def log(p, q):
assert curve.is_on_curve(p)
assert curve.is_on_curve(q)
sqrt_n = int(math.sqrt(curve.n)) + 1
# Compute the baby steps and store them in the 'precomputed' hash table.
r = None
precomputed = {None: 0}
for a in range(1, sqrt_n):
r = curve.add(r, p)
precomputed[r] = a
# Now compute the giant steps and check the hash table for any
# matching point.
r = q
s = curve.mult(sqrt_n, curve.neg(p))
for b in range(sqrt_n):
try:
a = precomputed[r]
except KeyError:
pass
else:
steps = sqrt_n + b
logarithm = a + sqrt_n * b
return logarithm, steps
r = curve.add(r, s)
raise AssertionError('logarithm not found')
def log(p, q):
assert curve.is_on_curve(p)
assert curve.is_on_curve(q)
sqrt_n = int(math.sqrt(curve.n)) + 1
# Compute the baby steps and store them in the 'precomputed' hash table.
r = None
precomputed = {None: 0}
for a in range(1, sqrt_n):
r = curve.add(r, p)
precomputed[r] = a
# Now compute the giant steps and check the hash table for any
# matching point.
r = q
s = curve.mult(sqrt_n, curve.neg(p))
for b in range(sqrt_n):
try:
a = precomputed[r]
except KeyError:
pass
else:
steps = sqrt_n + b
logarithm = a + sqrt_n * b
return logarithm, steps
r = curve.add(r, s)
raise AssertionError('logarithm not found')
def __init__(self, point1, point2):
self.point1 = point1
self.point2 = point2
self.add_a1 = random.randrange(1, curve.n)
self.add_b1 = random.randrange(1, curve.n)
self.add_x1 = curve.add(
curve.mult(self.add_a1, point1),
curve.mult(self.add_b1, point2),
)
self.add_a2 = random.randrange(1, curve.n)
self.add_b2 = random.randrange(1, curve.n)
self.add_x2 = curve.add(
curve.mult(self.add_a2, point1),
curve.mult(self.add_b2, point2),
)
def __init__(self, point1, point2):
self.point1 = point1
self.point2 = point2
self.add_a1 = random.randrange(1, curve.n)
self.add_b1 = random.randrange(1, curve.n)
self.add_x1 = curve.add(
curve.mult(self.add_a1, point1),
curve.mult(self.add_b1, point2),
)
self.add_a2 = random.randrange(1, curve.n)
self.add_b2 = random.randrange(1, curve.n)
self.add_x2 = curve.add(
curve.mult(self.add_a2, point1),
curve.mult(self.add_b2, point2),
)
def __iter__(self):
partition_size = curve.p // 3 + 1
x = None
a = 0
b = 0
while True:
if x is None:
i = 0
else:
i = x[0] // partition_size
if i == 0:
# x is either the point at infinity (None), or is in the first
# third of the plane (x[0] <= curve.p / 3).
a += self.add_a1
b += self.add_b1
x = curve.add(x, self.add_x1)
elif i == 1:
# x is in the second third of the plane
# (curve.p / 3 < x[0] <= curve.p * 2 / 3).
a *= 2
b *= 2
x = curve.double(x)
elif i == 2:
# x is in the last third of the plane (x[0] > curve.p * 2 / 3).
a += self.add_a2
b += self.add_b2
x = curve.add(x, self.add_x2)
else:
raise AssertionError(i)
a = a % curve.n
b = b % curve.n
yield x, a, b
def __iter__(self):
partition_size = curve.p // 3 + 1
x = None
a = 0
b = 0
while True:
if x is None:
i = 0
else:
i = x[0] // partition_size
if i == 0:
# x is either the point at infinity (None), or is in the first
# third of the plane (x[0] <= curve.p / 3).
a += self.add_a1
b += self.add_b1
x = curve.add(x, self.add_x1)
elif i == 1:
# x is in the second third of the plane
# (curve.p / 3 < x[0] <= curve.p * 2 / 3).
a *= 2
b *= 2
x = curve.double(x)
elif i == 2:
# x is in the last third of the plane (x[0] > curve.p * 2 / 3).
a += self.add_a2
b += self.add_b2
x = curve.add(x, self.add_x2)
else:
raise AssertionError(i)
a = a % curve.n
b = b % curve.n
yield x, a, b
def log(p, q):
assert curve.is_on_curve(p)
assert curve.is_on_curve(q)
start = random.randrange(curve.n)
r = curve.mult(start, p)
for x in range(curve.n):
if q == r:
logarithm = (start + x) % curve.n
steps = x + 1
return logarithm, steps
r = curve.add(r, p)
raise AssertionError('logarithm not found')
