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#! /usr/bin/env python
"""
Implementation of Elliptic-Curve Digital Signatures.
Classes and methods for elliptic-curve signatures:
private keys, public keys, signatures,
NIST prime-modulus curves with modulus lengths of
192, 224, 256, 384, and 521 bits.
Example:
# (In real-life applications, you would probably want to
# protect against defects in SystemRandom.)
from random import SystemRandom
randrange = SystemRandom().randrange
# Generate a public/private key pair using the NIST Curve P-192:
g = generator_192
n = g.order()
secret = randrange( 1, n )
pubkey = Public_key( g, g * secret )
privkey = Private_key( pubkey, secret )
# Signing a hash value:
hash = randrange( 1, n )
signature = privkey.sign( hash, randrange( 1, n ) )
# Verifying a signature for a hash value:
if pubkey.verifies( hash, signature ):
print_("Demo verification succeeded.")
else:
print_("*** Demo verification failed.")
# Verification fails if the hash value is modified:
if pubkey.verifies( hash-1, signature ):
print_("**** Demo verification failed to reject tampered hash.")
else:
print_("Demo verification correctly rejected tampered hash.")
Version of 2009.05.16.
Revision history:
2005.12.31 - Initial version.
2008.11.25 - Substantial revisions introducing new classes.
2009.05.16 - Warn against using random.randrange in real applications.
2009.05.17 - Use random.SystemRandom by default.
Written in 2005 by Peter Pearson and placed in the public domain.
"""
from six import int2byte, b
from . import ellipticcurve
from . import numbertheory
from .util import bit_length
class RSZeroError(RuntimeError):
pass
class InvalidPointError(RuntimeError):
pass
class Signature(object):
"""ECDSA signature.
"""
def __init__(self, r, s):
self.r = r
self.s = s
def recover_public_keys(self, hash, generator):
"""Returns two public keys for which the signature is valid
hash is signed hash
generator is the used generator of the signature
"""
curve = generator.curve()
n = generator.order()
r = self.r
s = self.s
e = hash
x = r
# Compute the curve point with x as x-coordinate
alpha = (pow(x, 3, curve.p()) + (curve.a() * x) + curve.b()) % curve.p()
beta = numbertheory.square_root_mod_prime(alpha, curve.p())
y = beta if beta % 2 == 0 else curve.p() - beta
# Compute the public key
R1 = ellipticcurve.PointJacobi(curve, x, y, 1, n)
Q1 = numbertheory.inverse_mod(r, n) * (s * R1 + (-e % n) * generator)
Pk1 = Public_key(generator, Q1)
# And the second solution
R2 = ellipticcurve.PointJacobi(curve, x, -y, 1, n)
Q2 = numbertheory.inverse_mod(r, n) * (s * R2 + (-e % n) * generator)
Pk2 = Public_key(generator, Q2)
return [Pk1, Pk2]
class Public_key(object):
"""Public key for ECDSA.
"""
def __init__(self, generator, point, verify=True):
"""
Low level ECDSA public key object.
:param generator: the Point that generates the group (the base point)
:param point: the Point that defines the public key
:param bool verify: if True check if point is valid point on curve
:raises InvalidPointError: if the point parameters are invalid or
point does not lie on the curve
"""
self.curve = generator.curve()
self.generator = generator
self.point = point
n = generator.order()
p = self.curve.p()
if not (0 <= point.x() < p) or not (0 <= point.y() < p):
raise InvalidPointError("The public point has x or y out of range.")
if verify and not self.curve.contains_point(point.x(), point.y()):
raise InvalidPointError("Point does not lie on the curve")
if not n:
raise InvalidPointError("Generator point must have order.")
# for curve parameters with base point with cofactor 1, all points
# that are on the curve are scalar multiples of the base point, so
# verifying that is not necessary. See Section 3.2.2.1 of SEC 1 v2
if verify and self.curve.cofactor() != 1 and \
not n * point == ellipticcurve.INFINITY:
raise InvalidPointError("Generator point order is bad.")
def __eq__(self, other):
if isinstance(other, Public_key):
"""Return True if the points are identical, False otherwise."""
return self.curve == other.curve \
and self.point == other.point
return NotImplemented
def verifies(self, hash, signature):
"""Verify that signature is a valid signature of hash.
Return True if the signature is valid.
"""
# From X9.62 J.3.1.
G = self.generator
n = G.order()
r = signature.r
s = signature.s
if r < 1 or r > n - 1:
return False
if s < 1 or s > n - 1:
return False
c = numbertheory.inverse_mod(s, n)
u1 = (hash * c) % n
u2 = (r * c) % n
if hasattr(G, "mul_add"):
xy = G.mul_add(u1, self.point, u2)
else:
xy = u1 * G + u2 * self.point
v = xy.x() % n
return v == r
class Private_key(object):
"""Private key for ECDSA.
"""
def __init__(self, public_key, secret_multiplier):
"""public_key is of class Public_key;
secret_multiplier is a large integer.
"""
self.public_key = public_key
self.secret_multiplier = secret_multiplier
def __eq__(self, other):
if isinstance(other, Private_key):
"""Return True if the points are identical, False otherwise."""
return self.public_key == other.public_key \
and self.secret_multiplier == other.secret_multiplier
return NotImplemented
def sign(self, hash, random_k):
"""Return a signature for the provided hash, using the provided
random nonce. It is absolutely vital that random_k be an unpredictable
number in the range [1, self.public_key.point.order()-1]. If
an attacker can guess random_k, he can compute our private key from a
single signature. Also, if an attacker knows a few high-order
bits (or a few low-order bits) of random_k, he can compute our private
key from many signatures. The generation of nonces with adequate
cryptographic strength is very difficult and far beyond the scope
of this comment.
May raise RuntimeError, in which case retrying with a new
random value k is in order.
"""
G = self.public_key.generator
n = G.order()
k = random_k % n
# Fix the bit-length of the random nonce,
# so that it doesn't leak via timing.
# This does not change that ks = k mod n
ks = k + n
kt = ks + n
if bit_length(ks) == bit_length(n):
p1 = kt * G
else:
p1 = ks * G
r = p1.x() % n
if r == 0:
raise RSZeroError("amazingly unlucky random number r")
s = (numbertheory.inverse_mod(k, n)
* (hash + (self.secret_multiplier * r) % n)) % n
if s == 0:
raise RSZeroError("amazingly unlucky random number s")
return Signature(r, s)
def int_to_string(x):
"""Convert integer x into a string of bytes, as per X9.62."""
assert x >= 0
if x == 0:
return b('\0')
result = []
while x:
ordinal = x & 0xFF
result.append(int2byte(ordinal))
x >>= 8
result.reverse()
return b('').join(result)
def string_to_int(s):
"""Convert a string of bytes into an integer, as per X9.62."""
result = 0
for c in s:
if not isinstance(c, int):
c = ord(c)
result = 256 * result + c
return result
def digest_integer(m):
"""Convert an integer into a string of bytes, compute
its SHA-1 hash, and convert the result to an integer."""
#
# I don't expect this function to be used much. I wrote
# it in order to be able to duplicate the examples
# in ECDSAVS.
#
from hashlib import sha1
return string_to_int(sha1(int_to_string(m)).digest())
def point_is_valid(generator, x, y):
"""Is (x,y) a valid public key based on the specified generator?"""
# These are the tests specified in X9.62.
n = generator.order()
curve = generator.curve()
p = curve.p()
if not (0 <= x < p) or not (0 <= y < p):
return False
if not curve.contains_point(x, y):
return False
if curve.cofactor() != 1 and \
not n * ellipticcurve.PointJacobi(curve, x, y, 1)\
== ellipticcurve.INFINITY:
return False
return True
# NIST Curve P-192:
_p = 6277101735386680763835789423207666416083908700390324961279
_r = 6277101735386680763835789423176059013767194773182842284081
# s = 0x3045ae6fc8422f64ed579528d38120eae12196d5L
# c = 0x3099d2bbbfcb2538542dcd5fb078b6ef5f3d6fe2c745de65L
_b = 0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1
_Gx = 0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012
_Gy = 0x07192b95ffc8da78631011ed6b24cdd573f977a11e794811
curve_192 = ellipticcurve.CurveFp(_p, -3, _b, 1)
generator_192 = ellipticcurve.PointJacobi(
curve_192, _Gx, _Gy, 1, _r, generator=True)
# NIST Curve P-224:
_p = 26959946667150639794667015087019630673557916260026308143510066298881
_r = 26959946667150639794667015087019625940457807714424391721682722368061
# s = 0xbd71344799d5c7fcdc45b59fa3b9ab8f6a948bc5L
# c = 0x5b056c7e11dd68f40469ee7f3c7a7d74f7d121116506d031218291fbL
_b = 0xb4050a850c04b3abf54132565044b0b7d7bfd8ba270b39432355ffb4
_Gx = 0xb70e0cbd6bb4bf7f321390b94a03c1d356c21122343280d6115c1d21
_Gy = 0xbd376388b5f723fb4c22dfe6cd4375a05a07476444d5819985007e34
curve_224 = ellipticcurve.CurveFp(_p, -3, _b, 1)
generator_224 = ellipticcurve.PointJacobi(
curve_224, _Gx, _Gy, 1, _r, generator=True)
# NIST Curve P-256:
_p = 115792089210356248762697446949407573530086143415290314195533631308867097853951
_r = 115792089210356248762697446949407573529996955224135760342422259061068512044369
# s = 0xc49d360886e704936a6678e1139d26b7819f7e90L
# c = 0x7efba1662985be9403cb055c75d4f7e0ce8d84a9c5114abcaf3177680104fa0dL
_b = 0x5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b
_Gx = 0x6b17d1f2e12c4247f8bce6e563a440f277037d812deb33a0f4a13945d898c296
_Gy = 0x4fe342e2fe1a7f9b8ee7eb4a7c0f9e162bce33576b315ececbb6406837bf51f5
curve_256 = ellipticcurve.CurveFp(_p, -3, _b, 1)
generator_256 = ellipticcurve.PointJacobi(
curve_256, _Gx, _Gy, 1, _r, generator=True)
# NIST Curve P-384:
_p = 39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319
_r = 39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942643
# s = 0xa335926aa319a27a1d00896a6773a4827acdac73L
# c = 0x79d1e655f868f02fff48dcdee14151ddb80643c1406d0ca10dfe6fc52009540a495e8042ea5f744f6e184667cc722483L
_b = 0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef
_Gx = 0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7
_Gy = 0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f
curve_384 = ellipticcurve.CurveFp(_p, -3, _b, 1)
generator_384 = ellipticcurve.PointJacobi(
curve_384, _Gx, _Gy, 1, _r, generator=True)
# NIST Curve P-521:
_p = 6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151
_r = 6864797660130609714981900799081393217269435300143305409394463459185543183397655394245057746333217197532963996371363321113864768612440380340372808892707005449
# s = 0xd09e8800291cb85396cc6717393284aaa0da64baL
# c = 0x0b48bfa5f420a34949539d2bdfc264eeeeb077688e44fbf0ad8f6d0edb37bd6b533281000518e19f1b9ffbe0fe9ed8a3c2200b8f875e523868c70c1e5bf55bad637L
_b = 0x051953eb9618e1c9a1f929a21a0b68540eea2da725b99b315f3b8b489918ef109e156193951ec7e937b1652c0bd3bb1bf073573df883d2c34f1ef451fd46b503f00
_Gx = 0xc6858e06b70404e9cd9e3ecb662395b4429c648139053fb521f828af606b4d3dbaa14b5e77efe75928fe1dc127a2ffa8de3348b3c1856a429bf97e7e31c2e5bd66
_Gy = 0x11839296a789a3bc0045c8a5fb42c7d1bd998f54449579b446817afbd17273e662c97ee72995ef42640c550b9013fad0761353c7086a272c24088be94769fd16650
curve_521 = ellipticcurve.CurveFp(_p, -3, _b, 1)
generator_521 = ellipticcurve.PointJacobi(
curve_521, _Gx, _Gy, 1, _r, generator=True)
# Certicom secp256-k1
_a = 0x0000000000000000000000000000000000000000000000000000000000000000
_b = 0x0000000000000000000000000000000000000000000000000000000000000007
_p = 0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f
_Gx = 0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798
_Gy = 0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8
_r = 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141
curve_secp256k1 = ellipticcurve.CurveFp(_p, _a, _b, 1)
generator_secp256k1 = ellipticcurve.PointJacobi(
curve_secp256k1, _Gx, _Gy, 1, _r, generator=True)
# Brainpool P-160-r1
_a = 0x340E7BE2A280EB74E2BE61BADA745D97E8F7C300
_b = 0x1E589A8595423412134FAA2DBDEC95C8D8675E58
_p = 0xE95E4A5F737059DC60DFC7AD95B3D8139515620F
_Gx = 0xBED5AF16EA3F6A4F62938C4631EB5AF7BDBCDBC3
_Gy = 0x1667CB477A1A8EC338F94741669C976316DA6321
_q = 0xE95E4A5F737059DC60DF5991D45029409E60FC09
curve_brainpoolp160r1 = ellipticcurve.CurveFp(_p, _a, _b, 1)
generator_brainpoolp160r1 = ellipticcurve.PointJacobi(
curve_brainpoolp160r1, _Gx, _Gy, 1, _q, generator=True)
# Brainpool P-192-r1
_a = 0x6A91174076B1E0E19C39C031FE8685C1CAE040E5C69A28EF
_b = 0x469A28EF7C28CCA3DC721D044F4496BCCA7EF4146FBF25C9
_p = 0xC302F41D932A36CDA7A3463093D18DB78FCE476DE1A86297
_Gx = 0xC0A0647EAAB6A48753B033C56CB0F0900A2F5C4853375FD6
_Gy = 0x14B690866ABD5BB88B5F4828C1490002E6773FA2FA299B8F
_q = 0xC302F41D932A36CDA7A3462F9E9E916B5BE8F1029AC4ACC1
curve_brainpoolp192r1 = ellipticcurve.CurveFp(_p, _a, _b, 1)
generator_brainpoolp192r1 = ellipticcurve.PointJacobi(
curve_brainpoolp192r1, _Gx, _Gy, 1, _q, generator=True)
# Brainpool P-224-r1
_a = 0x68A5E62CA9CE6C1C299803A6C1530B514E182AD8B0042A59CAD29F43
_b = 0x2580F63CCFE44138870713B1A92369E33E2135D266DBB372386C400B
_p = 0xD7C134AA264366862A18302575D1D787B09F075797DA89F57EC8C0FF
_Gx = 0x0D9029AD2C7E5CF4340823B2A87DC68C9E4CE3174C1E6EFDEE12C07D
_Gy = 0x58AA56F772C0726F24C6B89E4ECDAC24354B9E99CAA3F6D3761402CD
_q = 0xD7C134AA264366862A18302575D0FB98D116BC4B6DDEBCA3A5A7939F
curve_brainpoolp224r1 = ellipticcurve.CurveFp(_p, _a, _b, 1)
generator_brainpoolp224r1 = ellipticcurve.PointJacobi(
curve_brainpoolp224r1, _Gx, _Gy, 1, _q, generator=True)
# Brainpool P-256-r1
_a = 0x7D5A0975FC2C3057EEF67530417AFFE7FB8055C126DC5C6CE94A4B44F330B5D9
_b = 0x26DC5C6CE94A4B44F330B5D9BBD77CBF958416295CF7E1CE6BCCDC18FF8C07B6
_p = 0xA9FB57DBA1EEA9BC3E660A909D838D726E3BF623D52620282013481D1F6E5377
_Gx = 0x8BD2AEB9CB7E57CB2C4B482FFC81B7AFB9DE27E1E3BD23C23A4453BD9ACE3262
_Gy = 0x547EF835C3DAC4FD97F8461A14611DC9C27745132DED8E545C1D54C72F046997
_q = 0xA9FB57DBA1EEA9BC3E660A909D838D718C397AA3B561A6F7901E0E82974856A7
curve_brainpoolp256r1 = ellipticcurve.CurveFp(_p, _a, _b, 1)
generator_brainpoolp256r1 = ellipticcurve.PointJacobi(
curve_brainpoolp256r1, _Gx, _Gy, 1, _q, generator=True)
# Brainpool P-320-r1
_a = 0x3EE30B568FBAB0F883CCEBD46D3F3BB8A2A73513F5EB79DA66190EB085FFA9F492F375A97D860EB4
_b = 0x520883949DFDBC42D3AD198640688A6FE13F41349554B49ACC31DCCD884539816F5EB4AC8FB1F1A6
_p = 0xD35E472036BC4FB7E13C785ED201E065F98FCFA6F6F40DEF4F92B9EC7893EC28FCD412B1F1B32E27
_Gx = 0x43BD7E9AFB53D8B85289BCC48EE5BFE6F20137D10A087EB6E7871E2A10A599C710AF8D0D39E20611
_Gy = 0x14FDD05545EC1CC8AB4093247F77275E0743FFED117182EAA9C77877AAAC6AC7D35245D1692E8EE1
_q = 0xD35E472036BC4FB7E13C785ED201E065F98FCFA5B68F12A32D482EC7EE8658E98691555B44C59311
curve_brainpoolp320r1 = ellipticcurve.CurveFp(_p, _a, _b, 1)
generator_brainpoolp320r1 = ellipticcurve.PointJacobi(
curve_brainpoolp320r1, _Gx, _Gy, 1, _q, generator=True)
# Brainpool P-384-r1
_a = 0x7BC382C63D8C150C3C72080ACE05AFA0C2BEA28E4FB22787139165EFBA91F90F8AA5814A503AD4EB04A8C7DD22CE2826
_b = 0x04A8C7DD22CE28268B39B55416F0447C2FB77DE107DCD2A62E880EA53EEB62D57CB4390295DBC9943AB78696FA504C11
_p = 0x8CB91E82A3386D280F5D6F7E50E641DF152F7109ED5456B412B1DA197FB71123ACD3A729901D1A71874700133107EC53
_Gx = 0x1D1C64F068CF45FFA2A63A81B7C13F6B8847A3E77EF14FE3DB7FCAFE0CBD10E8E826E03436D646AAEF87B2E247D4AF1E
_Gy = 0x8ABE1D7520F9C2A45CB1EB8E95CFD55262B70B29FEEC5864E19C054FF99129280E4646217791811142820341263C5315
_q = 0x8CB91E82A3386D280F5D6F7E50E641DF152F7109ED5456B31F166E6CAC0425A7CF3AB6AF6B7FC3103B883202E9046565
curve_brainpoolp384r1 = ellipticcurve.CurveFp(_p, _a, _b, 1)
generator_brainpoolp384r1 = ellipticcurve.PointJacobi(
curve_brainpoolp384r1, _Gx, _Gy, 1, _q, generator=True)
# Brainpool P-512-r1
_a = 0x7830A3318B603B89E2327145AC234CC594CBDD8D3DF91610A83441CAEA9863BC2DED5D5AA8253AA10A2EF1C98B9AC8B57F1117A72BF2C7B9E7C1AC4D77FC94CA
_b = 0x3DF91610A83441CAEA9863BC2DED5D5AA8253AA10A2EF1C98B9AC8B57F1117A72BF2C7B9E7C1AC4D77FC94CADC083E67984050B75EBAE5DD2809BD638016F723
_p = 0xAADD9DB8DBE9C48B3FD4E6AE33C9FC07CB308DB3B3C9D20ED6639CCA703308717D4D9B009BC66842AECDA12AE6A380E62881FF2F2D82C68528AA6056583A48F3
_Gx = 0x81AEE4BDD82ED9645A21322E9C4C6A9385ED9F70B5D916C1B43B62EEF4D0098EFF3B1F78E2D0D48D50D1687B93B97D5F7C6D5047406A5E688B352209BCB9F822
_Gy = 0x7DDE385D566332ECC0EABFA9CF7822FDF209F70024A57B1AA000C55B881F8111B2DCDE494A5F485E5BCA4BD88A2763AED1CA2B2FA8F0540678CD1E0F3AD80892
_q = 0xAADD9DB8DBE9C48B3FD4E6AE33C9FC07CB308DB3B3C9D20ED6639CCA70330870553E5C414CA92619418661197FAC10471DB1D381085DDADDB58796829CA90069
curve_brainpoolp512r1 = ellipticcurve.CurveFp(_p, _a, _b, 1)
generator_brainpoolp512r1 = ellipticcurve.PointJacobi(
curve_brainpoolp512r1, _Gx, _Gy, 1, _q, generator=True)