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ecc/logs/pollardsrho.py
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#!/usr/bin/env python3 | |
# This script makes use of another module: common.py, which can be | |
# found on GitHub: | |
# | |
# https://github.com/andreacorbellini/ecc/blob/master/logs/common.py | |
# | |
# You must place that module on the same directory of this script | |
# prior to running it. | |
import random | |
from common import inverse_mod, tinycurve as curve | |
class PollardRhoSequence: | |
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 log(p, q, counter=None): | |
assert curve.is_on_curve(p) | |
assert curve.is_on_curve(q) | |
# Pollard's Rho may fail sometimes: it may find a1 == a2 and b1 == b2, | |
# leading to a division by zero error. Because PollardRhoSequence uses | |
# random coefficients, we have more chances of finding the logarithm | |
# if we try again, without affecting the asymptotic time complexity. | |
# We try at most three times before giving up. | |
for i in range(3): | |
sequence = PollardRhoSequence(p, q) | |
tortoise = iter(sequence) | |
hare = iter(sequence) | |
# The range is from 0 to curve.n - 1, but actually the algorithm will | |
# stop much sooner (either finding the logarithm, or failing with a | |
# division by zero). | |
for j in range(curve.n): | |
x1, a1, b1 = next(tortoise) | |
x2, a2, b2 = next(hare) | |
x2, a2, b2 = next(hare) | |
if x1 == x2: | |
if b1 == b2: | |
# This would lead to a division by zero. Try with | |
# another random sequence. | |
break | |
x = (a1 - a2) * inverse_mod(b2 - b1, curve.n) | |
logarithm = x % curve.n | |
steps = i * curve.n + j + 1 | |
return logarithm, steps | |
raise AssertionError('logarithm not found') | |
def main(): | |
x = random.randrange(1, curve.n) | |
p = curve.g | |
q = curve.mult(x, p) | |
print('Curve: {}'.format(curve)) | |
print('Curve order: {}'.format(curve.n)) | |
print('p = (0x{:x}, 0x{:x})'.format(*p)) | |
print('q = (0x{:x}, 0x{:x})'.format(*q)) | |
print(x, '* p = q') | |
y, steps = log(p, q) | |
print('log(p, q) =', y) | |
print('Took', steps, 'steps') | |
assert x == y | |
if __name__ == '__main__': | |
main() |