from hashlib import sha256def sha256d(string):if not isinstance(string, bytes):string = string.encode()return sha256(sha256(string).digest()).hexdigest()def inv_mod(b, p):if b < 0 or p <= b:b = b % pc , d = b, puc, vc, ud, vd, temp = 1, 0, 0, 1, 0while c != 0:temp = cq, c, d = d // c, d % c, tempuc, vc, ud, vd = ud - q * uc, vd - q * vc, uc, vcassert d == 1if ud > 0:return udelse:return ud + pdef leftmost_bit(x):assert x > 0result = 1while result <= x:result = 2 * resultreturn result // 2print(inv_mod(2, 23))
print(3*inv_mod(1, 23) % 23)def show_points(p, a, b):return [(x, y) for x in range(p) for y in range(p) if (y*y - (x*x*x + a*x + b)) % p == 0]print(show_points(p=29, a=4, b=20))def double(x, y, p, a, b):l = ((3 * x * x + a) * inv_mod(2 * y, p)) % px3 = (l * l - 2 * x) % py3 = (l * (x - x3) - y) % preturn x3, y3print(double(1, 4, p=5, a=2, b=3))def add(x1, y1, x2, y2, p, a, b):if x1 == x2 and y1 == y2:return double(x1, y1, p, a, b)l = ((y2 - y1) * inv_mod(x2 - x1, p)) % px3 = (l * l - x1 -x2) % py3 = (l * (x1 - x3) - y1) % preturn x3, y3print(add(1, 4, 3, 1, p=5, a=2, b=3))def get_bits(n):bits = []while n != 0:bits.append(n & 1)n >> 1return bitsclass CurveFp(object):def __init__(self, p, a, b):""" y^2 = x^3 + a*x + b (mod p)."""self.p = pself.a = aself.b = bdef contains_point(self, x, y):return (y * y - (x * x * x + self.a * x + self.b)) % self.p == 0def show_all_points(self):return [(x, y) for x in range(self.p) for y in range(self.p) if(y * y - (x * x * x + self.a * x + self.b)) % self.p == 0]def __repr__(self):return "Curve(p={0:d}, a={1:d}, b={2:d})".format(self.p, self.a, self.b)class Point(object):def __init__(self, curve, x, y, order=None):self.curve = curveself.x = xself.y = yself.order = order# self.curve is allowed to be None only for INFINITY:if self.curve:assert self.curve.contains_point(x, y)if order:assert self * order == INFINITYdef __eq__(self, other):"""Is this point equals to another"""if self.curve == other.curve \and self.x == other.x \and self.y == other.y:return Trueelse:return Falsedef __add__(self, other):"""Add one point to another point."""if other == INFINITY:return selfif self == INFINITY:return otherassert self.curve == other.curveif self.x == other.x:if (self.y + other.y) % self.curve.p == 0:return INFINITYelse:return self.double()p = self.curve.pl = ((other.y - self.y) * \inv_mod(other.x - self.x, p)) % px3 = (l * l - self.x - other.x) % py3 = (l * (self.x - x3) - self.y) % preturn Point(self.curve, x3, y3)def __mul__(self, other):e = otherif self.order:e = e % self.orderif e == 0:return INFINITYif self == INFINITY:return INFINITYe3 = 3 * enegative_self = Point(self.curve, self.x, -self.y, self.order)i = leftmost_bit(e3) // 2result = selfwhile i > 1:result = result.double()if (e3 & i) != 0 and (e & i) == 0:result = result + selfif (e3 & i) == 0 and (e & i) != 0:result = result + negative_selfi = i // 2return resultdef __rmul__(self, other):"""Multiply a point by an integer."""return self * otherdef __repr__(self):if self == INFINITY:return "infinity"return "({0},{1})".format(self.x, self.y)def double(self):"""the double point."""if self == INFINITY:return INFINITYp = self.curve.pa = self.curve.al = ((3 * self.x * self.x + a) * \inv_mod(2 * self.y, p)) % px3 = (l * l - 2 * self.x) % py3 = (l * (self.x - x3) - self.y) % preturn Point(self.curve, x3, y3)def invert(self):return Point(self.curve, self.x, -self.y % self.curve.p)INFINITY = Point(None, None, None)p, a, b = 29, 4, 20
curve = CurveFp(p, a, b)
p0 = Point(curve, 3, 1)
print(p0*2)
print(p0*20)

12
3
[(0, 7), (0, 22), (1, 5), (1, 24), (2, 6), (2, 23), (3, 1), (3, 28), (4, 10), (4, 19), (5, 7), (5, 22), (6, 12), (6, 17), (8, 10), (8, 19), (10, 4), (10, 25), (13, 6), (13, 23), (14, 6), (14, 23), (15, 2), (15, 27), (16, 2), (16, 27), (17, 10), (17, 19), (19, 13), (19, 16), (20, 3), (20, 26), (24, 7), (24, 22), (27, 2), (27, 27)]
(3, 1)
(2, 0)
(24,7)
(15,27)