def find_lambda(numerator, denominator, n):
"""
Вычисление значения l = numerator/denominator (mod n)
:return: l
"""
left = positive_mod(denominator, n)
right = positive_mod(numerator, n)
return int(crt((left, n), (0, right))[0] // left)
def verify(self, message, signature):
p, q, a, y = self.prover_key
w, s = signature
h = calc_hash(message, sha256)
v = pow(h, q - 2, q)
z1 = positive_mod(s * v, q)
z2 = positive_mod((q - w) * v, q)
u = positive_mod(positive_mod(pow(a, z1, p) * pow(y, z2, p), p), q)
passed = w == u
self.steps = locals()
return passed
def get_random_point(a, b, n):
"""
Получение случайной точки на эллиптической кривой ``y^2 = x^3 + a*x + b (mod n)``
:return: Точка (x, y)
"""
success = False
while not success:
try:
x = randint(1, n)
y = next(
sqrt_mod_iter(positive_mod(pow(x, 3, n) - (a * x) - b, n), n))
return x, y
except StopIteration:
pass
def add(first, second, curve):
"""
Операция сложения над точками кривой
:param first: первая точка
:param second: вторая точка
:param curve: параметры кривой
:return: новая точка
"""
if first == (0, 0):
return second
if second == (0, 0):
return first
x1, y1 = first
x2, y2 = second
if x1 == x2 and y1 != y2:
return 0, 0
a, b, n = curve
if x1 == x2:
l = find_lambda(3 * x1 * x1 + a, 2 * y1, n)
else:
l = find_lambda(y2 - y1, x2 - x1, n)
x = positive_mod(l * l - x1 - x2, n)
y = positive_mod(l * (x1 - x) - y1, n)
return x, y
def sign(self, message):
"""
Подтверждение авторства сообщения
"""
p, q, a, y = self.public_key
h = calc_hash(message, sha256)
s = w_ = w = k = 0
while s == 0:
while w == 0:
k = randint(1, q)
w = pow(a, k, p)
w_ = w % q
s = positive_mod(self._private_key * w_ + k * h, q)
self.steps = locals()
return w_, s
def verify(self, message, signature):
"""
Подтверждение авторства сообщения
"""
(a, b, n), (x_p, y_p), q, (x_q, y_q) = self.prover_key
r, s = signature
h = calc_hash(message, sha512)
e = h % q
e_1 = int(crt((e, q), (0, 1))[0]) // e # обратное значение
v = e_1 % q
z1 = (s * v) % q
z2 = positive_mod((q - r) * v, q)
x_c, y_c = add(mul((x_p, y_p), z1, (a, b, n)),
mul((x_q, y_q), z2, (a, b, n)), (a, b, n))
r_ = x_c % q
passed = r == r_
self.steps = locals()
return passed