def _pubkey_recovery(ec: Curve, c: int, sig: ECDS) -> Sequence[Point]:
"""Private function provided for testing purposes only."""
# ECDSA public key recovery operation according to SEC 1
# http://www.secg.org/sec1-v2.pdf
# See SEC 1 v.2 section 4.1.6
r, s = _to_sig(ec, sig)
# precomputations
r1 = mod_inv(r, ec.n)
r1s = r1 * s
r1e = -r1 * c
keys: Sequence[Point] = list()
for j in range(ec.h): # 1
x = r + j * ec.n # 1.1
try: #TODO: check test reporting 1, 2, 3, or 4 keys
x %= ec._p
R = x, ec.y_odd(x, 1) # 1.2, 1.3, and 1.4
# skip 1.5: in this function, c is an input
Q = double_mult(ec, r1s, R, r1e, ec.G) # 1.6.1
if Q[1] != 0 and _verhlp(ec, c, Q, sig): # 1.6.2
keys.append(Q)
R = ec.opposite(R) # 1.6.3
Q = double_mult(ec, r1s, R, r1e, ec.G)
if Q[1] != 0 and _verhlp(ec, c, Q, sig): # 1.6.2
keys.append(Q) # 1.6.2
except Exception: # R is not a curve point
pass
return keys
def point_from_octets(ec: Curve, o: octets) -> Point:
"""Return a tuple (Px, Py) that belongs to the curve
SEC 1 v.2, section 2.3.4
"""
if isinstance(o, str):
o = bytes.fromhex(o)
bsize = len(o) # bytes
if bsize == 1 and o[0] == 0x00: # infinity point
return Point()
if bsize == ec.psize + 1: # compressed point
if o[0] not in (0x02, 0x03):
m = f"{ec.psize+1} bytes, but not a compressed point"
raise ValueError(m)
Px = int.from_bytes(o[1:], 'big')
try:
Py = ec.y_odd(Px, o[0] % 2) # also check Px validity
return Point(Px, Py)
except:
raise ValueError("point not on curve")
else: # uncompressed point
if bsize != 2 * ec.psize + 1:
m = f"wrong byte-size ({bsize}) for a point: it "
m += f"should have be {ec.psize+1} or {2*ec.psize+1}"
raise ValueError(m)
if o[0] != 0x04:
raise ValueError("not an uncompressed point")
Px = int.from_bytes(o[1:ec.psize + 1], 'big')
P = Point(Px, int.from_bytes(o[ec.psize + 1:], 'big'))
if ec.is_on_curve(P):
return P
else:
raise ValueError("point not on curve")
def _batch_verify(ec: Curve, hf, ms: List[bytes], P: List[Point],
sig: List[ECSS]) -> bool:
t = 0
scalars: List(int) = list()
points: List[Point] = list()
for i in range(len(P)):
_ensure_msg_size(hf, ms[i])
ec.require_on_curve(P[i])
r, s = _to_sig(ec, sig[i])
e = _e(ec, hf, r, P[i], ms[i])
y = ec.y(r) # raises an error if y does not exist
# deterministically generated using a CSPRNG seeded by a cryptographic
# hash (e.g., SHA256) of all inputs of the algorithm, or randomly
# generated independently for each run of the batch verification
# algorithm FIXME
a = (1 if i == 0 else random.getrandbits(ec.nlen) % ec.n)
scalars.append(a)
points.append(_jac_from_aff((r, y)))
scalars.append(a * e % ec.n)
points.append(_jac_from_aff(P[i]))
t += a * s % ec.n
TJ = _mult_jac(ec, t, ec.GJ)
RHSJ = _multi_mult(ec, scalars, points)
# return T == RHS, checked in Jacobian coordinates
RHSZ2 = RHSJ[2] * RHSJ[2]
TZ2 = TJ[2] * TJ[2]
if (TJ[0] * RHSZ2) % ec._p != (RHSJ[0] * TZ2) % ec._p:
return False
return (TJ[1] * RHSZ2 * RHSJ[2]) % ec._p == (RHSJ[1] * TZ2 * TJ[2]) % ec._p
def _verhlp(ec: Curve, c: int, P: Point, sig: ECDS) -> bool:
# Private function for test/dev purposes
# Fail if r is not [1, n-1]
# Fail if s is not [1, n-1]
r, s = _to_sig(ec, sig) # 1
# Let P = point(pk); fail if point(pk) fails.
ec.require_on_curve(P)
if P[1] == 0:
raise ValueError("public key is infinite")
w = mod_inv(s, ec.n)
u = c * w
v = r * w # 4
# Let R = u*G + v*P.
RJ = _double_mult(ec, u, ec.GJ, v, (P[0], P[1], 1)) # 5
# Fail if infinite(R).
assert RJ[2] != 0, "how did you do that?!?" # 5
Rx = (RJ[0] * mod_inv(RJ[2] * RJ[2], ec._p)) % ec._p
x = Rx % ec.n # 6, 7
# Fail if r ≠ x(R) %n.
return r == x # 8
def _batch_verify(ec: Curve, hf: Callable[[Any], Any], ms: Sequence[bytes],
P: Sequence[Point], sig: Sequence[ECSS]) -> bool:
# the bitcoin proposed standard is only valid for curves
# whose prime p = 3 % 4
if not ec.pIsThreeModFour:
errmsg = 'curve prime p must be equal to 3 (mod 4)'
raise ValueError(errmsg)
batch_size = len(P)
if len(ms) != batch_size:
errMsg = f"mismatch between number of pubkeys ({batch_size}) "
errMsg += f"and number of messages ({len(ms)})"
raise ValueError(errMsg)
if len(sig) != batch_size:
errMsg = f"mismatch between number of pubkeys ({batch_size}) "
errMsg += f"and number of signatures ({len(sig)})"
raise ValueError(errMsg)
if batch_size == 1:
return _verify(ec, hf, ms[0], P[0], sig[0])
t = 0
scalars: Sequence(int) = list()
points: Sequence[Point] = list()
for i in range(batch_size):
r, s = _to_sig(ec, sig[i])
_ensure_msg_size(hf, ms[i])
ec.require_on_curve(P[i])
e = _e(ec, hf, r, P[i], ms[i])
# raises an error if y does not exist
# no need to check for quadratic residue
y = ec.y(r)
# a in [1, n-1]
# deterministically generated using a CSPRNG seeded by a
# cryptographic hash (e.g., SHA256) of all inputs of the
# algorithm, or randomly generated independently for each
# run of the batch verification algorithm
a = (1 if i == 0 else (1 + random.getrandbits(ec.nlen)) % ec.n)
scalars.append(a)
points.append(_jac_from_aff((r, y)))
scalars.append(a * e % ec.n)
points.append(_jac_from_aff(P[i]))
t += a * s
TJ = _mult_jac(ec, t, ec.GJ)
RHSJ = _multi_mult(ec, scalars, points)
# return T == RHS, checked in Jacobian coordinates
RHSZ2 = RHSJ[2] * RHSJ[2]
TZ2 = TJ[2] * TJ[2]
if (TJ[0] * RHSZ2) % ec._p != (RHSJ[0] * TZ2) % ec._p:
return False
return (TJ[1] * RHSZ2 * RHSJ[2]) % ec._p == (RHSJ[1] * TZ2 * TJ[2]) % ec._p
def octets_from_point(ec: Curve, Q: Point, compressed: bool) -> bytes:
"""Return a compressed (0x02, 0x03) or uncompressed (0x04) point as octets
SEC 1 v.2, section 2.3.3
"""
# check that Q is a point and that is on curve
ec.require_on_curve(Q)
if Q[1] == 0: # infinity point in affine coordinates
return b'\x00'
bPx = Q[0].to_bytes(ec.psize, byteorder='big')
if compressed:
return (b'\x03' if (Q[1] & 1) else b'\x02') + bPx
return b'\x04' + bPx + Q[1].to_bytes(ec.psize, byteorder='big')
def _verify(ec: Curve, hf: Callable[[Any], Any], mhd: bytes, P: Point,
sig: ECSS) -> bool:
# This raises Exceptions, while verify should always return True or False
# the bitcoin proposed standard is only valid for curves
# whose prime p = 3 % 4
if not ec.pIsThreeModFour:
errmsg = 'curve prime p must be equal to 3 (mod 4)'
raise ValueError(errmsg)
# Let r = int(sig[ 0:32]).
# Let s = int(sig[32:64]); fail if s is not [0, n-1].
r, s = _to_sig(ec, sig)
# The message mhd: a 32-byte array
_ensure_msg_size(hf, mhd)
# Let P = point(pk); fail if point(pk) fails.
ec.require_on_curve(P)
if P[1] == 0:
raise ValueError("public key is infinite")
# Let e = int(hf(bytes(r) || bytes(P) || mhd)) mod n.
e = _e(ec, hf, r, P, mhd)
# Let R = sG - eP.
# in Jacobian coordinates
R = _double_mult(ec, s, ec.GJ, -e, (P[0], P[1], 1))
# Fail if infinite(R).
if R[2] == 0:
raise ValueError("sG - eP is infinite")
# Fail if jacobi(R.y) ≠ 1.
if legendre_symbol(R[1] * R[2] % ec._p, ec._p) != 1:
raise ValueError("(sG - eP).y is not a quadratic residue")
# Fail if R.x ≠ r.
return R[0] == (R[2] * R[2] * r % ec._p)
def test_exceptions(self):
# good
Curve(11, 2, 7, (6, 9), 7, 2, False)
# p not odd
self.assertRaises(ValueError, Curve, 10, 2, 7, (6, 9), 7, 1, False)
# p not prime
self.assertRaises(ValueError, Curve, 15, 2, 7, (6, 9), 7, 1, False)
# a > p
self.assertRaises(ValueError, Curve, 11, 12, 7, (6, 9), 13, 1, False)
# b > p
self.assertRaises(ValueError, Curve, 11, 2, 12, (6, 9), 13, 1, False)
# zero discriminant
self.assertRaises(ValueError, Curve, 11, 7, 7, (6, 9), 7, 1, False)
# G not Tuple (int, int)
self.assertRaises(ValueError, Curve, 11, 2, 7, (6, 9, 1), 7, 1, False)
# G not on curve
self.assertRaises(ValueError, Curve, 11, 2, 7, (7, 9), 7, 1, False)
# n not prime
self.assertRaises(ValueError, Curve, 11, 2, 7, (6, 9), 8, 1, False)
# n not Hesse
self.assertRaises(ValueError, Curve, 11, 2, 7, (6, 9), 71, 1, True)
# h not as expected
self.assertRaises(ValueError, Curve, 11, 2, 7, (6, 9), 7, 1, True)
# Curve(11, 2, 7, (6, 9), 7, 1, 0, True)
# n not group order
self.assertRaises(ValueError, Curve, 11, 2, 7, (6, 9), 13, 1, False)
# n=p -> weak curve
# missing
# weak curve
self.assertRaises(UserWarning, Curve, 11, 2, 7, (6, 9), 7, 2, True)
# x-coordinate not in [0, p-1]
ec = CURVES['secp256k1']
self.assertRaises(ValueError, ec.y, ec.p)
# secp256k1.y(secp256k1.p)
# INF point does not generate a prime order subgroup
self.assertRaises(ValueError, Curve, 11, 2, 7, INF, 7, 2, False)
def test_jac():
ec = Curve(13, 0, 2, (1, 9), 19, 1, False)
assert ec._jac_equality(ec.GJ, _jac_from_aff(ec.G))
# q in [2, n-1]
q = 2 + secrets.randbelow(ec.n - 2)
Q = _mult_aff(q, ec.G, ec)
QJ = _mult_jac(q, ec.GJ, ec)
assert ec._jac_equality(QJ, _jac_from_aff(Q))
assert not ec._jac_equality(QJ, ec.negate(QJ))
assert not ec._jac_equality(QJ, ec.GJ)
def test_jac_equality() -> None:
ec = Curve(13, 0, 2, (1, 9), 19, 1, False)
assert ec._jac_equality(ec.GJ, _jac_from_aff(ec.G))
# q in [2, n-1], as the difference with ec.GJ is checked below
q = 2 + secrets.randbelow(ec.n - 2)
Q = _mult_aff(q, ec.G, ec)
QJ = _mult(q, ec.GJ, ec)
assert ec._jac_equality(QJ, _jac_from_aff(Q))
assert not ec._jac_equality(QJ, ec.negate_jac(QJ))
assert not ec._jac_equality(QJ, ec.GJ)
def _pubkey_recovery(ec: Curve, hf, e: int, sig: ECSS) -> Point:
# Private function provided for testing purposes only.
r, s = _to_sig(ec, sig)
K = r, ec.y_quadratic_residue(r, True)
# FIXME y_quadratic_residue in Jacobian coordinates?
if e == 0:
raise ValueError("invalid (zero) challenge e")
e1 = mod_inv(e, ec.n)
P = double_mult(ec, e1 * s, ec.G, -e1, K)
assert P[1] != 0, "how did you do that?!?"
return P
def verify_commit(c: bytes, ec: Curve, hf, receipt: Receipt) -> bool:
w, R = receipt
# w in [1..n-1] dsa
# w in [1..p-1] ssa
# different verify functions?
# verify R is a good point?
ch = hf(c).digest()
e = hf(octets_from_point(ec, R, True) + ch).digest()
e = int_from_bits(ec, e)
W = ec.add(R, mult(ec, e, ec.G))
# different verify functions?
# return w == W[0] # ECSS
return w == W[0] % ec.n # ECDS, FIXME: ECSSA
def _verhlp(ec: Curve, e: int, P: Point, sig: ECDS) -> bool:
"""Private function provided for testing purposes only."""
# Fail if r is not [1, n-1]
# Fail if s is not [1, n-1]
r, s = _to_sig(ec, sig) # 1
# Let P = point(pk); fail if point(pk) fails.
ec.require_on_curve(P)
if P[1] == 0:
raise ValueError("public key is infinite")
s1 = mod_inv(s, ec.n)
u1 = e*s1
u2 = r*s1 # 4
# Let R = u*G + v*P.
RJ = _double_mult(ec, u1, ec.GJ, u2, (P[0], P[1], 1)) # 5
# Fail if infinite(R).
assert RJ[2] != 0, "how did you do that?!?" # 5
Rx = (RJ[0]*mod_inv(RJ[2]*RJ[2], ec._p)) % ec._p
v = Rx % ec.n # 6, 7
# Fail if r ≠ x(R) %n.
return r == v # 8
def second_generator(ec: Curve, hf) -> Point:
"""Nothing-Up-My-Sleeve (NUMS) second generator H wrt ec.G
source: https://github.com/ElementsProject/secp256k1-zkp/blob/secp256k1-zkp/src/modules/rangeproof/main_impl.h
idea: https://crypto.stackexchange.com/questions/25581/second-generator-for-secp256k1-curve
Get the hash of G, then coerce it to a point (hx, hy).
The resulting point could not be a curvepoint: in this case keep on
incrementing hx until a valid curve point (hx, hy) is obtained.
"""
G_bytes = octets_from_point(ec, ec.G, False)
hd = hf(G_bytes).digest()
hx = int_from_bits(ec, hd)
isCurvePoint = False
while not isCurvePoint:
try:
hy = ec.y_odd(hx, False)
isCurvePoint = True
except:
hx += 1
return Point(hx, hy)
def test_exceptions():
# good curve
Curve(13, 0, 2, (1, 9), 19, 1, False)
with pytest.raises(ValueError, match="p is not prime: "):
Curve(15, 0, 2, (1, 9), 19, 1, False)
with pytest.raises(ValueError, match="negative a: "):
Curve(13, -1, 2, (1, 9), 19, 1, False)
with pytest.raises(ValueError, match="p <= a: "):
Curve(13, 13, 2, (1, 9), 19, 1, False)
with pytest.raises(ValueError, match="negative b: "):
Curve(13, 0, -2, (1, 9), 19, 1, False)
with pytest.raises(ValueError, match="p <= b: "):
Curve(13, 0, 13, (1, 9), 19, 1, False)
with pytest.raises(ValueError, match="zero discriminant"):
Curve(11, 7, 7, (1, 9), 19, 1, False)
err_msg = "Generator must a be a sequence\\[int, int\\]"
with pytest.raises(ValueError, match=err_msg):
Curve(13, 0, 2, (1, 9, 1), 19, 1, False)
with pytest.raises(ValueError, match="Generator is not on the curve"):
Curve(13, 0, 2, (2, 9), 19, 1, False)
with pytest.raises(ValueError, match="n is not prime: "):
Curve(13, 0, 2, (1, 9), 20, 1, False)
with pytest.raises(ValueError, match="n not in "):
Curve(13, 0, 2, (1, 9), 71, 1, False)
with pytest.raises(ValueError, match="INF point cannot be a generator"):
Curve(13, 0, 2, INF, 19, 1, False)
with pytest.raises(ValueError, match="n is not the group order: "):
Curve(13, 0, 2, (1, 9), 17, 1, False)
with pytest.raises(ValueError, match="invalid h: "):
Curve(13, 0, 2, (1, 9), 19, 2, False)
# n=p -> weak curve
# missing
with pytest.raises(UserWarning, match="weak curve"):
Curve(11, 2, 7, (6, 9), 7, 2, True)
from btclib.curve import Curve # low cardinality curves p<100 ec11_7 = Curve(11, 2, 7, (6, 9), 7, 2, 0, False) ec11_17 = Curve(11, 2, 4, (0, 9), 17, 1, 0, False) ec13_11 = Curve(13, 7, 6, (1, 1), 11, 1, 0, False) ec13_19 = Curve(13, 0, 2, (1, 9), 19, 1, 0, False) ec17_13 = Curve(17, 6, 8, (0, 12), 13, 2, 0, False) ec17_23 = Curve(17, 3, 5, (1, 14), 23, 1, 0, False) ec19_13 = Curve(19, 0, 2, (4, 16), 13, 2, 0, False) ec19_23 = Curve(19, 2, 9, (0, 16), 23, 1, 0, False) ec23_19 = Curve(23, 9, 7, (5, 4), 19, 1, 0, False) ec23_31 = Curve(23, 5, 1, (0, 1), 31, 1, 0, False) ec29_37 = Curve(29, 4, 9, (0, 26), 37, 1, 0, False) ec31_23 = Curve(31, 4, 7, (0, 10), 23, 1, 0, False) ec31_43 = Curve(31, 0, 3, (1, 2), 43, 1, 0, False) ec37_31 = Curve(37, 2, 8, (1, 23), 31, 1, 0, False) ec37_43 = Curve(37, 2, 9, (0, 34), 43, 1, 0, False) ec41_37 = Curve(41, 2, 6, (1, 38), 37, 1, 0, False) ec41_53 = Curve(41, 4, 4, (0, 2), 53, 1, 0, False) ec43_37 = Curve(43, 1, 5, (2, 31), 37, 1, 0, False) ec43_47 = Curve(43, 1, 3, (2, 23), 47, 1, 0, False) ec47_41 = Curve(47, 3, 9, (0, 3), 41, 1, 0, False) ec47_61 = Curve(47, 3, 5, (1, 3), 61, 1, 0, False) ec53_47 = Curve(53, 9, 4, (0, 51), 47, 1, 0, False) ec53_61 = Curve(53, 1, 8, (1, 13), 61, 1, 0, False) ec59_53 = Curve(59, 9, 3, (0, 48), 53, 1, 0, False) ec59_73 = Curve(59, 3, 3, (0, 48), 73, 1, 0, False) ec61_59 = Curve(61, 2, 5, (0, 35), 59, 1, 0, False) ec61_73 = Curve(61, 1, 9, (0, 58), 73, 1, 0, False) ec67_61 = Curve(67, 3, 8, (2, 25), 61, 1, 0, False)
"Tests for `btclib.secpoint` module."
import secrets
from typing import Dict
import pytest
from btclib.curve import Curve, _mult_aff
from btclib.curves import CURVES
from btclib.secpoint import bytes_from_point, point_from_octets
# test curves: very low cardinality
low_card_curves: Dict[str, Curve] = {}
# 13 % 4 = 1; 13 % 8 = 5
low_card_curves["ec13_11"] = Curve(13, 7, 6, (1, 1), 11, 1, False)
low_card_curves["ec13_19"] = Curve(13, 0, 2, (1, 9), 19, 1, False)
# 17 % 4 = 1; 17 % 8 = 1
low_card_curves["ec17_13"] = Curve(17, 6, 8, (0, 12), 13, 2, False)
low_card_curves["ec17_23"] = Curve(17, 3, 5, (1, 14), 23, 1, False)
# 19 % 4 = 3; 19 % 8 = 3
low_card_curves["ec19_13"] = Curve(19, 0, 2, (4, 16), 13, 2, False)
low_card_curves["ec19_23"] = Curve(19, 2, 9, (0, 16), 23, 1, False)
# 23 % 4 = 3; 23 % 8 = 7
low_card_curves["ec23_19"] = Curve(23, 9, 7, (5, 4), 19, 1, False)
low_card_curves["ec23_31"] = Curve(23, 5, 1, (0, 1), 31, 1, False)
all_curves: Dict[str, Curve] = {}
all_curves.update(low_card_curves)
all_curves.update(CURVES)
def test_exceptions(self):
# good
Curve(11, 2, 7, (6, 9), 7, 2, 0, False)
# p not odd
self.assertRaises(ValueError, Curve, 10, 2, 7, (6, 9), 7, 1, 0, False)
# p not prime
self.assertRaises(ValueError, Curve, 15, 2, 7, (6, 9), 7, 1, 0, False)
# required security level not in the allowed range
ec = secp112r1
p = ec._p
a = ec._a
b = ec._b
G = ec.G
n = ec.n
t = ec.t
h = ec.h
self.assertRaises(UserWarning, Curve, p, a, b, G, n, h, 273)
#Curve(p, a, b, G, n, h, 273)
# not enough bits for required security level
ec = secp160r1
p = ec._p
a = ec._a
b = ec._b
G = ec.G
n = ec.n
t = ec.t
h = ec.h
self.assertRaises(UserWarning, Curve, p, a, b, G, n, h, 2*t)
#Curve(p, a, b, G, n, h, 2*t)
# a > p
self.assertRaises(ValueError, Curve, 11, 12, 7, (6, 9), 13, 1, 0, False)
# b > p
self.assertRaises(ValueError, Curve, 11, 2, 12, (6, 9), 13, 1, 0, False)
# zero discriminant
self.assertRaises(ValueError, Curve, 11, 7, 7, (6, 9), 7, 1, 0, False)
# G not Tuple (int, int)
self.assertRaises(ValueError, Curve, 11, 2, 7, (6, 9, 1), 7, 1, 0, False)
# G not on curve
self.assertRaises(ValueError, Curve, 11, 2, 7, (7, 9), 7, 1, 0, False)
# n not prime
self.assertRaises(ValueError, Curve, 11, 2, 7, (6, 9), 8, 1, 0, False)
# n not Hesse
self.assertRaises(ValueError, Curve, 11, 2, 7, (6, 9), 71, 1, 0, True)
# h not as expected
self.assertRaises(ValueError, Curve, 11, 2, 7, (6, 9), 7, 1, 0, True)
#Curve(11, 2, 7, (6, 9), 7, 1, 0, True)
# n not group order
self.assertRaises(ValueError, Curve, 11, 2, 7, (6, 9), 13, 1, 0, False)
# n=p -> weak curve
# missing
# weak curve
self.assertRaises(UserWarning, Curve, 11, 2, 7, (6, 9), 7, 2, 0, True)
# x-coordinate not in [0, p-1]
self.assertRaises(ValueError, secp256k1.y, secp256k1._p)
# scroll down at the end of the file for 'relevant' code from btclib.curve import Curve # SEC 2 v.1 curves, removed from SEC 2 v.2 as insecure ones # http://www.secg.org/SEC2-Ver-1.0.pdf __p = (2**128 - 3) // 76439 __a = 0xDB7C2ABF62E35E668076BEAD2088 __b = 0x659EF8BA043916EEDE8911702B22 __Gx = 0x09487239995A5EE76B55F9C2F098 __Gy = 0xA89CE5AF8724C0A23E0E0FF77500 __n = 0xDB7C2ABF62E35E7628DFAC6561C5 __h = 1 secp112r1 = Curve(__p, __a, __b, (__Gx, __Gy), __n, __h, 56, True) __p = (2**128 - 3) // 76439 __a = 0x6127C24C05F38A0AAAF65C0EF02C __b = 0x51DEF1815DB5ED74FCC34C85D709 __Gx = 0x4BA30AB5E892B4E1649DD0928643 __Gy = 0xADCD46F5882E3747DEF36E956E97 __n = 0x36DF0AAFD8B8D7597CA10520D04B __h = 4 secp112r2 = Curve(__p, __a, __b, (__Gx, __Gy), __n, __h, 56, False) __p = 2**128 - 2**97 - 1 __a = 0xFFFFFFFDFFFFFFFFFFFFFFFFFFFFFFFC __b = 0xE87579C11079F43DD824993C2CEE5ED3 __Gx = 0x161FF7528B899B2D0C28607CA52C5B86 __Gy = 0xCF5AC8395BAFEB13C02DA292DDED7A83
