class EccPoint(object):
"""A class to abstract a point over an Elliptic Curve.
:ivar x: The X-coordinate of the ECC point
:vartype x: integer
:ivar y: The Y-coordinate of the ECC point
:vartype y: integer
"""
def __init__(self, x, y):
self._x = Integer(x)
self._y = Integer(y)
# Buffers
self._common = Integer(0)
self._tmp1 = Integer(0)
self._x3 = Integer(0)
self._y3 = Integer(0)
def set(self, point):
self._x = Integer(point._x)
self._y = Integer(point._y)
return self
def __eq__(self, point):
return self._x == point._x and self._y == point._y
def __neg__(self):
if self.is_point_at_infinity():
return self.point_at_infinity()
return EccPoint(self._x, _curve.p - self._y)
def copy(self):
return EccPoint(self._x, self._y)
def is_point_at_infinity(self):
return not (self._x or self._y)
@staticmethod
def point_at_infinity():
return EccPoint(0, 0)
@property
def x(self):
if self.is_point_at_infinity():
raise ValueError("Point at infinity")
return self._x
@property
def y(self):
if self.is_point_at_infinity():
raise ValueError("Point at infinity")
return self._y
def double(self):
"""Double this point (in-place operation).
:Return:
:class:`EccPoint` : this same object (to enable chaining)
"""
if not self._y:
return self.point_at_infinity()
common = self._common
tmp1 = self._tmp1
x3 = self._x3
y3 = self._y3
# common = (pow(self._x, 2, _curve.p) * 3 - 3) * (self._y << 1).inverse(_curve.p) % _curve.p
common.set(self._x)
common.inplace_pow(2, _curve.p)
common *= 3
common -= 3
tmp1.set(self._y)
tmp1 <<= 1
tmp1.inplace_inverse(_curve.p)
common *= tmp1
common %= _curve.p
# x3 = (pow(common, 2, _curve.p) - 2 * self._x) % _curve.p
x3.set(common)
x3.inplace_pow(2, _curve.p)
x3 -= self._x
x3 -= self._x
while x3.is_negative():
x3 += _curve.p
# y3 = ((self._x - x3) * common - self._y) % _curve.p
y3.set(self._x)
y3 -= x3
y3 *= common
y3 -= self._y
y3 %= _curve.p
self._x.set(x3)
self._y.set(y3)
return self
def __iadd__(self, point):
"""Add a second point to this one"""
if self.is_point_at_infinity():
return self.set(point)
if point.is_point_at_infinity():
return self
if self == point:
return self.double()
if self._x == point._x:
return self.set(self.point_at_infinity())
common = self._common
tmp1 = self._tmp1
x3 = self._x3
y3 = self._y3
# common = (point._y - self._y) * (point._x - self._x).inverse(_curve.p) % _curve.p
common.set(point._y)
common -= self._y
tmp1.set(point._x)
tmp1 -= self._x
tmp1.inplace_inverse(_curve.p)
common *= tmp1
common %= _curve.p
# x3 = (pow(common, 2, _curve.p) - self._x - point._x) % _curve.p
x3.set(common)
x3.inplace_pow(2, _curve.p)
x3 -= self._x
x3 -= point._x
while x3.is_negative():
x3 += _curve.p
# y3 = ((self._x - x3) * common - self._y) % _curve.p
y3.set(self._x)
y3 -= x3
y3 *= common
y3 -= self._y
y3 %= _curve.p
self._x.set(x3)
self._y.set(y3)
return self
def __add__(self, point):
"""Return a new point, the addition of this one and another"""
result = self.copy()
result += point
return result
def __mul__(self, scalar):
"""Return a new point, the scalar product of this one"""
if scalar < 0:
raise ValueError("Scalar multiplication only defined for non-negative integers")
# Trivial results
if scalar == 0 or self.is_point_at_infinity():
return self.point_at_infinity()
elif scalar == 1:
return self.copy()
# Scalar randomization
scalar_blind = Integer.random(exact_bits=64) * _curve.order + scalar
# Montgomery key ladder
r = [self.point_at_infinity().copy(), self.copy()]
bit_size = int(scalar_blind.size_in_bits())
scalar_int = int(scalar_blind)
for i in range(bit_size, -1, -1):
di = scalar_int >> i & 1
r[di ^ 1] += r[di]
r[di].double()
return r[0]
class EccPoint(object):
"""A class to abstract a point over an Elliptic Curve.
:undocumented: __init__, __eq__, __neg__, __iadd__, __add__, __mul__
"""
def __init__(self, x, y):
self._x = Integer(x)
self._y = Integer(y)
# Buffers
self._common = Integer(0)
self._tmp1 = Integer(0)
self._x3 = Integer(0)
self._y3 = Integer(0)
def set(self, point):
self._x = Integer(point._x)
self._y = Integer(point._y)
return self
def __eq__(self, point):
return self._x == point._x and self._y == point._y
def __neg__(self):
if self.is_point_at_infinity():
return self.point_at_infinity()
return EccPoint(self._x, _curve.p - self._y)
def copy(self):
return EccPoint(self._x, self._y)
def is_point_at_infinity(self):
return not (self._x or self._y)
@staticmethod
def point_at_infinity():
return EccPoint(0, 0)
@property
def x(self):
"""The X-coordinate of the ECC point"""
if self.is_point_at_infinity():
raise ValueError("Point at infinity")
return self._x
@property
def y(self):
"""The Y-coordinate of the ECC point"""
if self.is_point_at_infinity():
raise ValueError("Point at infinity")
return self._y
def double(self):
"""Double this point"""
if not self._y:
return self.point_at_infinity()
common = self._common
tmp1 = self._tmp1
x3 = self._x3
y3 = self._y3
# common = (pow(self._x, 2, _curve.p) * 3 - 3) * (self._y << 1).inverse(_curve.p) % _curve.p
common.set(self._x)
common.inplace_pow(2, _curve.p)
common *= 3
common -= 3
tmp1.set(self._y)
tmp1 <<= 1
tmp1.inplace_inverse(_curve.p)
common *= tmp1
common %= _curve.p
# x3 = (pow(common, 2, _curve.p) - 2 * self._x) % _curve.p
x3.set(common)
x3.inplace_pow(2, _curve.p)
x3 -= self._x
x3 -= self._x
while x3.is_negative():
x3 += _curve.p
# y3 = ((self._x - x3) * common - self._y) % _curve.p
y3.set(self._x)
y3 -= x3
y3 *= common
y3 -= self._y
y3 %= _curve.p
self._x.set(x3)
self._y.set(y3)
return self
def __iadd__(self, point):
"""Add a second point to this one"""
if self.is_point_at_infinity():
return self.set(point)
if point.is_point_at_infinity():
return self
if self == point:
return self.double()
if self._x == point._x:
return self.set(self.point_at_infinity())
common = self._common
tmp1 = self._tmp1
x3 = self._x3
y3 = self._y3
# common = (point._y - self._y) * (point._x - self._x).inverse(_curve.p) % _curve.p
common.set(point._y)
common -= self._y
tmp1.set(point._x)
tmp1 -= self._x
tmp1.inplace_inverse(_curve.p)
common *= tmp1
common %= _curve.p
# x3 = (pow(common, 2, _curve.p) - self._x - point._x) % _curve.p
x3.set(common)
x3.inplace_pow(2, _curve.p)
x3 -= self._x
x3 -= point._x
while x3.is_negative():
x3 += _curve.p
# y3 = ((self._x - x3) * common - self._y) % _curve.p
y3.set(self._x)
y3 -= x3
y3 *= common
y3 -= self._y
y3 %= _curve.p
self._x.set(x3)
self._y.set(y3)
return self
def __add__(self, point):
"""Return a new point, the addition of this one and another"""
result = self.copy()
result += point
return result
def __mul__(self, scalar):
"""Return a new point, the scalar product of this one"""
if scalar < 0:
raise ValueError("Scalar multiplication only defined for non-negative integers")
# Trivial results
if scalar == 0 or self.is_point_at_infinity():
return self.point_at_infinity()
elif scalar == 1:
return self.copy()
# Scalar randomization
scalar_blind = Integer.random(exact_bits=64) * _curve.order + scalar
# Montgomery key ladder
r = [self.point_at_infinity().copy(), self.copy()]
bit_size = int(scalar_blind.size_in_bits())
scalar_int = int(scalar_blind)
for i in range(bit_size, -1, -1):
di = scalar_int >> i & 1
r[di ^ 1] += r[di]
r[di].double()
return r[0]
def lucas_test(candidate):
"""Perform a Lucas primality test on an integer.
The test is specified in Section C.3.3 of `FIPS PUB 186-4`__.
:Parameters:
candidate : integer
The number to test for primality.
:Returns:
``Primality.COMPOSITE`` or ``Primality.PROBABLY_PRIME``.
.. __: http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf
"""
if not isinstance(candidate, Integer):
candidate = Integer(candidate)
# Step 1
if candidate in (1, 2, 3, 5):
return PROBABLY_PRIME
if candidate.is_even() or candidate.is_perfect_square():
return COMPOSITE
# Step 2
def alternate():
value = 5
while True:
yield value
if value > 0:
value += 2
else:
value -= 2
value = -value
for D in alternate():
if candidate in (D, -D):
continue
js = Integer.jacobi_symbol(D, candidate)
if js == 0:
return COMPOSITE
if js == -1:
break
# Found D. P=1 and Q=(1-D)/4 (note that Q is guaranteed to be an integer)
# Step 3
# This is \delta(n) = n - jacobi(D/n)
K = candidate + 1
# Step 4
r = K.size_in_bits() - 1
# Step 5
# U_1=1 and V_1=P
U_i = Integer(1)
V_i = Integer(1)
U_temp = Integer(0)
V_temp = Integer(0)
# Step 6
for i in range(r - 1, -1, -1):
# Square
# U_temp = U_i * V_i % candidate
U_temp.set(U_i)
U_temp *= V_i
U_temp %= candidate
# V_temp = (((V_i ** 2 + (U_i ** 2 * D)) * K) >> 1) % candidate
V_temp.set(U_i)
V_temp *= U_i
V_temp *= D
V_temp.multiply_accumulate(V_i, V_i)
if V_temp.is_odd():
V_temp += candidate
V_temp >>= 1
V_temp %= candidate
# Multiply
if K.get_bit(i):
# U_i = (((U_temp + V_temp) * K) >> 1) % candidate
U_i.set(U_temp)
U_i += V_temp
if U_i.is_odd():
U_i += candidate
U_i >>= 1
U_i %= candidate
# V_i = (((V_temp + U_temp * D) * K) >> 1) % candidate
V_i.set(V_temp)
V_i.multiply_accumulate(U_temp, D)
if V_i.is_odd():
V_i += candidate
V_i >>= 1
V_i %= candidate
else:
U_i.set(U_temp)
V_i.set(V_temp)
# Step 7
if U_i == 0:
return PROBABLY_PRIME
return COMPOSITE
def lucas_test(candidate):
"""Perform a Lucas primality test on an integer.
The test is specified in Section C.3.3 of `FIPS PUB 186-4`__.
:Parameters:
candidate : integer
The number to test for primality.
:Returns:
``Primality.COMPOSITE`` or ``Primality.PROBABLY_PRIME``.
.. __: http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf
"""
if not isinstance(candidate, Integer):
candidate = Integer(candidate)
# Step 1
if candidate in (1, 2, 3, 5):
return PROBABLY_PRIME
if candidate.is_even() or candidate.is_perfect_square():
return COMPOSITE
# Step 2
def alternate():
value = 5
while True:
yield value
if value > 0:
value += 2
else:
value -= 2
value = -value
for D in alternate():
if candidate in (D, -D):
continue
js = Integer.jacobi_symbol(D, candidate)
if js == 0:
return COMPOSITE
if js == -1:
break
# Found D. P=1 and Q=(1-D)/4 (note that Q is guaranteed to be an integer)
# Step 3
# This is \delta(n) = n - jacobi(D/n)
K = candidate + 1
# Step 4
r = K.size_in_bits() - 1
# Step 5
# U_1=1 and V_1=P
U_i = Integer(1)
V_i = Integer(1)
U_temp = Integer(0)
V_temp = Integer(0)
# Step 6
for i in range(r - 1, -1, -1):
# Square
# U_temp = U_i * V_i % candidate
U_temp.set(U_i)
U_temp *= V_i
U_temp %= candidate
# V_temp = (((V_i ** 2 + (U_i ** 2 * D)) * K) >> 1) % candidate
V_temp.set(U_i)
V_temp *= U_i
V_temp *= D
V_temp.multiply_accumulate(V_i, V_i)
if V_temp.is_odd():
V_temp += candidate
V_temp >>= 1
V_temp %= candidate
# Multiply
if K.get_bit(i):
# U_i = (((U_temp + V_temp) * K) >> 1) % candidate
U_i.set(U_temp)
U_i += V_temp
if U_i.is_odd():
U_i += candidate
U_i >>= 1
U_i %= candidate
# V_i = (((V_temp + U_temp * D) * K) >> 1) % candidate
V_i.set(V_temp)
V_i.multiply_accumulate(U_temp, D)
if V_i.is_odd():
V_i += candidate
V_i >>= 1
V_i %= candidate
else:
U_i.set(U_temp)
V_i.set(V_temp)
# Step 7
if U_i == 0:
return PROBABLY_PRIME
return COMPOSITE
