# andreacorbellini/ecc

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8d364f3 Dec 14, 2015
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 #!/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))