive</sub>(bytes(d) || m)) mod n''<ref>Note that in general, taking the output of a hash function modulo the curve order will produce an unacceptably biased result. However, for the secp256k1 curve, the order is sufficiently close to ''2<sup>256</sup>'' that this bias is not observable (''1 - n / 2<sup>256</sup>'' is around ''1.27 * 2<sup>-128</sup>'').</ref>.
 * Fail if ''k' = 0''.
 * Let ''R = k'G''.
 * Let ''k = k' '' if ''jacobi(y(R)) = 1'', otherwise let ''k = n - k' ''.
-* Let ''e = int(hash(bytes(R) || bytes(P) || m)) mod n''.
+* Let ''e = int(hash<sub>BIPSchnorr</sub>(bytes(R) || bytes(P) || m)) mod n''.
 * The signature is ''bytes(R) || bytes((k + ed) mod n)''.
 
 '''Above deterministic derivation of ''R'' is designed specifically for this signing algorithm and may not be secure when used in other signature schemes.'''

bip-schnorr/reference.py

@@ -5,6 +5,12 @@ p = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F
 n = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141
 G = (0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798, 0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8)
 
+# This implementation can be sped up by storing the midstate after hashing
+# tag_hash instead of rehashing it all the time.
+def tagged_hash(tag, msg):
+    tag_hash = hashlib.sha256(tag.encode()).digest()
+    return hashlib.sha256(tag_hash + tag_hash + msg).digest()
+
 def point_add(P1, P2):
     if (P1 is None):
         return P2
@@ -61,12 +67,12 @@ def schnorr_sign(msg, seckey0):
         raise ValueError('The secret key must be an integer in the range 1..n-1.')
     P = point_mul(G, seckey0)
     seckey = seckey0 if (jacobi(P[1]) == 1) else n - seckey0
-    k0 = int_from_bytes(hash_sha256(bytes_from_int(seckey) + msg)) % n
+    k0 = int_from_bytes(tagged_hash("BIPSchnorrDerive", bytes_from_int(seckey) + msg)) % n
     if k0 == 0:
         raise RuntimeError('Failure. This happens only with negligible probability.')
     R = point_mul(G, k0)
     k = n - k0 if (jacobi(R[1]) != 1) else k0
-    e = int_from_bytes(hash_sha256(bytes_from_point(R) + bytes_from_point(P) + msg)) % n
+    e = int_from_bytes(tagged_hash("BIPSchnorr", bytes_from_point(R) + bytes_from_point(P) + msg)) % n
     return bytes_from_point(R) + bytes_from_int((k + e * seckey) % n)
 
 def schnorr_verify(msg, pubkey, sig):
@@ -83,7 +89,7 @@ def schnorr_verify(msg, pubkey, sig):
     s = int_from_bytes(sig[32:64])
     if (r >= p or s >= n):
         return False
-    e = int_from_bytes(hash_sha256(sig[0:32] + pubkey + msg)) % n
+    e = int_from_bytes(tagged_hash("BIPSchnorr", sig[0:32] + pubkey + msg)) % n
     R = point_add(point_mul(G, s), point_mul(P, n - e))
     if R is None or jacobi(R[1]) != 1 or R[0] != r:
         return False

bip-schnorr/test-vectors.csv

@@ -1,15 +1,15 @@
 index,secret key,public key,message,signature,verification result,comment
-0,0000000000000000000000000000000000000000000000000000000000000001,79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798,0000000000000000000000000000000000000000000000000000000000000000,787A848E71043D280C50470E8E1532B2DD5D20EE912A45DBDD2BD1DFBF187EF6166FCCEE14F021B31AF22A90D0639CC010C2B764C304A8FFF266ABBC01A0A880,TRUE,
-1,B7E151628AED2A6ABF7158809CF4F3C762E7160F38B4DA56A784D9045190CFEF,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,3C4D3ADB1F05C3E948216ECB6007D1534D97D35D5589EDB226AD0370C0293C780AEA9BA361509DA2FDF7BC6E5E38926E40B6ACCD24EA9E06B9DA8244A1D0B691,TRUE,
-2,C90FDAA22168C234C4C6628B80DC1CD129024E088A67CC74020BBEA63B14E5C7,FAC2114C2FBB091527EB7C64ECB11F8021CB45E8E7809D3C0938E4B8C0E5F84B,5E2D58D8B3BCDF1ABADEC7829054F90DDA9805AAB56C77333024B9D0A508B75C,33380A613013115518CD6030ABC8220B2DED273FD3ABD13DF0882DC6A928E5AFAC72DF3248F8FAC522EF1A819EE5EE5BEA81D92D0E19A6FAB228E5D1D61BF833,TRUE,
-3,0B432B2677937381AEF05BB02A66ECD012773062CF3FA2549E44F58ED2401710,25D1DFF95105F5253C4022F628A996AD3A0D95FBF21D468A1B33F8C160D8F517,FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF,7C4E8B0C2575A77A01AF41EE6678EF9C41F97CC4D15FF6D6CA45D73BC6FF7FF4919D246589AF2FA6306F4B7A392857E9C4A17CB21DBE72C38A48C99979EEDCB8,TRUE,test fails if msg is reduced modulo p or n
-4,,D69C3509BB99E412E68B0FE8544E72837DFA30746D8BE2AA65975F29D22DC7B9,4DF3C3F68FCC83B27E9D42C90431A72499F17875C81A599B566C9889B9696703,00000000000000000000003B78CE563F89A0ED9414F5AA28AD0D96D6795F9C63F84A709CFFD89AD94FCBD808D41BD26BF62F263AA253527134DDC4A4715BF491,TRUE,
-5,,EEFDEA4CDB677750A420FEE807EACF21EB9898AE79B9768766E4FAA04A2D4A34,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,3C4D3ADB1F05C3E948216ECB6007D1534D97D35D5589EDB226AD0370C0293C780AEA9BA361509DA2FDF7BC6E5E38926E40B6ACCD24EA9E06B9DA8244A1D0B691,FALSE,public key not on the curve
-6,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,F9308A019258C31049344F85F89D5229B531C845836F99B08601F113BCE036F98631F4EF21D5F231A1000ED069E0348ED057EDF4FB1B6672009EDE9DBB2DEE14,FALSE,incorrect R residuosity
-7,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,07636CBE3092D11AFCA4DD1D32E50F039EB5FF5FB2AB72A7DC2BA0F3A4E7ED418C312ED6ABF6A41446D28789DF4AA43A15E166F001D072536ADC6E49C21DC419,FALSE,negated message
-8,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,3C4D3ADB1F05C3E948216ECB6007D1534D97D35D5589EDB226AD0370C0293C78F515645C9EAF625D02084391A1C76D9079F830198A5E023505F7DC482E658AB0,FALSE,negated s value
-9,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,0000000000000000000000000000000000000000000000000000000000000000221A96FEEBA7AD29F11B675DB394948A83A220FD8FE181A7667BDBEE178011B1,FALSE,sG - eP is infinite. Test fails in single verification if jacobi(y(inf)) is defined as 1 and x(inf) as 0
-10,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,00000000000000000000000000000000000000000000000000000000000000011A39648F31350D8E591106B43B9EB364F8617BCD52DC96A3FA8C4E50347F31DE,FALSE,sG - eP is infinite. Test fails in single verification if jacobi(y(inf)) is defined as 1 and x(inf) as 1
-11,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,4A298DACAE57395A15D0795DDBFD1DCB564DA82B0F269BC70A74F8220429BA1D0AEA9BA361509DA2FDF7BC6E5E38926E40B6ACCD24EA9E06B9DA8244A1D0B691,FALSE,sig[0:32] is not an X coordinate on the curve
-12,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F0AEA9BA361509DA2FDF7BC6E5E38926E40B6ACCD24EA9E06B9DA8244A1D0B691,FALSE,sig[0:32] is equal to field size
-13,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,3C4D3ADB1F05C3E948216ECB6007D1534D97D35D5589EDB226AD0370C0293C78FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141,FALSE,sig[32:64] is equal to curve order
+0,0000000000000000000000000000000000000000000000000000000000000001,79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798,0000000000000000000000000000000000000000000000000000000000000000,528F745793E8472C0329742A463F59E58F3A3F1A4AC09C28F6F8514D4D0322A258BD08398F82CF67B812AB2C7717CE566F877C2F8795C846146978E8F04782AE,TRUE,
+1,B7E151628AED2A6ABF7158809CF4F3C762E7160F38B4DA56A784D9045190CFEF,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,667C2F778E0616E611BD0C14B8A600C5884551701A949EF0EBFD72D452D64E844160BCFC3F466ECB8FACD19ADE57D8699D74E7207D78C6AEDC3799B52A8E0598,TRUE,
+2,C90FDAA22168C234C4C6628B80DC1CD129024E088A67CC74020BBEA63B14E5C9,DD308AFEC5777E13121FA72B9CC1B7CC0139715309B086C960E18FD969774EB8,5E2D58D8B3BCDF1ABADEC7829054F90DDA9805AAB56C77333024B9D0A508B75C,2D941B38E32624BF0AC7669C0971B990994AF6F9B18426BF4F4E7EC10E6CDF386CF646C6DDAFCFA7F1993EEB2E4D66416AEAD1DDAE2F22D63CAD901412D116C6,TRUE,
+3,0B432B2677937381AEF05BB02A66ECD012773062CF3FA2549E44F58ED2401710,25D1DFF95105F5253C4022F628A996AD3A0D95FBF21D468A1B33F8C160D8F517,FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF,8BD2C11604B0A87A443FCC2E5D90E5328F934161B18864FB48CE10CB59B45FB9B5B2A0F129BD88F5BDC05D5C21E5C57176B913002335784F9777A24BD317CD36,TRUE,test fails if msg is reduced modulo p or n
+4,,D69C3509BB99E412E68B0FE8544E72837DFA30746D8BE2AA65975F29D22DC7B9,4DF3C3F68FCC83B27E9D42C90431A72499F17875C81A599B566C9889B9696703,00000000000000000000003B78CE563F89A0ED9414F5AA28AD0D96D6795F9C63EE374AC7FAE927D334CCB190F6FB8FD27A2DDC639CCEE46D43F113A4035A2C7F,TRUE,
+5,,EEFDEA4CDB677750A420FEE807EACF21EB9898AE79B9768766E4FAA04A2D4A34,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,667C2F778E0616E611BD0C14B8A600C5884551701A949EF0EBFD72D452D64E844160BCFC3F466ECB8FACD19ADE57D8699D74E7207D78C6AEDC3799B52A8E0598,FALSE,public key not on the curve
+6,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,F9308A019258C31049344F85F89D5229B531C845836F99B08601F113BCE036F9935554D1AA5F0374E5CDAACB3925035C7C169B27C4426DF0A6B19AF3BAEAB138,FALSE,incorrect R residuosity
+7,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,10AC49A6A2EBF604189C5F40FC75AF2D42D77DE9A2782709B1EB4EAF1CFE9108D7003B703A3499D5E29529D39BA040A44955127140F81A8A89A96F992AC0FE79,FALSE,negated message
+8,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,667C2F778E0616E611BD0C14B8A600C5884551701A949EF0EBFD72D452D64E84BE9F4303C0B9913470532E6521A827951D39F5C631CFD98CE39AC4D7A5A83BA9,FALSE,negated s value
+9,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,000000000000000000000000000000000000000000000000000000000000000099D2F0EBC2996808208633CD9926BF7EC3DAB73DAAD36E85B3040A698E6D1CE0,FALSE,sG - eP is infinite. Test fails in single verification if jacobi(y(inf)) is defined as 1 and x(inf) as 0
+10,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,000000000000000000000000000000000000000000000000000000000000000124E81D89F01304695CE943F7D5EBD00EF726A0864B4FF33895B4E86BEADC5456,FALSE,sG - eP is infinite. Test fails in single verification if jacobi(y(inf)) is defined as 1 and x(inf) as 1
+11,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,4A298DACAE57395A15D0795DDBFD1DCB564DA82B0F269BC70A74F8220429BA1D4160BCFC3F466ECB8FACD19ADE57D8699D74E7207D78C6AEDC3799B52A8E0598,FALSE,sig[0:32] is not an X coordinate on the curve
+12,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F4160BCFC3F466ECB8FACD19ADE57D8699D74E7207D78C6AEDC3799B52A8E0598,FALSE,sig[0:32] is equal to field size
+13,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,667C2F778E0616E611BD0C14B8A600C5884551701A949EF0EBFD72D452D64E84FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141,FALSE,sig[32:64] is equal to curve order

bip-schnorr/test-vectors.py

@@ -26,7 +26,7 @@ def vector1():
     return (bytes_from_int(seckey), pubkey, msg, sig, "TRUE", None)
 
 def vector2():
-    seckey = 0xC90FDAA22168C234C4C6628B80DC1CD129024E088A67CC74020BBEA63B14E5C7
+    seckey = 0xC90FDAA22168C234C4C6628B80DC1CD129024E088A67CC74020BBEA63B14E5C9
     msg = bytes_from_int(0x5E2D58D8B3BCDF1ABADEC7829054F90DDA9805AAB56C77333024B9D0A508B75C)
     sig = schnorr_sign(msg, seckey)
 
@@ -53,7 +53,7 @@ def schnorr_sign_fixed_nonce(msg, seckey0, k):
     P = point_mul(G, seckey0)
     seckey = seckey0 if (jacobi(P[1]) == 1) else n - seckey0
     R = point_mul(G, k)
-    e = int_from_bytes(hash_sha256(bytes_from_point(R) + bytes_from_point(P) + msg)) % n
+    e = int_from_bytes(tagged_hash("BIPSchnorr", bytes_from_point(R) + bytes_from_point(P) + msg)) % n
     return bytes_from_point(R) + bytes_from_int((k + e * seckey) % n)
 
 # Creates a singature with a small x(R) by using k = 1/2

bip-taproot.mediawiki

@@ -41,7 +41,7 @@ As a result we choose this combination of technologies:
 * Taproot's advantages become apparent under the assumption that most applications involve outputs that could be spent by all parties agreeing. That's where '''Schnorr''' signatures come in, as they permit [https://eprint.iacr.org/2018/068 key aggregation]: a public key can be constructed from multiple participant public keys, and which requires cooperation between all participants to sign for. Such multi-party public keys and signatures are indistinguishable from their single-party equivalents. This means that under this Taproot assumption, the all-parties-agree case can be handled using the key-based spending path, which is both private and efficient using Taproot. This can be generalized to arbitrary M-of-N policies, as Schnorr signatures support threshold signing, at the cost of more complex setup protocols.
 * As Schnorr signatures also permit '''batch validation''', allowing multiple signatures to be validated together more efficiently than validating each one independently, we make sure all parts of the design are compatible with this.
 * Where unused bits appear as a result of the above changes, they are reserved for mechanisms for '''future extensions'''. As a result, every script in the Merkle tree has an associated version such that new script versions can be introduced with a soft fork while remaining compatible with bip-taproot. Additionally, future soft forks can make use of the currently unused <code>annex</code> in the witness (see [[#Rationale]]).
-* While the core semantics of the '''signature hashing algorithm''' are not changed, a number of improvements are included in this proposal. The new signature hashing algorithm fixes the verification capabilities of offline signing devices by including amount and scriptPubKey in the digest, avoids unnecessary hashing, introduces '''tagged hashes''' and defines a default sighash byte.
+* While the core semantics of the '''signature hashing algorithm''' are not changed, a number of improvements are included in this proposal. The new signature hashing algorithm fixes the verification capabilities of offline signing devices by including amount and scriptPubKey in the digest, avoids unnecessary hashing, uses '''tagged hashes'''<ref>'''Why use tagged hashes?''' So far, nowhere in the Bitcoin protocol are hashes used where the input of SHA256 starts with two (non-double) SHA256 hashes, making collisions with existing uses of hash functions infeasible.</ref> (according to bip-schnorr) and defines a default sighash byte.
 
 Not included in this proposal are additional features like new sighash modes or opcodes that can be included with no loss in effectiveness as a future extension. Also not included is cross-input aggregation, as it [https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2018-March/015838.html interacts] in complex ways with upgrade mechanisms and solutions to that are still [https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2018-October/016461.html in flux].
 
@@ -51,15 +51,6 @@ This section specifies the Taproot consensus rules. Validity is defined by exclu
 
 The notation below follows that of bip-schnorr.
 
-=== Tagged hashes ===
-
-Cryptographic hash functions are used for multiple purposes in the specification below and in Bitcoin in general. To make sure hashes used in one context can't be reinterpreted in another one, all hash functions are tweaked with a context-dependent tag name, in such a way that collisions across contexts can be assumed to be infeasible.
-
-In the text below, ''hash<sub>tag</sub>(m)'' is a shorthand for ''SHA256(SHA256(tag) || SHA256(tag) || m)'', where ''tag'' is a UTF-8 encoded tag name.
-* So far, nowhere in the Bitcoin protocol are hashes used where the input of SHA256 starts with two (non-double) SHA256 hashes, making collisions with existing uses of hash functions infeasible.
-* Because the prefix ''SHA256(tag) || SHA256(tag)'' is a 64-byte long context-specific constant, optimized implementations are possible (identical to SHA256 itself, but with a modified initial state).
-* Using SHA256 of the tag name itself is reasonably simple and efficient for implementations that don't choose to use the optimization above.
-
 === Script validation rules ===
 
 A Taproot output is a SegWit output (native or P2SH-nested, see [https://github.com/bitcoin/bips/blob/master/bip-0141.mediawiki BIP141]) with version number 1, and a 32-byte witness program.
@@ -173,15 +164,9 @@ Satisfying any of these conditions is sufficient to spend the output.
 * If one or more of the spending conditions consist of just a single key (after aggregation), the most likely one should be made the internal key. If no such condition exists, it may be worthwhile adding one that consists of an aggregation of all keys participating in all scripts combined; effectively adding an "everyone agrees" branch. If that is inacceptable, pick as internal key a point with unknown discrete logarithm. One example of such a point is ''H = point(0x0250929b74c1a04954b78b4b6035e97a5e078a5a0f28ec96d547bfee9ace803ac0)'' which is [https://github.com/ElementsProject/secp256k1-zkp/blob/11af7015de624b010424273be3d91f117f172c82/src/modules/rangeproof/main_impl.h#L16 constructed] by taking the hash of the standard uncompressed encoding of secp256k1 generator ''G'' as X coordinate. In order to avoid leaking the information that key path spending is not possible it is recommended to pick a fresh integer ''r'' in the range ''0...n-1'' uniformly at random and use ''H + rG'' as internal key. It is possible to prove that this internal key is does not have a known discrete logarithm with respect to ''G'' by revealing ''r'' to a verifier who can then reconstruct how the internal key was created.
 * The remaining scripts should be organized into the leaves of a binary tree. This can be a balanced tree if each of the conditions these scripts correspond to are equally likely. If probabilities for each condition are known, consider constructing the tree as a Huffman tree.
 
-'''Computing the output script''' Once the spending conditions are split into an internal key <code>internal_pubkey</code> and a binary tree whose leaves are (leaf_version, script) tuples, the following Python3 algorithm can be used to compute the output script. In the code below, <code>ser_script</code> prefixes its input with a CCompactSize-encoded length. Public key objects hold 32-byte public keys according to bip-schnorr, have a method <code>get_bytes</code> to get the byte array and a method <code>tweak_add</code> which returns a new public key corresponding to the sum of the public key point and a multiple of the secp256k1 generator (similar to BIP32's derivation). The second return value of <code>tweak_add</code> is a boolean indicating the quadratic residuosity of the Y coordinate of the resulting point.
+'''Computing the output script''' Once the spending conditions are split into an internal key <code>internal_pubkey</code> and a binary tree whose leaves are (leaf_version, script) tuples, the following Python3 algorithm can be used to compute the output script. In the code below, <code>ser_script</code> prefixes its input with a CCompactSize-encoded length. Public key objects hold 32-byte public keys according to bip-schnorr, have a method <code>get_bytes</code> to get the byte array and a method <code>tweak_add</code> which returns a new public key corresponding to the sum of the public key point and a multiple of the secp256k1 generator (similar to BIP32's derivation). The second return value of <code>tweak_add</code> is a boolean indicating the quadratic residuosity of the Y coordinate of the resulting point. <code>tagged_hash</code> computes the tagged hash according to bip-schnorr.
 
 <source lang="python">
-import hashlib
-
-def tagged_hash(tag, msg):
-    tag_hash = hashlib.sha256(tag.encode()).digest()
-    return hashlib.sha256(tag_hash + tag_hash + msg).digest()
-
 def taproot_tree_helper(script_tree):
     if isinstance(script_tree, tuple):
         leaf_version, script = script_tree