"""
# tag::example1[]
>>> from ecc import FieldElement, Point
>>> 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)
>>> print(p1)
Point(192,105)_0_7 FieldElement(223)

# end::example1[]
# tag::example3[]
>>> from ecc import FieldElement, Point
>>> 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)
>>> print(p1+p2)
Point(170,142)_0_7 FieldElement(223)

# end::example3[]
# tag::example4[]
>>> from ecc import FieldElement, Point
>>> prime = 223
>>> a = FieldElement(0, prime)
>>> b = FieldElement(7, prime)
>>> x = FieldElement(47, prime)
>>> y = FieldElement(71, prime)
>>> p = Point(x, y, a, b)
>>> for s in range(1,21):
...     result = s*p
...     print('{}*(47,71)=({},{})'.format(s,result.x.num,result.y.num))
1*(47,71)=(47,71)
2*(47,71)=(36,111)
3*(47,71)=(15,137)
4*(47,71)=(194,51)
5*(47,71)=(126,96)
6*(47,71)=(139,137)
7*(47,71)=(92,47)
8*(47,71)=(116,55)
9*(47,71)=(69,86)
10*(47,71)=(154,150)
11*(47,71)=(154,73)
12*(47,71)=(69,137)
13*(47,71)=(116,168)
14*(47,71)=(92,176)
15*(47,71)=(139,86)
16*(47,71)=(126,127)
17*(47,71)=(194,172)
18*(47,71)=(15,86)
19*(47,71)=(36,112)
20*(47,71)=(47,152)

# end::example4[]
# tag::example5[]
>>> 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)
>>> print(7*p)
Point(infinity)

# end::example5[]
# tag::example6[]
>>> gx = 0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798
>>> gy = 0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8
>>> p = 2**256 - 2**32 - 977
>>> print(gy**2 % p == (gx**3 + 7) % p)
True

# end::example6[]
# tag::example7[]
>>> from ecc import FieldElement, Point
>>> 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)
>>> print(n*G)
Point(infinity)

# end::example7[]
# tag::example8[]
>>> from ecc import G, N
>>> print(N*G)
S256Point(infinity)

# end::example8[]
# tag::example9[]
>>> from ecc import S256Point, G, N
>>> z = 0xbc62d4b80d9e36da29c16c5d4d9f11731f36052c72401a76c23c0fb5a9b74423
>>> r = 0x37206a0610995c58074999cb9767b87af4c4978db68c06e8e6e81d282047a7c6
>>> s = 0x8ca63759c1157ebeaec0d03cecca119fc9a75bf8e6d0fa65c841c8e2738cdaec
>>> px = 0x04519fac3d910ca7e7138f7013706f619fa8f033e6ec6e09370ea38cee6a7574
>>> py = 0x82b51eab8c27c66e26c858a079bcdf4f1ada34cec420cafc7eac1a42216fb6c4
>>> point = S256Point(px, py)
>>> s_inv = pow(s, N-2, N)  # <1>
>>> u = z * s_inv % N  # <2>
>>> v = r * s_inv % N  # <3>
>>> print((u*G + v*point).x.num == r)  # <4>
True

# end::example9[]
# tag::example10[]
>>> from ecc import S256Point, G, N
>>> from helper import hash256
>>> e = int.from_bytes(hash256(b'my secret'), 'big')  # <1>
>>> z = int.from_bytes(hash256(b'my message'), 'big')  # <2>
>>> k = 1234567890  # <3>
>>> r = (k*G).x.num  # <4>
>>> k_inv = pow(k, N-2, N)
>>> s = (z+r*e) * k_inv % N  # <5>
>>> point = e*G  # <6>
>>> print(point)
S256Point(028d003eab2e428d11983f3e97c3fa0addf3b42740df0d211795ffb3be2f6c52, \
0ae987b9ec6ea159c78cb2a937ed89096fb218d9e7594f02b547526d8cd309e2)
>>> print(hex(z))
0x231c6f3d980a6b0fb7152f85cee7eb52bf92433d9919b9c5218cb08e79cce78
>>> print(hex(r))
0x2b698a0f0a4041b77e63488ad48c23e8e8838dd1fb7520408b121697b782ef22
>>> print(hex(s))
0xbb14e602ef9e3f872e25fad328466b34e6734b7a0fcd58b1eb635447ffae8cb9

# end::example10[]
"""