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handle non-prime order curves more gracefully #344

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merged 4 commits into from
Aug 21, 2024
Merged

handle non-prime order curves more gracefully #344

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8254038
handle non-prime order curves more gracefully
tomato42 Aug 6, 2024
3d00b32
work with curves that have x=0, y=0 as point on the curve
tomato42 Aug 8, 2024
7426df8
better test coverage for ellipticcurve
tomato42 Aug 16, 2024
de9141c
formatting fixups
tomato42 Aug 16, 2024
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57 changes: 34 additions & 23 deletions src/ecdsa/ellipticcurve.py
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Original file line number Diff line number Diff line change
Expand Up @@ -633,7 +633,7 @@ def __eq__(self, other):
"""
x1, y1, z1 = self.__coords
if other is INFINITY:
return not y1 or not z1
return not z1
if isinstance(other, Point):
x2, y2, z2 = other.x(), other.y(), 1
elif isinstance(other, PointJacobi):
Expand Down Expand Up @@ -723,11 +723,13 @@ def scale(self):

def to_affine(self):
"""Return point in affine form."""
_, y, z = self.__coords
if not y or not z:
_, _, z = self.__coords
p = self.__curve.p()
if not (z % p):
return INFINITY
self.scale()
x, y, z = self.__coords
assert z == 1
return Point(self.__curve, x, y, self.__order)

@staticmethod
Expand Down Expand Up @@ -759,7 +761,7 @@ def _double_with_z_1(self, X1, Y1, p, a):
# http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#doubling-mdbl-2007-bl
XX, YY = X1 * X1 % p, Y1 * Y1 % p
if not YY:
return 0, 0, 1
return 0, 0, 0
YYYY = YY * YY % p
S = 2 * ((X1 + YY) ** 2 - XX - YYYY) % p
M = 3 * XX + a
Expand All @@ -773,13 +775,13 @@ def _double(self, X1, Y1, Z1, p, a):
"""Add a point to itself, arbitrary z."""
if Z1 == 1:
return self._double_with_z_1(X1, Y1, p, a)
if not Y1 or not Z1:
return 0, 0, 1
if not Z1:
return 0, 0, 0
# after:
# http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#doubling-dbl-2007-bl
XX, YY = X1 * X1 % p, Y1 * Y1 % p
if not YY:
return 0, 0, 1
return 0, 0, 0
YYYY = YY * YY % p
ZZ = Z1 * Z1 % p
S = 2 * ((X1 + YY) ** 2 - XX - YYYY) % p
Expand All @@ -795,14 +797,14 @@ def double(self):
"""Add a point to itself."""
X1, Y1, Z1 = self.__coords

if not Y1:
if not Z1:
return INFINITY

p, a = self.__curve.p(), self.__curve.a()

X3, Y3, Z3 = self._double(X1, Y1, Z1, p, a)

if not Y3 or not Z3:
if not Z3:
return INFINITY
return PointJacobi(self.__curve, X3, Y3, Z3, self.__order)

Expand Down Expand Up @@ -886,10 +888,10 @@ def __radd__(self, other):

def _add(self, X1, Y1, Z1, X2, Y2, Z2, p):
"""add two points, select fastest method."""
if not Y1 or not Z1:
return X2, Y2, Z2
if not Y2 or not Z2:
return X1, Y1, Z1
if not Z1:
return X2 % p, Y2 % p, Z2 % p
if not Z2:
return X1 % p, Y1 % p, Z1 % p
if Z1 == Z2:
if Z1 == 1:
return self._add_with_z_1(X1, Y1, X2, Y2, p)
Expand Down Expand Up @@ -917,7 +919,7 @@ def __add__(self, other):

X3, Y3, Z3 = self._add(X1, Y1, Z1, X2, Y2, Z2, p)

if not Y3 or not Z3:
if not Z3:
return INFINITY
return PointJacobi(self.__curve, X3, Y3, Z3, self.__order)

Expand All @@ -927,7 +929,7 @@ def __rmul__(self, other):

def _mul_precompute(self, other):
"""Multiply point by integer with precomputation table."""
X3, Y3, Z3, p = 0, 0, 1, self.__curve.p()
X3, Y3, Z3, p = 0, 0, 0, self.__curve.p()
_add = self._add
for X2, Y2 in self.__precompute:
if other % 2:
Expand All @@ -940,7 +942,7 @@ def _mul_precompute(self, other):
else:
other //= 2

if not Y3 or not Z3:
if not Z3:
return INFINITY
return PointJacobi(self.__curve, X3, Y3, Z3, self.__order)

Expand All @@ -959,7 +961,7 @@ def __mul__(self, other):

self = self.scale()
X2, Y2, _ = self.__coords
X3, Y3, Z3 = 0, 0, 1
X3, Y3, Z3 = 0, 0, 0
p, a = self.__curve.p(), self.__curve.a()
_double = self._double
_add = self._add
Expand All @@ -972,7 +974,7 @@ def __mul__(self, other):
elif i > 0:
X3, Y3, Z3 = _add(X3, Y3, Z3, X2, Y2, 1, p)

if not Y3 or not Z3:
if not Z3:
return INFINITY

return PointJacobi(self.__curve, X3, Y3, Z3, self.__order)
Expand Down Expand Up @@ -1001,7 +1003,7 @@ def mul_add(self, self_mul, other, other_mul):
other_mul = other_mul % self.__order

# (X3, Y3, Z3) is the accumulator
X3, Y3, Z3 = 0, 0, 1
X3, Y3, Z3 = 0, 0, 0
p, a = self.__curve.p(), self.__curve.a()

# as we have 6 unique points to work with, we can't scale all of them,
Expand All @@ -1025,7 +1027,7 @@ def mul_add(self, self_mul, other, other_mul):
# when the self and other sum to infinity, we need to add them
# one by one to get correct result but as that's very unlikely to
# happen in regular operation, we don't need to optimise this case
if not pApB_Y or not pApB_Z:
if not pApB_Z:
return self * self_mul + other * other_mul

# gmp object creation has cumulatively higher overhead than the
Expand Down Expand Up @@ -1070,7 +1072,7 @@ def mul_add(self, self_mul, other, other_mul):
assert B > 0
X3, Y3, Z3 = _add(X3, Y3, Z3, pApB_X, pApB_Y, pApB_Z, p)

if not Y3 or not Z3:
if not Z3:
return INFINITY

return PointJacobi(self.__curve, X3, Y3, Z3, self.__order)
Expand Down Expand Up @@ -1154,6 +1156,8 @@ def __eq__(self, other):

Note: only points that lay on the same curve can be equal.
"""
if other is INFINITY:
return self.__x is None or self.__y is None
if isinstance(other, Point):
return (
self.__curve == other.__curve
Expand Down Expand Up @@ -1220,7 +1224,12 @@ def leftmost_bit(x):
# From X9.62 D.3.2:

e3 = 3 * e
negative_self = Point(self.__curve, self.__x, -self.__y, self.__order)
negative_self = Point(
self.__curve,
self.__x,
(-self.__y) % self.__curve.p(),
self.__order,
)
i = leftmost_bit(e3) // 2
result = self
# print("Multiplying %s by %d (e3 = %d):" % (self, other, e3))
Expand All @@ -1247,7 +1256,6 @@ def __str__(self):

def double(self):
"""Return a new point that is twice the old."""

if self == INFINITY:
return INFINITY

Expand All @@ -1261,6 +1269,9 @@ def double(self):
* numbertheory.inverse_mod(2 * self.__y, p)
) % p

if not l:
return INFINITY

x3 = (l * l - 2 * self.__x) % p
y3 = (l * (self.__x - x3) - self.__y) % p

Expand Down
38 changes: 38 additions & 0 deletions src/ecdsa/test_ellipticcurve.py
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Original file line number Diff line number Diff line change
Expand Up @@ -83,6 +83,11 @@ def test_inequality_curves(self):
c192 = CurveFp(p, -3, b)
self.assertNotEqual(self.c_23, c192)

def test_inequality_curves_by_b_only(self):
a = CurveFp(23, 1, 0)
b = CurveFp(23, 1, 1)
self.assertNotEqual(a, b)

def test_usability_in_a_hashed_collection_curves(self):
{self.c_23: None}

Expand Down Expand Up @@ -184,6 +189,33 @@ def test_double(self):
self.assertEqual(p3.x(), x3)
self.assertEqual(p3.y(), y3)

def test_double_to_infinity(self):
p1 = Point(self.c_23, 11, 20)
p2 = p1.double()
self.assertEqual((p2.x(), p2.y()), (4, 0))
self.assertNotEqual(p2, INFINITY)
p3 = p2.double()
self.assertEqual(p3, INFINITY)
self.assertIs(p3, INFINITY)

def test_add_self_to_infinity(self):
p1 = Point(self.c_23, 11, 20)
p2 = p1 + p1
self.assertEqual((p2.x(), p2.y()), (4, 0))
self.assertNotEqual(p2, INFINITY)
p3 = p2 + p2
self.assertEqual(p3, INFINITY)
self.assertIs(p3, INFINITY)

def test_mul_to_infinity(self):
p1 = Point(self.c_23, 11, 20)
p2 = p1 * 2
self.assertEqual((p2.x(), p2.y()), (4, 0))
self.assertNotEqual(p2, INFINITY)
p3 = p2 * 2
self.assertEqual(p3, INFINITY)
self.assertIs(p3, INFINITY)

def test_multiply(self):
x1, y1, m, x3, y3 = (3, 10, 2, 7, 12)
p1 = Point(self.c_23, x1, y1)
Expand Down Expand Up @@ -224,6 +256,12 @@ def test_inequality_points_diff_types(self):
c = CurveFp(100, -3, 100)
self.assertNotEqual(self.g_23, c)

def test_inequality_diff_y(self):
p1 = Point(self.c_23, 6, 4)
p2 = Point(self.c_23, 6, 19)

self.assertNotEqual(p1, p2)

def test_to_bytes_from_bytes(self):
p = Point(self.c_23, 3, 10)

Expand Down
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