def extended_legendre_symbol(a):
"""
Legendre Symbol on the Extended Field
Args:
a: The Target
Returns:
Legendre Symbol of a
"""
from six.moves import xrange
from ecpy.utils import legendre_symbol
m = a.field.degree()
p = a.field.p
b = pow(a, sum([p**i for i in xrange(0, m)]), p)
return legendre_symbol(b, p)
def extended_legendre_symbol(a):
"""
Legendre Symbol on the Extended Field
Args:
a: The Target
Returns:
Legendre Symbol of a
"""
from six.moves import xrange
from ecpy.utils import legendre_symbol
m = a.field.degree()
p = a.field.p
b = pow(a, sum([p**i for i in xrange(0, m)]), p)
return legendre_symbol(b, p)
def __modular_square_root(a, m):
from ecpy.utils import is_prime, legendre_symbol, modinv
if is_prime(m):
if legendre_symbol(a, m) == -1:
return []
# Tonelli-Shanks Algorithm
if m % 4 == 3:
r = pow(a, (m + 1) // 4, m)
return [r, m - r]
s = _find_power_divisor(2, m - 1)
q = (m - 1) // 2**s
z = 0
while legendre_symbol(z, m) != -1:
z = random.randint(1, m)
c = pow(z, q, m)
r = pow(a, (q + 1) // 2, m)
t = pow(a, q, m)
l = s
while True:
if t % m == 1:
assert (r ** 2) % m == a
return [r, m - r]
i = _find_power(2, t, 1, m)
power = l - i - 1
if power < 0:
power = modinv(2**-power, m)
else:
power = 2**power
b = pow(c, power, m)
r = (r * b) % m
t = (t * (b**2)) % m
c = pow(b, 2, m)
l = i
if m == 2:
return a
if m % 4 == 3:
r = pow(a, (m + 1) // 4, m)
return [r, m - r]
if m % 8 == 5:
v = pow(2 * a, (m - 5) // 8, m)
i = pow(2 * a * v, 2, m)
r = a * v * (i - 1) % m
return [r, m - r]
if m % 8 == 1:
e = _find_power_divisor(2, m - 1)
q = (m - 1) // 2**e
z = 1
while pow(z, 2**(e - 1), m) == 1:
x = random.randint(1, m)
z = pow(x, q, m)
y = z
r = e
x = pow(a, (q - 1) // 2, m)
v = a * x % m
w = v * x % m
while True:
if w == 1:
return [v, m - v]
k = _find_power(2, w, 1, m)
d = pow(y, 2**(r - k - 1), m)
y = pow(d, 2, m)
r = k
v = d * v % m
w = w * y % m
def __modular_square_root(a, m):
from ecpy.utils import is_prime, legendre_symbol, modinv
if is_prime(m):
if legendre_symbol(a, m) == -1:
return []
# Tonelli-Shanks Algorithm
if m % 4 == 3:
r = pow(a, (m + 1) // 4, m)
return [r, m - r]
s = _find_power_divisor(2, m - 1)
q = (m - 1) // 2**s
z = 0
while legendre_symbol(z, m) != -1:
z = random.randint(1, m)
c = pow(z, q, m)
r = pow(a, (q + 1) // 2, m)
t = pow(a, q, m)
l = s
while True:
if t % m == 1:
assert (r**2) % m == a
return [r, m - r]
i = _find_power(2, t, 1, m)
power = l - i - 1
if power < 0:
power = modinv(2**-power, m)
else:
power = 2**power
b = pow(c, power, m)
r = (r * b) % m
t = (t * (b**2)) % m
c = pow(b, 2, m)
l = i
if m == 2:
return a
if m % 4 == 3:
r = pow(a, (m + 1) // 4, m)
return [r, m - r]
if m % 8 == 5:
v = pow(2 * a, (m - 5) // 8, m)
i = pow(2 * a * v, 2, m)
r = a * v * (i - 1) % m
return [r, m - r]
if m % 8 == 1:
e = _find_power_divisor(2, m - 1)
q = (m - 1) // 2**e
z = 1
while pow(z, 2**(e - 1), m) == 1:
x = random.randint(1, m)
z = pow(x, q, m)
y = z
r = e
x = pow(a, (q - 1) // 2, m)
v = a * x % m
w = v * x % m
while True:
if w == 1:
return [v, m - v]
k = _find_power(2, w, 1, m)
d = pow(y, 2**(r - k - 1), m)
y = pow(d, 2, m)
r = k
v = d * v % m
w = w * y % m