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executable file 182 lines (127 sloc) 4.34 KB
#!/usr/bin/env python3
import collections
import random
EllipticCurve = collections.namedtuple('EllipticCurve', 'name p a b g n h')
curve = EllipticCurve(
'secp256k1',
# Field characteristic.
p=0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f,
# Curve coefficients.
a=0,
b=7,
# Base point.
g=(0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798,
0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8),
# Subgroup order.
n=0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141,
# Subgroup cofactor.
h=1,
)
# Modular arithmetic ##########################################################
def inverse_mod(k, p):
"""Returns the inverse of k modulo p.
This function returns the only integer x such that (x * k) % p == 1.
k must be non-zero and p must be a prime.
"""
if k == 0:
raise ZeroDivisionError('division by zero')
if k < 0:
# k ** -1 = p - (-k) ** -1 (mod p)
return p - inverse_mod(-k, p)
# Extended Euclidean algorithm.
s, old_s = 0, 1
t, old_t = 1, 0
r, old_r = p, k
while r != 0:
quotient = old_r // r
old_r, r = r, old_r - quotient * r
old_s, s = s, old_s - quotient * s
old_t, t = t, old_t - quotient * t
gcd, x, y = old_r, old_s, old_t
assert gcd == 1
assert (k * x) % p == 1
return x % p
# Functions that work on curve points #########################################
def is_on_curve(point):
"""Returns True if the given point lies on the elliptic curve."""
if point is None:
# None represents the point at infinity.
return True
x, y = point
return (y * y - x * x * x - curve.a * x - curve.b) % curve.p == 0
def point_neg(point):
"""Returns -point."""
assert is_on_curve(point)
if point is None:
# -0 = 0
return None
x, y = point
result = (x, -y % curve.p)
assert is_on_curve(result)
return result
def point_add(point1, point2):
"""Returns the result of point1 + point2 according to the group law."""
assert is_on_curve(point1)
assert is_on_curve(point2)
if point1 is None:
# 0 + point2 = point2
return point2
if point2 is None:
# point1 + 0 = point1
return point1
x1, y1 = point1
x2, y2 = point2
if x1 == x2 and y1 != y2:
# point1 + (-point1) = 0
return None
if x1 == x2:
# This is the case point1 == point2.
m = (3 * x1 * x1 + curve.a) * inverse_mod(2 * y1, curve.p)
else:
# This is the case point1 != point2.
m = (y1 - y2) * inverse_mod(x1 - x2, curve.p)
x3 = m * m - x1 - x2
y3 = y1 + m * (x3 - x1)
result = (x3 % curve.p,
-y3 % curve.p)
assert is_on_curve(result)
return result
def scalar_mult(k, point):
"""Returns k * point computed using the double and point_add algorithm."""
assert is_on_curve(point)
if k % curve.n == 0 or point is None:
return None
if k < 0:
# k * point = -k * (-point)
return scalar_mult(-k, point_neg(point))
result = None
addend = point
while k:
if k & 1:
# Add.
result = point_add(result, addend)
# Double.
addend = point_add(addend, addend)
k >>= 1
assert is_on_curve(result)
return result
# Keypair generation and ECDHE ################################################
def make_keypair():
"""Generates a random private-public key pair."""
private_key = random.randrange(1, curve.n)
public_key = scalar_mult(private_key, curve.g)
return private_key, public_key
print('Curve:', curve.name)
# Alice generates her own keypair.
alice_private_key, alice_public_key = make_keypair()
print("Alice's private key:", hex(alice_private_key))
print("Alice's public key: (0x{:x}, 0x{:x})".format(*alice_public_key))
# Bob generates his own key pair.
bob_private_key, bob_public_key = make_keypair()
print("Bob's private key:", hex(bob_private_key))
print("Bob's public key: (0x{:x}, 0x{:x})".format(*bob_public_key))
# Alice and Bob exchange their public keys and calculate the shared secret.
s1 = scalar_mult(alice_private_key, bob_public_key)
s2 = scalar_mult(bob_private_key, alice_public_key)
assert s1 == s2
print('Shared secret: (0x{:x}, 0x{:x})'.format(*s1))