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proxmark3/tools/recover_pk.py
iceman1001 57b20f9392 added prime256v1 / secp256r1 curve
2020-12-10 11:41:04 +01:00

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#!/usr/bin/env python3
# MIT License
# Copyright (c) 2020 @doegox
import binascii
import sys
debug = False
#######################################################################
# Using external sslcrypto library:
# import sslcrypto
# ... sslcrypto.ecc.get_curve()
# But to get this script autonomous, i.e. for CI, we embedded the
# code snippets we needed:
#######################################################################
# code snippets from JacobianCurve:
# This code is public domain. Everyone has the right to do whatever they want with it for any purpose.
# Copyright (c) 2013 Vitalik Buterin
class JacobianCurve:
def __init__(self, p, n, a, b, g):
self.p = p
self.n = n
self.a = a
self.b = b
self.g = g
self.n_length = len(bin(self.n).replace("0b", ""))
def to_jacobian(self, p):
return p[0], p[1], 1
def jacobian_double(self, p):
if not p[1]:
return 0, 0, 0
ysq = (p[1] ** 2) % self.p
s = (4 * p[0] * ysq) % self.p
m = (3 * p[0] ** 2 + self.a * p[2] ** 4) % self.p
nx = (m ** 2 - 2 * s) % self.p
ny = (m * (s - nx) - 8 * ysq ** 2) % self.p
nz = (2 * p[1] * p[2]) % self.p
return nx, ny, nz
def jacobian_add(self, p, q):
if not p[1]:
return q
if not q[1]:
return p
u1 = (p[0] * q[2] ** 2) % self.p
u2 = (q[0] * p[2] ** 2) % self.p
s1 = (p[1] * q[2] ** 3) % self.p
s2 = (q[1] * p[2] ** 3) % self.p
if u1 == u2:
if s1 != s2:
return (0, 0, 1)
return self.jacobian_double(p)
h = u2 - u1
r = s2 - s1
h2 = (h * h) % self.p
h3 = (h * h2) % self.p
u1h2 = (u1 * h2) % self.p
nx = (r ** 2 - h3 - 2 * u1h2) % self.p
ny = (r * (u1h2 - nx) - s1 * h3) % self.p
nz = (h * p[2] * q[2]) % self.p
return (nx, ny, nz)
def from_jacobian(self, p):
z = inverse(p[2], self.p)
return (p[0] * z ** 2) % self.p, (p[1] * z ** 3) % self.p
def jacobian_shamir(self, a, n, b, m):
ab = self.jacobian_add(a, b)
if n < 0 or n >= self.n:
n %= self.n
if m < 0 or m >= self.n:
m %= self.n
res = 0, 0, 1 # point on infinity
for i in range(self.n_length - 1, -1, -1):
res = self.jacobian_double(res)
has_n = n & (1 << i)
has_m = m & (1 << i)
if has_n:
if has_m == 0:
res = self.jacobian_add(res, a)
if has_m != 0:
res = self.jacobian_add(res, ab)
else:
if has_m == 0:
res = self.jacobian_add(res, (0, 0, 1)) # Try not to leak
if has_m != 0:
res = self.jacobian_add(res, b)
return res
def fast_shamir(self, a, n, b, m):
return self.from_jacobian(self.jacobian_shamir(self.to_jacobian(a), n, self.to_jacobian(b), m))
#######################################################################
# code snippets from sslcrypto
# MIT License
# Copyright (c) 2019 Ivan Machugovskiy
import hmac
import os
import hashlib
import struct
def int_to_bytes(raw, length):
data = []
for _ in range(length):
data.append(raw % 256)
raw //= 256
return bytes(data[::-1])
def bytes_to_int(data):
raw = 0
for byte in data:
raw = raw * 256 + byte
return raw
def legendre(a, p):
res = pow(a, (p - 1) // 2, p)
if res == p - 1:
return -1
else:
return res
def inverse(a, n):
if a == 0:
return 0
lm, hm = 1, 0
low, high = a % n, n
while low > 1:
r = high // low
nm, new = hm - lm * r, high - low * r
lm, low, hm, high = nm, new, lm, low
return lm % n
def square_root_mod_prime(n, p):
if n == 0:
return 0
if p == 2:
return n # We should never get here but it might be useful
if legendre(n, p) != 1:
raise ValueError("No square root")
# Optimizations
if p % 4 == 3:
return pow(n, (p + 1) // 4, p)
# 1. By factoring out powers of 2, find Q and S such that p - 1 =
# Q * 2 ** S with Q odd
q = p - 1
s = 0
while q % 2 == 0:
q //= 2
s += 1
# 2. Search for z in Z/pZ which is a quadratic non-residue
z = 1
while legendre(z, p) != -1:
z += 1
m, c, t, r = s, pow(z, q, p), pow(n, q, p), pow(n, (q + 1) // 2, p)
while True:
if t == 0:
return 0
elif t == 1:
return r
# Use repeated squaring to find the least i, 0 < i < M, such
# that t ** (2 ** i) = 1
t_sq = t
i = 0
for i in range(1, m):
t_sq = t_sq * t_sq % p
if t_sq == 1:
break
else:
raise ValueError("Should never get here")
# Let b = c ** (2 ** (m - i - 1))
b = pow(c, 2 ** (m - i - 1), p)
m = i
c = b * b % p
t = t * b * b % p
r = r * b % p
return r
# name: (nid, p, n, a, b, (Gx, Gy)),
CURVES = {
"secp128r1": (
706,
0xFFFFFFFDFFFFFFFFFFFFFFFFFFFFFFFF,
0xFFFFFFFE0000000075A30D1B9038A115,
0xFFFFFFFDFFFFFFFFFFFFFFFFFFFFFFFC,
0xE87579C11079F43DD824993C2CEE5ED3,
(
0x161FF7528B899B2D0C28607CA52C5B86,
0xCF5AC8395BAFEB13C02DA292DDED7A83
)
),
"secp224r1": (
713,
0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF000000000000000000000001,
0xFFFFFFFFFFFFFFFFFFFFFFFFFFFF16A2E0B8F03E13DD29455C5C2A3D,
0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFFFFFFFFFFFFFFFFFFFE,
0xB4050A850C04B3ABF54132565044B0B7D7BFD8BA270B39432355FFB4,
(
0xB70E0CBD6BB4BF7F321390B94A03C1D356C21122343280D6115C1D21,
0xBD376388B5F723FB4C22DFE6CD4375A05A07476444D5819985007E34
)
),
## openssl uses the name: prime256v1
"secp256r1": (
415,
0xFFFFFFFF00000001000000000000000000000000FFFFFFFFFFFFFFFFFFFFFFFF,
0xFFFFFFFF00000000FFFFFFFFFFFFFFFFBCE6FAADA7179E84F3B9CAC2FC632551,
-3,
0x5AC635D8AA3A93E7B3EbBD55769886BC651D06B0CC53B0F63BCE3C3E27D2604B,
(
0x6B17D1F2E12c4247F8BCE6E563A440F277037D812DEB33A0F4A13945D898C296,
0x4FE342E2FE1A7F9B8EE7EB4A7C0F9E162BCE33576B315ECECBB6406837BF51F5
)
),
}
def get_curve(name):
if name not in CURVES:
raise ValueError("Unknown curve {}".format(name))
nid, p, n, a, b, g = CURVES[name]
params = {"p": p, "n": n, "a": a, "b": b, "g": g}
return EllipticCurve(nid, p, n, a, b, g)
class EllipticCurve:
def __init__(self, nid, p, n, a, b, g):
self.p, self.n, self.a, self.b, self.g = p, n, a, b, g
self.jacobian = JacobianCurve(self.p, self.n, self.a, self.b, self.g)
self.public_key_length = (len(bin(p).replace("0b", "")) + 7) // 8
self.order_bitlength = len(bin(n).replace("0b", ""))
def _int_to_bytes(self, raw, len=None):
return int_to_bytes(raw, len or self.public_key_length)
def _subject_to_int(self, subject):
return bytes_to_int(subject[:(self.order_bitlength + 7) // 8])
def recover(self, signature, data, hash="sha256"):
# Sanity check: is this signature recoverable?
if len(signature) != 1 + 2 * self.public_key_length:
raise ValueError("Cannot recover an unrecoverable signature")
subject = self._digest(data, hash)
z = self._subject_to_int(subject)
recid = signature[0] - 27 if signature[0] < 31 else signature[0] - 31
r = bytes_to_int(signature[1:self.public_key_length + 1])
s = bytes_to_int(signature[self.public_key_length + 1:])
# Verify bounds
if not 0 <= recid < 2 * (self.p // self.n + 1):
raise ValueError("Invalid recovery ID")
if r >= self.n:
raise ValueError("r is out of bounds")
if s >= self.n:
raise ValueError("s is out of bounds")
rinv = inverse(r, self.n)
u1 = (-z * rinv) % self.n
u2 = (s * rinv) % self.n
# Recover R
rx = r + (recid // 2) * self.n
if rx >= self.p:
raise ValueError("Rx is out of bounds")
# Almost copied from decompress_point
ry_square = (pow(rx, 3, self.p) + self.a * rx + self.b) % self.p
try:
ry = square_root_mod_prime(ry_square, self.p)
except Exception:
raise ValueError("Invalid recovered public key") from None
# Ensure the point is correct
if ry % 2 != recid % 2:
# Fix Ry sign
ry = self.p - ry
x, y = self.jacobian.fast_shamir(self.g, u1, (rx, ry), u2)
x, y = self._int_to_bytes(x), self._int_to_bytes(y)
is_compressed = signature[0] >= 31
if is_compressed:
return bytes([0x02 + (y[-1] % 2)]) + x
else:
return bytes([0x04]) + x + y
def _digest(self, data, hash):
if hash is None:
return data
elif callable(hash):
return hash(data)
elif hash == "sha1":
return hashlib.sha1(data).digest()
elif hash == "sha256":
return hashlib.sha256(data).digest()
elif hash == "sha512":
return hashlib.sha512(data).digest()
else:
raise ValueError("Unknown hash/derivation method")
#######################################################################
def recover(data, signature, alghash=None):
recovered = set()
if len(signature) == 32:
curve = get_curve("secp128r1")
recoverable = False
elif len(signature) == 33:
curve = get_curve("secp128r1")
recoverable = True
elif len(signature) == 56:
curve = get_curve("secp224r1")
recoverable = False
elif len(signature) == 57:
curve = get_curve("secp224r1")
recoverable = True
elif len(signature) == 64:
curve = get_curve("secp256r1")
recoverable = False
elif len(signature) == 65:
curve = get_curve("secp256r1")
recoverable = True
else:
print("Unsupported signature size %i" % len(signature))
exit(1)
if (recoverable):
try:
pk = curve.recover(signature, data, hash=alghash)
recovered.add(pk)
if debug:
print("Possible Pk: ", binascii.hexlify(pk))
except:
pass
else:
for i in range(2):
# Brute force RECID
recid = bytes([27+i])
try:
pk = curve.recover(recid + signature, data, hash=alghash)
recovered.add(pk)
if debug:
print("Possible Pk: ", binascii.hexlify(pk))
except:
pass
return recovered
def recover_multiple(uids, sigs, alghash=None):
recovered = set()
assert len(uids) == len(sigs)
for i in range(len(uids)):
data = binascii.unhexlify(uids[i])
if debug:
print("UID (%2i): " % len(data), binascii.hexlify(data))
signature = binascii.unhexlify(sigs[i])
if debug:
print("Signature (%2i): " % len(signature), binascii.hexlify(signature))
recovered_tmp = recover(data, signature, alghash)
if i == 0:
if recovered_tmp == set():
break
else:
recovered = recovered_tmp
else:
recovered &= recovered_tmp
return recovered
def selftests():
tests = [
{'name': "Mifare Ultralight EV1",
'samples': ["04C1285A373080", "CEA2EB0B3C95D0844A95B824A7553703B3702378033BF0987899DB70151A19E7",
"04C2285A373080", "A561506723D422D29ED9F93E60D20B9ED1E05CC1BF81DA19FE500CA0B81CC0ED"],
'pk': "0490933BDCD6E99B4E255E3DA55389A827564E11718E017292FAF23226A96614B8" },
{'name': "NTAG21x",
'samples': ["04E10CDA993C80", "8B76052EE42F5567BEB53238B3E3F9950707C0DCC956B5C5EFCFDB709B2D82B3",
"04DB0BDA993C80", "6048EFD9417CD10F6B7F1818D471A7FE5B46868D2EABDC6307A1E0AAE139D8D0"],
'pk': "04494E1A386D3D3CFE3DC10E5DE68A499B1C202DB5B132393E89ED19FE5BE8BC61" },
{'name': "Mifare Classic EV1",
'samples': ["0433619AB35780", "B9FAE369EC21C980650D87ED9AE9B1610E859131B4B8699C647548AB68D249BB",
"524374E2", "F8758CE30A58553A9985C458FB9C7D340FCFB04847B928A0667939272BC58B5E",
"53424B8A", "B4F533E8C06C021E242EFE8558C1672ED7022E5AE4E7AA2D46113B0AB6928AFC"],
'pk': "044F6D3F294DEA5737F0F46FFEE88A356EED95695DD7E0C27A591E6F6F65962BAF" },
{'name': "DESFire Light",
'samples': ["0439556ACB6480", "D5BD0978106E1E38B513642335966AB21E9F950DCFCFAB45FF13D0DC3CA4C2AE7E0D671DF1240937D040DAC4601C5F66ED62C546EE03ED08",
"043B156ACB6480", "76B46932BF2FCF4931A24C755F5CB1686B914F1856177686B864BDAD58EFA6A7493E5C2232F3ADDAA434EA4647BFD1D385BDA6115E77D74C"],
'pk': "040E98E117AAA36457F43173DC920A8757267F44CE4EC5ADD3C54075571AEBBF7B942A9774A1D94AD02572427E5AE0A2DD36591B1FB34FCF3D" },
{'name': "DESFire EV2",
'samples': ["042A41CAE45380", "B2769F8DDB575AEA2A680ADCA8FFED4FAB81A1E9908E2B82FE0FABB697BBD9B23835C416970E75768F12902ACA491349E94E6589EAF4F508",
"045640CAE45380", "D34B53A8C2C100D700DEA1C4C0D0DE4409F3A418CD8D57C4F41F146E42AD9A55F014199ABBF5CA259C7799DB0AE20D5E77D4950AC7E95D33"],
'pk': "04B304DC4C615F5326FE9383DDEC9AA892DF3A57FA7FFB3276192BC0EAA252ED45A865E3B093A3D0DCE5BE29E92F1392CE7DE321E3E5C52B3A" },
{'name': "DESFire EV3",
'samples': ["04448BD2DB6B80", "5CBB5632795C8F15263FEFB095B51C7B541AFD914A1AE44EF6FB8AF605EDF13DBFEE6C3A2DB372245E671DFE0D42CB1F0D0B8FE67A89D2F6",
"04445DD2DB6B80", "166BFD9F9BFAA451172566101580DF9894F582C4A4E258C15037AD2F35A475CF1D7FB817618623A6569F991931AFB2766984E21A18512A6D"],
'pk': "041DB46C145D0A36539C6544BD6D9B0AA62FF91EC48CBC6ABAE36E0089A46F0D08C8A715EA40A63313B92E90DDC1730230E0458A33276FB743" },
# TODO one more Mifare Plus EV1...
{'name': "Mifare Plus EV1",
'samples': ["042A2B221C5080", "BAC40CD88E9193C58ADA5055350C4F648EB5A7AEC4FCF9BD4CDD7B1C558DE5F59C6636F26286ED48622AAA2331D4DF1CEE23B57B94BDA631"],
'pk': "044409ADC42F91A8394066BA83D872FB1D16803734E911170412DDF8BAD1A4DADFD0416291AFE1C748253925DA39A5F39A1C557FFACD34C62E" },
{'name': "NTAG413DNA",
'samples': ["042468222F5C80", "B9211E320F321BD1D0E158E10FF15109B389638BAE15D9909D7725BF1250ED236D66F1AF75C94D60330E4E92535F5E6997675281A5687173",
"042938222F5C80", "18B642797D1FD71806146A7A6EC778D3FDD04F39C4A3B36A592BD1A114DC44E5528380FA766C0B7EA32B284AFBE84300B620369F0686D8CC"],
'pk': "04bb5d514f7050025c7d0f397310360eec91eaf792e96fc7e0f496cb4e669d414f877b7b27901fe67c2e3b33cd39d1c797715189ac951c2add" },
{'name': "NTAG424DNA",
'samples': ["0463474AA26A80", "27E9A50E6CA4BA9037C02F7D20A80D0284D0C1D83C67F5A5AC1D8A4EF86C9508417E4E9C6F85AA7920F0ABDED984CAF20467D66EA54BBF08",
"04C46C222A6380", "344A806EBF704C05C19215D2F840529CE365AAD2D08A469A95896D75D477D9FAB02A0C827E9F215BD8EB0E56A3A9A008FB75D706AABBD4DA"],
'pk': "048A9B380AF2EE1B98DC417FECC263F8449C7625CECE82D9B916C992DA209D68422B81EC20B65A66B5102A61596AF3379200599316A00A1410" },
{'name': "Vivokey Spark1",
# ! tag signature bytes output by pm3 must be read right to left: echo $sig |sed 's/\(..\)/\1\n/g'|tac|tr -d '\n' (and it uses a SHA256)
'samples': ["E0040118009C870C", "4B4E03E1211952EF6A5F9D84AB218CD4D7549D0CDF8CA8779F9AD16C9A9CBF3B",
"E0040118009B4D62", "25CF13747C3389EC7889DE916E3747584978511CC78B51CFB1883B494CBED7AB"],
'pk': "04d64bb732c0d214e7ec580736acf847284b502c25c0f7f2fa86aace1dada4387a" },
# ! tag UID is considered inversed: E0040118009B5FEE => EE5F9B00180104E0
# TODO one more ICODE-DNA...
{'name': "ICODE DNA, ICODE SLIX2",
'samples': ["EE5F9B00180104E0", "32D9E7579CD77E6F1FA11419231E874826984C5F189FDE1421684563A9663377"],
'pk': "048878A2A2D3EEC336B4F261A082BD71F9BE11C4E2E896648B32EFA59CEA6E59F0" },
# ! uses secp256r1 , SHA-256,
# {'name': "Minecraft Earth",
# 'samples': ["aa", "DF0E506DFF8FCFC4B7B979D917644445F1230D2C7CDC342AFA842CA240C210BE7275F62073A9670F2DCEFC602CBEE771C2B4CD4A04F3D1EA11F49ABDF7E8B721"],
# 'pk': "" },
]
succeeded = True
for t in tests:
print("Testing %-25s" % (t['name']+":"), end="")
recovered = recover_multiple(t['samples'][::2], t['samples'][1::2])
recovered |= recover_multiple(t['samples'][::2], t['samples'][1::2], alghash="sha256")
if (len(recovered) == 1):
pk = recovered.pop()
pk = binascii.hexlify(pk).decode('utf8')
if pk.lower() == t['pk'].lower():
print("[OK]")
else:
succeeded = False
print("[FAIL], got %s" % pk.lower())
elif len(t['samples'])//2 == 1:
pks = [binascii.hexlify(pk).decode('utf8').lower() for pk in list(recovered)]
if t['pk'].lower() in pks:
print("[OK] (partial)")
else:
succeeded = False
print("[FAIL], got %s" % pks)
else:
succeeded = False
print("[FAIL]")
print("Tests: [%s]" % ["FAIL", "OK"][succeeded])
if __name__ == "__main__":
if len(sys.argv) == 2 and sys.argv[1] == "selftests":
selftests()
exit(0)
if len(sys.argv) < 3 or len(sys.argv) % 2 == 0:
print("Usage: \n%s UID SIGN [UID SIGN] [...]" % sys.argv[0])
print("Example: \n%s 04ee45daa34084 ebb6102bff74b087d18a57a54bc375159a04ea9bc61080b7f4a85afe1587d73b" % sys.argv[0])
exit(1)
print("Assuming no hash was used in the signature generation:")
recovered = recover_multiple(sys.argv[1:][::2], sys.argv[1:][1::2])
print("Possible uncompressed Pk(s):")
for pk in list(recovered):
print(binascii.hexlify(pk).decode('utf8'))
print("Assuming SHA-256 was used in the signature generation:")
recovered = recover_multiple(sys.argv[1:][::2], sys.argv[1:][1::2], alghash="sha256")
print("Possible uncompressed Pk(s):")
for pk in list(recovered):
print(binascii.hexlify(pk).decode('utf8'))
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