def test_on_curve(self):
# tests the following points whether they are on the curve or not
# on curve y^2=x^3-7 over F_223:
# (192,105) (17,56) (200,119) (1,193) (42,99)
# the ones that aren't should raise a RuntimeError
prime = 223
a = FieldElement(0, prime)
b = FieldElement(7, prime)
valid_points = ((192, 105), (17, 56), (1, 193))
invalid_points = ((200, 119), (42, 99))
# iterate over valid points
for x_raw, y_raw in valid_points:
# Initialize points this way:
# x = FieldElement(192, prime)
# y = FieldElement(105, prime)
# Point(x, y, a, b)
x = FieldElement(x_raw, prime)
y = FieldElement(y_raw, prime)
# Creating the point should not result in an error
Point(x, y, a, b)
# iterate over invalid points
for x_raw, y_raw in invalid_points:
# Initialize points this way:
# x = FieldElement(192, prime)
# y = FieldElement(105, prime)
# Point(x, y, a, b)
x = FieldElement(x_raw, prime)
y = FieldElement(y_raw, prime)
# check that creating the point results in a RuntimeError
# with self.assertRaises(RuntimeError):
# Point(x, y, a, b)
with self.assertRaises(RuntimeError):
Point(x, y, a, b)
def test_add0(self):
a = Point(x=None, y=None, a=5, b=7)
b = Point(x=2, y=5, a=5, b=7)
c = Point(x=2, y=-5, a=5, b=7)
self.assertEqual(a+b, b)
self.assertEqual(b+a, b)
self.assertEqual(b+c, a)
def test_example_2(self):
p1 = Point(-1, -1, 5, 7)
p2 = Point(-1, 1, 5, 7)
inf = Point(None, None, 5, 7)
self.assertEqual(p1 + inf, p1)
self.assertEqual(inf + p2, p2)
self.assertEqual(p1 + p2, inf)
def test_exercise_4(self):
prime = 223
a = FieldElement(0, prime)
b = FieldElement(7, prime)
x1 = FieldElement(num=192, prime=prime)
y1 = FieldElement(num=105, prime=prime)
p = Point(x1, y1, a, b)
p2 = p + p
self.assertEqual(p2.x.num, 49)
self.assertEqual(p2.y.num, 71)
self.assertEqual(p2.x.prime, prime)
x1 = FieldElement(num=143, prime=prime)
y1 = FieldElement(num=98, prime=prime)
p = Point(x1, y1, a, b)
p2 = p + p
self.assertEqual(p2.x.num, 64)
self.assertEqual(p2.y.num, 168)
self.assertEqual(p2.x.prime, prime)
x1 = FieldElement(num=47, prime=prime)
y1 = FieldElement(num=71, prime=prime)
p = Point(x1, y1, a, b)
p2 = p + p
self.assertEqual(p2.x.num, 36)
self.assertEqual(p2.y.num, 111)
self.assertEqual(p2.x.prime, prime)
p2 = p + p + p + p
self.assertEqual(p2.x.num, 194)
self.assertEqual(p2.y.num, 51)
self.assertEqual(p2.x.prime, prime)
p2 = p + p + p + p + p + p + p + p
self.assertEqual(p2.x.num, 116)
self.assertEqual(p2.y.num, 55)
self.assertEqual(p2.x.prime, prime)
p2 = p + p + p + p + p + p + p + p + p + p + p + p + p + p + p + p + p + p + p + p + p
self.assertEqual(p2.x, None)
def test_add1(self):
# tests the following additions on curve y^2=x^3-7 over F_223:
# (192,105) + (17,56)
# (47,71) + (117,141)
# (143,98) + (76,66)
prime = 223
a = FieldElement(0, prime)
b = FieldElement(7, prime)
additions = (
# (x1, y1, x2, y2, x3, y3)
(192, 105, 17, 56, 170, 142),
(47, 71, 117, 141, 60, 139),
(143, 98, 76, 66, 47, 71),
)
# iterate over the additions
for x1_raw, y1_raw, x2_raw, y2_raw, x3_raw, y3_raw in additions:
# Initialize points this way:
# x = FieldElement(192, prime)
# y = FieldElement(105, prime)
# p1 = Point(x, y, a, b)
x1 = FieldElement(x1_raw, prime)
y1 = FieldElement(y1_raw, prime)
p1 = Point(x1, y1, a, b)
x2 = FieldElement(x2_raw, prime)
y2 = FieldElement(y2_raw, prime)
p2 = Point(x2, y2, a, b)
x3 = FieldElement(x3_raw, prime)
y3 = FieldElement(y3_raw, prime)
p3 = Point(x3, y3, a, b)
# check that p1 + p2 == p3
self.assertEqual(p1+p2, p3)
def test_example_2(self):
prime = 137
a = FieldElement(0, prime)
b = FieldElement(7, prime)
p1 = Point(FieldElement(73, prime), FieldElement(128, prime), a, b)
p2 = Point(FieldElement(46, prime), FieldElement(22, prime), a, b)
p3 = p1 + p2
self.assertEqual(p3.x.num, 99)
self.assertEqual(p3.y.num, 49)
def test_add_Point(self):
# add a point to another point
p1 = Point(2, 5, 5, 7)
p2 = Point(-1, -1, 5, 7)
p3 = Point(3, -7, 5, 7)
self.assertEqual(p1 + p2, p3)
# add a point to itself
p4 = Point(-1.1100000000000003, 0.2870000000000008, 5, 7)
self.assertEqual(p1 + p1, p4)
def test_example_4(self):
expected = [(47, 71), (36, 111), (15, 137), (194, 51), (126, 96), (139, 137), (92, 47), (116, 55), (69, 86), (154, 150), (154, 73), (69, 137), (116, 168), (92, 176), (139, 86), (126, 127), (194, 172), (15, 86), (36, 112), (47, 152)]
prime = 223
a = FieldElement(0, prime)
b = FieldElement(7, prime)
g = Point(FieldElement(47, prime), FieldElement(71, prime), a, b)
inf = Point(None, None, a, b)
total = g
count = 1
while total != inf:
self.assertEqual((total.x.num, total.y.num), expected.pop(0))
total += g
count += 1
self.assertEqual(total, inf)
def test_exercise_1(self):
prime = 223
a = FieldElement(0, prime)
b = FieldElement(7, prime)
points = ((192, 105), (17, 56), (200, 119), (1, 193), (42, 99))
expected = [True, True, False, True, False]
for x_raw, y_raw in points:
x = FieldElement(x_raw, prime)
y = FieldElement(y_raw, prime)
if expected.pop(0):
self.assertTrue(Point(x, y, a, b))
else:
with self.assertRaises(ValueError):
Point(x, y, a, b)
def test_exercise_5(self):
prime = 223
a = FieldElement(0, prime)
b = FieldElement(7, prime)
x = FieldElement(15, prime)
y = FieldElement(86, prime)
p = Point(x, y, a, b)
inf = Point(None, None, a, b)
product = p
count = 1
while product != inf:
product += p
count += 1
self.assertEqual(count, 7)
def test_example_3(self):
prime = 223
a = FieldElement(num=0, prime=prime)
b = FieldElement(num=7, prime=prime)
x1 = FieldElement(num=192, prime=prime)
y1 = FieldElement(num=105, prime=prime)
x2 = FieldElement(num=17, prime=prime)
y2 = FieldElement(num=56, prime=prime)
p1 = Point(x1, y1, a, b)
p2 = Point(x2, y2, a, b)
p3 = p1 + p2
self.assertEqual(p3.x.num, 170)
self.assertEqual(p3.y.num, 142)
self.assertEqual(p3.x.prime, 223)
def test_exercise_2(self):
prime = 223
a = FieldElement(0, prime)
b = FieldElement(7, prime)
additions = ((192, 105, 17, 56), (47, 71, 117, 141), (143, 98, 76, 66))
expected = [(170, 142), (60, 139), (47, 71)]
for x1_raw, y1_raw, x2_raw, y2_raw in additions:
x1 = FieldElement(x1_raw, prime)
y1 = FieldElement(y1_raw, prime)
p1 = Point(x1, y1, a, b)
x2 = FieldElement(x2_raw, prime)
y2 = FieldElement(y2_raw, prime)
p2 = Point(x2, y2, a, b)
p3 = p1 + p2
self.assertEqual((p3.x.num, p3.y.num), expected.pop(0))
def t_on_curve(self):
prime = 223
a = FieldElement(0, prime)
b = FieldElement(7, prime)
valid_points = ((192, 105), (17, 56), (1, 193))
invalid_points = ((200, 119), (42, 99))
for x_raw, y_raw in valid_points:
x = FieldElement(x_raw, prime)
y = FieldElement(y_raw, prime)
Point(x, y, a, b)
for x_raw, y_raw in invalid_points:
x = FieldElement(x_raw, prime)
y = FieldElement(y_raw, prime)
with self.assertRaises(ValueError):
Point(x, y, a, b)
def test_example_4(self):
prime = 223
a = FieldElement(0, prime)
b = FieldElement(7, prime)
x = FieldElement(47, prime)
y = FieldElement(71, prime)
p = Point(x, y, a, b)
want = (
(None, None),
(47, 71),
(36, 111),
(15, 137),
(194, 51),
(126, 96),
(139, 137),
(92, 47),
(116, 55),
(69, 86),
(154, 150),
(154, 73),
(69, 137),
(116, 168),
(92, 176),
(139, 86),
(126, 127),
(194, 172),
(15, 86),
(36, 112),
(47, 152),
)
for s in range(1, 21):
result = s * p
self.assertEqual((result.x.num, result.y.num), want[s])
def test_Point_scalar_multiply(self):
prime = 223
a = FieldElement(0, prime)
b = FieldElement(7, prime)
x = FieldElement(47, prime)
y = FieldElement(71, prime)
p = Point(x, y, a, b)
x6 = FieldElement(139, prime)
y6 = FieldElement(137, prime)
p6 = Point(x6, y6, a, b)
result = 6 * p
self.assertEqual(result, p6)
result = 21 * p
self.assertEqual(result.x, None)
self.assertEqual(result.y, None)
def test_example_3(self):
prime = 137
a = FieldElement(0, prime)
b = FieldElement(7, prime)
p = Point(FieldElement(73, prime), FieldElement(128, prime), a, b)
p2 = p + p
self.assertEqual(p2.x.num, 103)
self.assertEqual(p2.y.num, 76)
def test_exercise_3(self):
prime = 223
a = FieldElement(0, prime)
b = FieldElement(7, prime)
multiplications = ((2, 192, 105), (2, 143, 98), (2, 47, 71), (4, 47, 71), (8, 47, 71), (21, 47, 71))
expected = [(49, 71), (64, 168), (36, 111), (194, 51), (116, 55), None]
for n, x_raw, y_raw in multiplications:
x = FieldElement(x_raw, prime)
y = FieldElement(y_raw, prime)
p = Point(x, y, a, b)
product = Point(None, None, a, b)
for _ in range(n):
product = product + p
if product.x is None:
self.assertEqual(None, expected.pop(0))
else:
self.assertEqual((product.x.num, product.y.num), expected.pop(0))
def test_example_1(self):
a = FieldElement(num=0, prime=223)
b = FieldElement(num=7, prime=223)
x = FieldElement(num=192, prime=223)
y = FieldElement(num=105, prime=223)
p1 = Point(x, y, a, b)
self.assertEqual(p1.x.num, 192)
self.assertEqual(p1.y.num, 105)
self.assertEqual(p1.x.prime, 223)
def test_example_5(self):
prime = 223
a = FieldElement(0, prime)
b = FieldElement(7, prime)
x = FieldElement(15, prime)
y = FieldElement(86, prime)
p = Point(x, y, a, b)
p2 = 7 * p
self.assertEqual(p2.x, None)
def t_add(self):
prime = 223
a = FieldElement(0, prime)
b = FieldElement(7, prime)
additions = (
(192, 105, 17, 56, 170, 142),
(47, 71, 117, 141, 60, 139),
(143, 98, 76, 66, 47, 71),
)
for x1_raw, y1_raw, x2_raw, y2_raw, x3_raw, y3_raw in additions:
x1 = FieldElement(x1_raw, prime)
y1 = FieldElement(y1_raw, prime)
p1 = Point(x1, y1, a, b)
x2 = FieldElement(x2_raw, prime)
y2 = FieldElement(y2_raw, prime)
p2 = Point(x2, y2, a, b)
x3 = FieldElement(x3_raw, prime)
y3 = FieldElement(y3_raw, prime)
p3 = Point(x3, y3, a, b)
self.assertEqual(p1 + p2, p3)
def test_rmul(self):
# tests the following scalar multiplications
# 2*(192,105)
# 2*(143,98)
# 2*(47,71)
# 4*(47,71)
# 8*(47,71)
# 21*(47,71)
prime = 223
a = FieldElement(0, prime)
b = FieldElement(7, prime)
multiplications = (
# (coefficient, x1, y1, x2, y2)
(2, 192, 105, 49, 71),
(2, 143, 98, 64, 168),
(2, 47, 71, 36, 111),
(4, 47, 71, 194, 51),
(8, 47, 71, 116, 55),
(21, 47, 71, None, None),
)
# iterate over the multiplications
for s, x1_raw, y1_raw, x2_raw, y2_raw in multiplications:
# Initialize points this way:
# x = FieldElement(192, prime)
# y = FieldElement(105, prime)
# p = Point(x, y, a, b)
x1 = FieldElement(x1_raw, prime)
y1 = FieldElement(y1_raw, prime)
p1 = Point(x1, y1, a, b)
# initialize the second point based on whether it's the point at infinity
if x2_raw is None:
p2 = Point(None, None, a, b)
else:
x2 = FieldElement(x2_raw, prime)
y2 = FieldElement(y2_raw, prime)
p2 = Point(x2, y2, a, b)
# check that the product is equal to the expected point
self.assertEqual(s * p1, p2)
def test_example_7(self):
gx = 0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798
gy = 0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8
p = 2**256 - 2**32 - 977
n = 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141
x = FieldElement(gx, p)
y = FieldElement(gy, p)
seven = FieldElement(7, p)
zero = FieldElement(0, p)
G = Point(x, y, zero, seven)
p2 = n * G
self.assertEqual(p2.x, None)
def test_add_Point_over_FF(self):
a = FieldElement(0, 223)
b = FieldElement(7, 223)
x1 = FieldElement(192, 223)
y1 = FieldElement(105, 223)
p1 = Point(x1, y1, a, b)
x2 = FieldElement(17, 223)
y2 = FieldElement(56, 223)
p2 = Point(x2, y2, a, b)
x3 = FieldElement(170, 223)
y3 = FieldElement(142, 223)
p3 = Point(x3, y3, a, b)
self.assertEqual(p1 + p2, p3)
x4 = FieldElement(60, 223)
y4 = FieldElement(139, 223)
p4 = Point(x4, y4, a, b)
x5 = FieldElement(220, 223)
y5 = FieldElement(181, 223)
p5 = Point(x5, y5, a, b)
self.assertEqual(p3 + p4, p5)
x6 = FieldElement(49, 223)
y6 = FieldElement(71, 223)
p6 = Point(x6, y6, a, b)
self.assertEqual(p1 + p1, p6)
def test_on_curve(self):
prime = 223
a = FieldElement(0, prime)
b = FieldElement(7, prime)
valid_points = ((192, 105), (17, 56), (1, 193))
invalid_points = ((200, 119), (42, 99))
# iterate over valid points
for x_raw, y_raw in valid_points:
x = FieldElement(x_raw, prime)
y = FieldElement(y_raw, prime)
# Creating the point should not result in an error
Point(x, y, a, b)
# iterate over invalid points
for x_raw, y_raw in invalid_points:
x = FieldElement(x_raw, prime)
y = FieldElement(y_raw, prime)
# check that creating the point results in a ValueError
with self.assertRaises(ValueError):
Point(x, y, a, b)
def t_rmul(self):
prime = 223
a = FieldElement(0, prime)
b = FieldElement(7, prime)
multiplications = (
(2, 192, 105, 49, 71),
(2, 143, 98, 64, 168),
(2, 47, 71, 36, 111),
(4, 47, 71, 194, 51),
(8, 47, 71, 116, 55),
(21, 47, 71, None, None),
)
for s, x1_raw, y1_raw, x2_raw, y2_raw in multiplications:
x1 = FieldElement(x1_raw, prime)
y1 = FieldElement(y1_raw, prime)
p1 = Point(x1, y1, a, b)
if x2_raw is None:
p2 = Point(None, None, a, b)
else:
x2 = FieldElement(x2_raw, prime)
y2 = FieldElement(y2_raw, prime)
p2 = Point(x2, y2, a, b)
self.assertEqual(s * p1, p2)
def test_exercise_2(self):
prime = 223
a = FieldElement(0, prime)
b = FieldElement(7, prime)
p1 = Point(FieldElement(170, prime), FieldElement(142, prime), a, b)
p2 = Point(FieldElement(60, prime), FieldElement(139, prime), a, b)
p3 = p1 + p2
self.assertEqual(p3.x.num, 220)
self.assertEqual(p3.y.num, 181)
self.assertEqual(p3.x.prime, prime)
p1 = Point(FieldElement(47, prime), FieldElement(71, prime), a, b)
p2 = Point(FieldElement(17, prime), FieldElement(56, prime), a, b)
p3 = p1 + p2
self.assertEqual(p3.x.num, 215)
self.assertEqual(p3.y.num, 68)
self.assertEqual(p3.x.prime, prime)
p1 = Point(FieldElement(143, prime), FieldElement(98, prime), a, b)
p2 = Point(FieldElement(76, prime), FieldElement(66, prime), a, b)
p3 = p1 + p2
self.assertEqual(p3.x.num, 47)
self.assertEqual(p3.y.num, 71)
self.assertEqual(p3.x.prime, prime)
from ecc import FieldElement, Point
prime = 223
a = FieldElement(0, prime)
b = FieldElement(7, prime)
x = FieldElement(15, prime)
y = FieldElement(86, prime)
p = Point(x, y, a, b)
inf = Point(None, None, a, b)
product = p
count = 1
while product != inf:
product += p
count += 1
print(count)
from ecc import Point p1=Point(2,5,5,7) p2=Point(-1,-1,5,7) print(p1+p2)
'''
#code
>>> import ecc, helper
>>> from ecc import FieldElement, G, Point, S256Point
>>> from helper import encode_base58, hash160, hash256, little_endian_to_int
#endcode
#code
>>> # Verify curve Example
>>> prime = 137
>>> x, y = 73, 128
>>> print(y**2 % prime == (x**3 + 7) % prime)
True
#endcode
#exercise
Find out which points are valid on the curve \\( y^2 = x^3 + 7: F_{223} \\)
```
(192,105), (17,56), (200,119), (1,193), (42,99)
```
---
>>> prime = 223
>>> a = FieldElement(0, prime)
>>> b = FieldElement(7, prime)
>>> points = ((192,105), (17,56), (200,119), (1,193), (42,99))
>>> # iterate over points
>>> for x_raw, y_raw in points: #/
... # Initialize points this way:
... # x = FieldElement(x_raw, prime)
... # y = FieldElement(y_raw, prime)
from ecc import FieldElement, Point prime = 223 a = FieldElement(0, prime) b = FieldElement(7, prime) p1 = Point(FieldElement(170, prime), FieldElement(142, prime), a, b) p2 = Point(FieldElement(60, prime), FieldElement(139, prime), a, b) print(p1 + p2) p1 = Point(FieldElement(47, prime), FieldElement(71, prime), a, b) p2 = Point(FieldElement(17, prime), FieldElement(56, prime), a, b) print(p1 + p2) p1 = Point(FieldElement(143, prime), FieldElement(98, prime), a, b) p2 = Point(FieldElement(76, prime), FieldElement(66, prime), a, b) print(p1 + p2)
