BIP 340: Use synthetic nonces in reference code and test vectors

bip-0340/reference.py

@@ -51,6 +51,9 @@ def bytes_from_int(x):
 def bytes_from_point(P):
     return bytes_from_int(x(P))
 
+def xor_bytes(b0, b1):
+    return bytes(x ^ y for (x, y) in zip(b0, b1))
+
 def lift_x_square_y(b):
     x = int_from_bytes(b)
     if x >= p:
@@ -90,15 +93,18 @@ def pubkey_gen(seckey):
     P = point_mul(G, x)
     return bytes_from_point(P)
 
-def schnorr_sign(msg, seckey0):
+def schnorr_sign(msg, seckey0, aux_rand):
     if len(msg) != 32:
         raise ValueError('The message must be a 32-byte array.')
     seckey0 = int_from_bytes(seckey0)
     if not (1 <= seckey0 <= n - 1):
         raise ValueError('The secret key must be an integer in the range 1..n-1.')
+    if len(aux_rand) != 32:
+        raise ValueError('aux_rand must be 32 bytes instead of %i.' % len(aux_rand))
     P = point_mul(G, seckey0)
     seckey = seckey0 if has_even_y(P) else n - seckey0
-    k0 = int_from_bytes(tagged_hash("BIP340/nonce", bytes_from_int(seckey) + bytes_from_point(P) + msg)) % n
+    t = xor_bytes(bytes_from_int(seckey), tagged_hash("BIP340/aux", aux_rand))
+    k0 = int_from_bytes(tagged_hash("BIP340/nonce", t + bytes_from_point(P) + msg)) % n
     if k0 == 0:
         raise RuntimeError('Failure. This happens only with negligible probability.')
     R = point_mul(G, k0)
@@ -137,7 +143,7 @@ def test_vectors():
         reader = csv.reader(csvfile)
         reader.__next__()
         for row in reader:
-            (index, seckey, pubkey, msg, sig, result, comment) = row
+            (index, seckey, pubkey, aux_rand, msg, sig, result, comment) = row
             pubkey = bytes.fromhex(pubkey)
             msg = bytes.fromhex(msg)
             sig = bytes.fromhex(sig)
@@ -150,7 +156,8 @@ def test_vectors():
                     print(' * Failed key generation.')
                     print('   Expected key:', pubkey.hex().upper())
                     print('     Actual key:', pubkey_actual.hex().upper())
-                sig_actual = schnorr_sign(msg, seckey)
+                aux_rand = bytes.fromhex(aux_rand)
+                sig_actual = schnorr_sign(msg, seckey, aux_rand)
                 if sig == sig_actual:
                     print(' * Passed signing test.')
                 else:

bip-0340/test-vectors.csv

@@ -1,16 +1,16 @@
-index,secret key,public key,message,signature,verification result,comment
-0,0000000000000000000000000000000000000000000000000000000000000003,F9308A019258C31049344F85F89D5229B531C845836F99B08601F113BCE036F9,0000000000000000000000000000000000000000000000000000000000000000,514F0E96BB9AD56A245A7F4ED1030D4DE3FB0F5DE285116514292B2F910C979201D5C686A9D968E169C3ED1C2249C81F2BD27D53C42D15FA275EA6445389410A,TRUE,
-1,B7E151628AED2A6ABF7158809CF4F3C762E7160F38B4DA56A784D9045190CFEF,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,74556372D3369E8C53E6B84B5D7EE9AE0220EB37A6EA5501EF828FBFBA90A864092EF727796DACA51118BE8FBD70B3EC50536E65DB6F3B3B3FE1049862018B02,TRUE,
-2,C90FDAA22168C234C4C6628B80DC1CD129024E088A67CC74020BBEA63B14E5C9,DD308AFEC5777E13121FA72B9CC1B7CC0139715309B086C960E18FD969774EB8,7E2D58D8B3BCDF1ABADEC7829054F90DDA9805AAB56C77333024B9D0A508B75C,FAD73AE779EDD67BA40772867FEF9F20F151EB4BFDDECC53B90DD3017FC5D6035670DB8C83BA96EAF51C069B2AA7CEEF556787AE897F84F8D822C4ED7115B851,TRUE,
-3,0B432B2677937381AEF05BB02A66ECD012773062CF3FA2549E44F58ED2401710,25D1DFF95105F5253C4022F628A996AD3A0D95FBF21D468A1B33F8C160D8F517,FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF,60DC1A7E50D2269424060FF66361601075EB4B516DE89BF1D91B1D6AD78900DACDA5AC4B697491430CAA7604C8D819B2150DEC26E8D01E2981DDA071D7556CD3,TRUE,test fails if msg is reduced modulo p or n
-4,,D69C3509BB99E412E68B0FE8544E72837DFA30746D8BE2AA65975F29D22DC7B9,4DF3C3F68FCC83B27E9D42C90431A72499F17875C81A599B566C9889B9696703,00000000000000000000003B78CE563F89A0ED9414F5AA28AD0D96D6795F9C630EC50E5363E227ACAC6F542CE1C0B186657E0E0D1A6FFE283A33438DE4738419,TRUE,
-5,,EEFDEA4CDB677750A420FEE807EACF21EB9898AE79B9768766E4FAA04A2D4A34,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,74556372D3369E8C53E6B84B5D7EE9AE0220EB37A6EA5501EF828FBFBA90A864092EF727796DACA51118BE8FBD70B3EC50536E65DB6F3B3B3FE1049862018B02,FALSE,public key not on the curve
-6,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,F9308A019258C31049344F85F89D5229B531C845836F99B08601F113BCE036F995A579DA959FA739FCE39E8BD16FECB5CDCF97060B2C73CDE60E87ABCA1AA5D9,FALSE,has_square_y(R) is false
-7,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,0B3E00AF0641F28B4B52F7E7AD3DDEB9BD313F9E382563BA9C9A8274F45D3D72D8F733F2901432C8DD99C739B0C1EE4030E79A94318278EC4E7160A65CDE8015,FALSE,negated message
-8,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,74556372D3369E8C53E6B84B5D7EE9AE0220EB37A6EA5501EF828FBFBA90A864F6D108D88692535AEEE74170428F4C126A5B6E80D3D965007FF159F46E34B63F,FALSE,negated s value
-9,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,00000000000000000000000000000000000000000000000000000000000000009915EE59F07F9DBBAEDC31BFCC9B34AD49DE669CD24773BCED77DDA36D073EC8,FALSE,sG - eP is infinite. Test fails in single verification if has_square_y(inf) is defined as true and x(inf) as 0
-10,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,0000000000000000000000000000000000000000000000000000000000000001C7EC918B2B9CF34071BB54BED7EB4BB6BAB148E9A7E36E6B228F95DFA08B43EC,FALSE,sG - eP is infinite. Test fails in single verification if has_square_y(inf) is defined as true and x(inf) as 1
-11,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,4A298DACAE57395A15D0795DDBFD1DCB564DA82B0F269BC70A74F8220429BA1D092EF727796DACA51118BE8FBD70B3EC50536E65DB6F3B3B3FE1049862018B02,FALSE,sig[0:32] is not an X coordinate on the curve
-12,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F092EF727796DACA51118BE8FBD70B3EC50536E65DB6F3B3B3FE1049862018B02,FALSE,sig[0:32] is equal to field size
-13,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,74556372D3369E8C53E6B84B5D7EE9AE0220EB37A6EA5501EF828FBFBA90A864FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141,FALSE,sig[32:64] is equal to curve order
-14,,FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC30,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,74556372D3369E8C53E6B84B5D7EE9AE0220EB37A6EA5501EF828FBFBA90A864092EF727796DACA51118BE8FBD70B3EC50536E65DB6F3B3B3FE1049862018B02,FALSE,public key is not a valid X coordinate because it exceeds the field size
+index,secret key,public key,aux_rand,message,signature,verification result,comment
+0,0000000000000000000000000000000000000000000000000000000000000003,F9308A019258C31049344F85F89D5229B531C845836F99B08601F113BCE036F9,0000000000000000000000000000000000000000000000000000000000000000,0000000000000000000000000000000000000000000000000000000000000000,067E337AD551B2276EC705E43F0920926A9CE08AC68159F9D258C9BBA412781C9F059FCDF4824F13B3D7C1305316F956704BB3FEA2C26142E18ACD90A90C947E,TRUE,
+1,B7E151628AED2A6ABF7158809CF4F3C762E7160F38B4DA56A784D9045190CFEF,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,0000000000000000000000000000000000000000000000000000000000000001,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,0E12B8C520948A776753A96F21ABD7FDC2D7D0C0DDC90851BE17B04E75EF86A47EF0DA46C4DC4D0D1BCB8668C2CE16C54C7C23A6716EDE303AF86774917CF928,TRUE,
+2,C90FDAA22168C234C4C6628B80DC1CD129024E088A67CC74020BBEA63B14E5C9,DD308AFEC5777E13121FA72B9CC1B7CC0139715309B086C960E18FD969774EB8,C87AA53824B4D7AE2EB035A2B5BBBCCC080E76CDC6D1692C4B0B62D798E6D906,7E2D58D8B3BCDF1ABADEC7829054F90DDA9805AAB56C77333024B9D0A508B75C,FC012F9FB8FE00A358F51EF93DCE0DC0C895F6E9A87C6C4905BC820B0C3677616B8737D14E703AF8E16E22E5B8F26227D41E5128F82D86F747244CC289C74D1D,TRUE,
+3,0B432B2677937381AEF05BB02A66ECD012773062CF3FA2549E44F58ED2401710,25D1DFF95105F5253C4022F628A996AD3A0D95FBF21D468A1B33F8C160D8F517,FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF,FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF,FC132D4E426DFF535AEC0FA7083AC5118BC1D5FFFD848ABD8290C23F271CA0DD11AEDCEA3F55DA9BD677FE29C9DDA0CF878BCE43FDE0E313D69D1AF7A5AE8369,TRUE,test fails if msg is reduced modulo p or n
+4,,D69C3509BB99E412E68B0FE8544E72837DFA30746D8BE2AA65975F29D22DC7B9,,4DF3C3F68FCC83B27E9D42C90431A72499F17875C81A599B566C9889B9696703,00000000000000000000003B78CE563F89A0ED9414F5AA28AD0D96D6795F9C630EC50E5363E227ACAC6F542CE1C0B186657E0E0D1A6FFE283A33438DE4738419,TRUE,
+5,,EEFDEA4CDB677750A420FEE807EACF21EB9898AE79B9768766E4FAA04A2D4A34,,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,7036D6BFE1837AE919631039A2CF652A295DFAC9A8BBB0806014B2F48DD7C807941607B563ABBA414287F374A332BA3636DE009EE1EF551A17796B72B68B8A24,FALSE,public key not on the curve
+6,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,F9308A019258C31049344F85F89D5229B531C845836F99B08601F113BCE036F995A579DA959FA739FCE39E8BD16FECB5CDCF97060B2C73CDE60E87ABCA1AA5D9,FALSE,has_square_y(R) is false
+7,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,F8704654F4687B7365ED32E796DE92761390A3BCC495179BFE073817B7ED32824E76B987F7C1F9A751EF5C343F7645D3CFFC7D570B9A7192EBF1898E1344E3BF,FALSE,negated message
+8,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,7036D6BFE1837AE919631039A2CF652A295DFAC9A8BBB0806014B2F48DD7C8076BE9F84A9C5445BEBD780C8B5CCD45C883D0DC47CD594B21A858F31A19AAB71D,FALSE,negated s value
+9,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,00000000000000000000000000000000000000000000000000000000000000009915EE59F07F9DBBAEDC31BFCC9B34AD49DE669CD24773BCED77DDA36D073EC8,FALSE,sG - eP is infinite. Test fails in single verification if has_square_y(inf) is defined as true and x(inf) as 0
+10,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,0000000000000000000000000000000000000000000000000000000000000001C7EC918B2B9CF34071BB54BED7EB4BB6BAB148E9A7E36E6B228F95DFA08B43EC,FALSE,sG - eP is infinite. Test fails in single verification if has_square_y(inf) is defined as true and x(inf) as 1
+11,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,4A298DACAE57395A15D0795DDBFD1DCB564DA82B0F269BC70A74F8220429BA1D941607B563ABBA414287F374A332BA3636DE009EE1EF551A17796B72B68B8A24,FALSE,sig[0:32] is not an X coordinate on the curve
+12,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F941607B563ABBA414287F374A332BA3636DE009EE1EF551A17796B72B68B8A24,FALSE,sig[0:32] is equal to field size
+13,,DFF1D77F2A671C5F36183726DB2341BE58FEAE1DA2DECED843240F7B502BA659,,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,7036D6BFE1837AE919631039A2CF652A295DFAC9A8BBB0806014B2F48DD7C807FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141,FALSE,sig[32:64] is equal to curve order
+14,,FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC30,,243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89,7036D6BFE1837AE919631039A2CF652A295DFAC9A8BBB0806014B2F48DD7C807941607B563ABBA414287F374A332BA3636DE009EE1EF551A17796B72B68B8A24,FALSE,public key is not a valid X coordinate because it exceeds the field size

bip-0340/test-vectors.py

@@ -4,7 +4,8 @@ from reference import *
 def vector0():
     seckey = bytes_from_int(3)
     msg = bytes_from_int(0)
-    sig = schnorr_sign(msg, seckey)
+    aux_rand = bytes_from_int(0)
+    sig = schnorr_sign(msg, seckey, aux_rand)
     pubkey = pubkey_gen(seckey)
 
     # We should have at least one test vector where the seckey needs to be
@@ -21,18 +22,21 @@ def vector0():
     pubkey_point = lift_x_even_y(pubkey)
     assert(not has_square_y(pubkey_point))
 
-    return (seckey, pubkey, msg, sig, "TRUE", None)
+    return (seckey, pubkey, aux_rand, msg, sig, "TRUE", None)
 
 def vector1():
     seckey = bytes_from_int(0xB7E151628AED2A6ABF7158809CF4F3C762E7160F38B4DA56A784D9045190CFEF)
     msg = bytes_from_int(0x243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89)
-    sig = schnorr_sign(msg, seckey)
-    return (seckey, pubkey_gen(seckey), msg, sig, "TRUE", None)
+    aux_rand = bytes_from_int(1)
+
+    sig = schnorr_sign(msg, seckey, aux_rand)
+    return (seckey, pubkey_gen(seckey), aux_rand, msg, sig, "TRUE", None)
 
 def vector2():
     seckey = bytes_from_int(0xC90FDAA22168C234C4C6628B80DC1CD129024E088A67CC74020BBEA63B14E5C9)
     msg = bytes_from_int(0x7E2D58D8B3BCDF1ABADEC7829054F90DDA9805AAB56C77333024B9D0A508B75C)
-    sig = schnorr_sign(msg, seckey)
+    aux_rand = bytes_from_int(0xC87AA53824B4D7AE2EB035A2B5BBBCCC080E76CDC6D1692C4B0B62D798E6D906)
+    sig = schnorr_sign(msg, seckey, aux_rand)
 
     # The point reconstructed from the public key has a square Y coordinate.
     pubkey = pubkey_gen(seckey)
@@ -44,7 +48,7 @@ def vector2():
     R = lift_x_square_y(sig[0:32])
     assert(not is_square(R[0]))
 
-    return (seckey, pubkey, msg, sig, "TRUE", None)
+    return (seckey, pubkey, aux_rand, msg, sig, "TRUE", None)
 
 def vector3():
     seckey = bytes_from_int(0x0B432B2677937381AEF05BB02A66ECD012773062CF3FA2549E44F58ED2401710)
@@ -55,8 +59,10 @@ def vector3():
     assert(y(P) % 2 != 0)
 
     msg = bytes_from_int(0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF)
-    sig = schnorr_sign(msg, seckey)
-    return (seckey, pubkey_gen(seckey), msg, sig, "TRUE", "test fails if msg is reduced modulo p or n")
+    aux_rand = bytes_from_int(0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF)
+
+    sig = schnorr_sign(msg, seckey, aux_rand)
+    return (seckey, pubkey_gen(seckey), aux_rand, msg, sig, "TRUE", "test fails if msg is reduced modulo p or n")
 
 # Signs with a given nonce. This can be INSECURE and is only INTENDED FOR
 # GENERATING TEST VECTORS. Results in an invalid signature if y(kG) is not
@@ -79,10 +85,11 @@ def vector4():
     seckey = bytes_from_int(0x763758E5CBEEDEE4F7D3FC86F531C36578933228998226672F13C4F0EBE855EB)
     msg = bytes_from_int(0x4DF3C3F68FCC83B27E9D42C90431A72499F17875C81A599B566C9889B9696703)
     sig = insecure_schnorr_sign_fixed_nonce(msg, seckey, one_half)
-    return (None, pubkey_gen(seckey), msg, sig, "TRUE", None)
+    return (None, pubkey_gen(seckey), None, msg, sig, "TRUE", None)
 
 default_seckey = bytes_from_int(0xB7E151628AED2A6ABF7158809CF4F3C762E7160F38B4DA56A784D9045190CFEF)
 default_msg = bytes_from_int(0x243F6A8885A308D313198A2E03707344A4093822299F31D0082EFA98EC4E6C89)
+default_aux_rand = bytes_from_int(0xC87AA53824B4D7AE2EB035A2B5BBBCCC080E76CDC6D1692C4B0B62D798E6D906)
 
 # Public key is not on the curve
 def vector5():
@@ -90,12 +97,12 @@ def vector5():
     # public key.
     seckey = default_seckey
     msg = default_msg
-    sig = schnorr_sign(msg, seckey)
+    sig = schnorr_sign(msg, seckey, default_aux_rand)
 
     pubkey = bytes_from_int(0xEEFDEA4CDB677750A420FEE807EACF21EB9898AE79B9768766E4FAA04A2D4A34)
     assert(lift_x_even_y(pubkey) is None)
 
-    return (None, pubkey, msg, sig, "FALSE", "public key not on the curve")
+    return (None, pubkey, None, msg, sig, "FALSE", "public key not on the curve")
 
 def vector6():
     seckey = default_seckey
@@ -107,21 +114,21 @@ def vector6():
     R = point_mul(G, k)
     assert(not has_square_y(R))
 
-    return (None, pubkey_gen(seckey), msg, sig, "FALSE", "has_square_y(R) is false")
+    return (None, pubkey_gen(seckey), None, msg, sig, "FALSE", "has_square_y(R) is false")
 
 def vector7():
     seckey = default_seckey
     msg = int_from_bytes(default_msg)
     neg_msg = bytes_from_int(n - msg)
-    sig = schnorr_sign(neg_msg, seckey)
-    return (None, pubkey_gen(seckey), bytes_from_int(msg), sig, "FALSE", "negated message")
+    sig = schnorr_sign(neg_msg, seckey, default_aux_rand)
+    return (None, pubkey_gen(seckey), None, bytes_from_int(msg), sig, "FALSE", "negated message")
 
 def vector8():
     seckey = default_seckey
     msg = default_msg
-    sig = schnorr_sign(msg, seckey)
+    sig = schnorr_sign(msg, seckey, default_aux_rand)
     sig = sig[0:32] + bytes_from_int(n - int_from_bytes(sig[32:64]))
-    return (None, pubkey_gen(seckey), msg, sig, "FALSE", "negated s value")
+    return (None, pubkey_gen(seckey), None, msg, sig, "FALSE", "negated s value")
 
 def bytes_from_point_inf0(P):
     if P == None:
@@ -140,7 +147,7 @@ def vector9():
     sig = insecure_schnorr_sign_fixed_nonce(msg, seckey, k)
     bytes_from_point.__code__ = bytes_from_point_tmp
 
-    return (None, pubkey_gen(seckey), msg, sig, "FALSE", "sG - eP is infinite. Test fails in single verification if has_square_y(inf) is defined as true and x(inf) as 0")
+    return (None, pubkey_gen(seckey), None, msg, sig, "FALSE", "sG - eP is infinite. Test fails in single verification if has_square_y(inf) is defined as true and x(inf) as 0")
 
 def bytes_from_point_inf1(P):
     if P == None:
@@ -159,7 +166,7 @@ def vector10():
     sig = insecure_schnorr_sign_fixed_nonce(msg, seckey, k)
     bytes_from_point.__code__ = bytes_from_point_tmp
 
-    return (None, pubkey_gen(seckey), msg, sig, "FALSE", "sG - eP is infinite. Test fails in single verification if has_square_y(inf) is defined as true and x(inf) as 1")
+    return (None, pubkey_gen(seckey), None, msg, sig, "FALSE", "sG - eP is infinite. Test fails in single verification if has_square_y(inf) is defined as true and x(inf) as 1")
 
 # It's cryptographically impossible to create a test vector that fails if run
 # in an implementation which merely misses the check that sig[0:32] is an X
@@ -167,14 +174,14 @@ def vector10():
 def vector11():
     seckey = default_seckey
     msg = default_msg
-    sig = schnorr_sign(msg, seckey)
+    sig = schnorr_sign(msg, seckey, default_aux_rand)
 
     # Replace R's X coordinate with an X coordinate that's not on the curve
     x_not_on_curve = bytes_from_int(0x4A298DACAE57395A15D0795DDBFD1DCB564DA82B0F269BC70A74F8220429BA1D)
     assert(lift_x_square_y(x_not_on_curve) is None)
     sig = x_not_on_curve + sig[32:64]
 
-    return (None, pubkey_gen(seckey), msg, sig, "FALSE", "sig[0:32] is not an X coordinate on the curve")
+    return (None, pubkey_gen(seckey), None, msg, sig, "FALSE", "sig[0:32] is not an X coordinate on the curve")
 
 # It's cryptographically impossible to create a test vector that fails if run
 # in an implementation which merely misses the check that sig[0:32] is smaller
@@ -182,12 +189,12 @@ def vector11():
 def vector12():
     seckey = default_seckey
     msg = default_msg
-    sig = schnorr_sign(msg, seckey)
+    sig = schnorr_sign(msg, seckey, default_aux_rand)
 
     # Replace R's X coordinate with an X coordinate that's equal to field size
     sig = bytes_from_int(p) + sig[32:64]
 
-    return (None, pubkey_gen(seckey), msg, sig, "FALSE", "sig[0:32] is equal to field size")
+    return (None, pubkey_gen(seckey), None, msg, sig, "FALSE", "sig[0:32] is equal to field size")
 
 # It's cryptographically impossible to create a test vector that fails if run
 # in an implementation which merely misses the check that sig[32:64] is smaller
@@ -195,12 +202,12 @@ def vector12():
 def vector13():
     seckey = default_seckey
     msg = default_msg
-    sig = schnorr_sign(msg, seckey)
+    sig = schnorr_sign(msg, seckey, default_aux_rand)
 
     # Replace s with a number that's equal to the curve order
     sig = sig[0:32] + bytes_from_int(n)
 
-    return (None, pubkey_gen(seckey), msg, sig, "FALSE", "sig[32:64] is equal to curve order")
+    return (None, pubkey_gen(seckey), None, msg, sig, "FALSE", "sig[32:64] is equal to curve order")
 
 # Test out of range pubkey
 # It's cryptographically impossible to create a test vector that fails if run
@@ -212,8 +219,7 @@ def vector14():
     # public key.
     seckey = default_seckey
     msg = default_msg
-    sig = schnorr_sign(msg, seckey)
-
+    sig = schnorr_sign(msg, seckey, default_aux_rand)
     pubkey_int = p + 1
     pubkey = bytes_from_int(pubkey_int)
     assert(lift_x_even_y(pubkey) is None)
@@ -221,7 +227,7 @@ def vector14():
     # pubkey would be valid
     assert(lift_x_even_y(bytes_from_int(pubkey_int % p)) is not None)
 
-    return (None, pubkey, msg, sig, "FALSE", "public key is not a valid X coordinate because it exceeds the field size")
+    return (None, pubkey, None, msg, sig, "FALSE", "public key is not a valid X coordinate because it exceeds the field size")
 
 vectors = [
         vector0(),
@@ -242,14 +248,14 @@ vectors = [
     ]
 
 # Converts the byte strings of a test vector into hex strings
-def bytes_to_hex(seckey, pubkey, msg, sig, result, comment):
-    return (seckey.hex().upper() if seckey is not None else None, pubkey.hex().upper(), msg.hex().upper(), sig.hex().upper(), result, comment)
+def bytes_to_hex(seckey, pubkey, aux_rand, msg, sig, result, comment):
+    return (seckey.hex().upper() if seckey is not None else None, pubkey.hex().upper(), aux_rand.hex().upper() if aux_rand is not None else None, msg.hex().upper(), sig.hex().upper(), result, comment)
 
-vectors = list(map(lambda vector: bytes_to_hex(vector[0], vector[1], vector[2], vector[3], vector[4], vector[5]), vectors))
+vectors = list(map(lambda vector: bytes_to_hex(vector[0], vector[1], vector[2], vector[3], vector[4], vector[5], vector[6]), vectors))
 
 def print_csv(vectors):
     writer = csv.writer(sys.stdout)
-    writer.writerow(("index", "secret key", "public key", "message", "signature", "verification result", "comment"))
+    writer.writerow(("index", "secret key", "public key", "aux_rand", "message", "signature", "verification result", "comment"))
     for (i,v) in enumerate(vectors):
         writer.writerow((i,)+v)