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387 lines
13 KiB
387 lines
13 KiB
# Copyright (c) 2019 Pieter Wuille
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# Distributed under the MIT software license, see the accompanying
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# file COPYING or http://www.opensource.org/licenses/mit-license.php.
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"""Test-only secp256k1 elliptic curve implementation
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WARNING: This code is slow, uses bad randomness, does not properly protect
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keys, and is trivially vulnerable to side channel attacks. Do not use for
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anything but tests."""
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import random
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def modinv(a, n):
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"""Compute the modular inverse of a modulo n
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See https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#Modular_integers.
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"""
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t1, t2 = 0, 1
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r1, r2 = n, a
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while r2 != 0:
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q = r1 // r2
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t1, t2 = t2, t1 - q * t2
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r1, r2 = r2, r1 - q * r2
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if r1 > 1:
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return None
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if t1 < 0:
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t1 += n
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return t1
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def jacobi_symbol(n, k):
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"""Compute the Jacobi symbol of n modulo k
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See http://en.wikipedia.org/wiki/Jacobi_symbol
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For our application k is always prime, so this is the same as the Legendre symbol."""
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assert k > 0 and k & 1, "jacobi symbol is only defined for positive odd k"
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n %= k
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t = 0
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while n != 0:
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while n & 1 == 0:
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n >>= 1
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r = k & 7
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t ^= (r == 3 or r == 5)
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n, k = k, n
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t ^= (n & k & 3 == 3)
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n = n % k
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if k == 1:
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return -1 if t else 1
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return 0
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def modsqrt(a, p):
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"""Compute the square root of a modulo p when p % 4 = 3.
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The Tonelli-Shanks algorithm can be used. See https://en.wikipedia.org/wiki/Tonelli-Shanks_algorithm
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Limiting this function to only work for p % 4 = 3 means we don't need to
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iterate through the loop. The highest n such that p - 1 = 2^n Q with Q odd
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is n = 1. Therefore Q = (p-1)/2 and sqrt = a^((Q+1)/2) = a^((p+1)/4)
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secp256k1's is defined over field of size 2**256 - 2**32 - 977, which is 3 mod 4.
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"""
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if p % 4 != 3:
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raise NotImplementedError("modsqrt only implemented for p % 4 = 3")
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sqrt = pow(a, (p + 1)//4, p)
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if pow(sqrt, 2, p) == a % p:
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return sqrt
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return None
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class EllipticCurve:
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def __init__(self, p, a, b):
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"""Initialize elliptic curve y^2 = x^3 + a*x + b over GF(p)."""
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self.p = p
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self.a = a % p
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self.b = b % p
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def affine(self, p1):
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"""Convert a Jacobian point tuple p1 to affine form, or None if at infinity.
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An affine point is represented as the Jacobian (x, y, 1)"""
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x1, y1, z1 = p1
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if z1 == 0:
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return None
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inv = modinv(z1, self.p)
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inv_2 = (inv**2) % self.p
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inv_3 = (inv_2 * inv) % self.p
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return ((inv_2 * x1) % self.p, (inv_3 * y1) % self.p, 1)
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def negate(self, p1):
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"""Negate a Jacobian point tuple p1."""
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x1, y1, z1 = p1
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return (x1, (self.p - y1) % self.p, z1)
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def on_curve(self, p1):
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"""Determine whether a Jacobian tuple p is on the curve (and not infinity)"""
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x1, y1, z1 = p1
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z2 = pow(z1, 2, self.p)
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z4 = pow(z2, 2, self.p)
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return z1 != 0 and (pow(x1, 3, self.p) + self.a * x1 * z4 + self.b * z2 * z4 - pow(y1, 2, self.p)) % self.p == 0
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def is_x_coord(self, x):
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"""Test whether x is a valid X coordinate on the curve."""
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x_3 = pow(x, 3, self.p)
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return jacobi_symbol(x_3 + self.a * x + self.b, self.p) != -1
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def lift_x(self, x):
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"""Given an X coordinate on the curve, return a corresponding affine point."""
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x_3 = pow(x, 3, self.p)
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v = x_3 + self.a * x + self.b
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y = modsqrt(v, self.p)
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if y is None:
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return None
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return (x, y, 1)
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def double(self, p1):
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"""Double a Jacobian tuple p1
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See https://en.wikibooks.org/wiki/Cryptography/Prime_Curve/Jacobian_Coordinates - Point Doubling"""
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x1, y1, z1 = p1
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if z1 == 0:
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return (0, 1, 0)
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y1_2 = (y1**2) % self.p
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y1_4 = (y1_2**2) % self.p
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x1_2 = (x1**2) % self.p
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s = (4*x1*y1_2) % self.p
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m = 3*x1_2
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if self.a:
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m += self.a * pow(z1, 4, self.p)
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m = m % self.p
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x2 = (m**2 - 2*s) % self.p
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y2 = (m*(s - x2) - 8*y1_4) % self.p
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z2 = (2*y1*z1) % self.p
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return (x2, y2, z2)
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def add_mixed(self, p1, p2):
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"""Add a Jacobian tuple p1 and an affine tuple p2
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See https://en.wikibooks.org/wiki/Cryptography/Prime_Curve/Jacobian_Coordinates - Point Addition (with affine point)"""
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x1, y1, z1 = p1
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x2, y2, z2 = p2
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assert(z2 == 1)
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# Adding to the point at infinity is a no-op
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if z1 == 0:
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return p2
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z1_2 = (z1**2) % self.p
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z1_3 = (z1_2 * z1) % self.p
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u2 = (x2 * z1_2) % self.p
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s2 = (y2 * z1_3) % self.p
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if x1 == u2:
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if (y1 != s2):
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# p1 and p2 are inverses. Return the point at infinity.
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return (0, 1, 0)
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# p1 == p2. The formulas below fail when the two points are equal.
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return self.double(p1)
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h = u2 - x1
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r = s2 - y1
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h_2 = (h**2) % self.p
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h_3 = (h_2 * h) % self.p
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u1_h_2 = (x1 * h_2) % self.p
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x3 = (r**2 - h_3 - 2*u1_h_2) % self.p
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y3 = (r*(u1_h_2 - x3) - y1*h_3) % self.p
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z3 = (h*z1) % self.p
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return (x3, y3, z3)
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def add(self, p1, p2):
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"""Add two Jacobian tuples p1 and p2
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See https://en.wikibooks.org/wiki/Cryptography/Prime_Curve/Jacobian_Coordinates - Point Addition"""
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x1, y1, z1 = p1
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x2, y2, z2 = p2
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# Adding the point at infinity is a no-op
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if z1 == 0:
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return p2
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if z2 == 0:
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return p1
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# Adding an Affine to a Jacobian is more efficient since we save field multiplications and squarings when z = 1
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if z1 == 1:
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return self.add_mixed(p2, p1)
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if z2 == 1:
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return self.add_mixed(p1, p2)
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z1_2 = (z1**2) % self.p
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z1_3 = (z1_2 * z1) % self.p
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z2_2 = (z2**2) % self.p
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z2_3 = (z2_2 * z2) % self.p
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u1 = (x1 * z2_2) % self.p
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u2 = (x2 * z1_2) % self.p
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s1 = (y1 * z2_3) % self.p
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s2 = (y2 * z1_3) % self.p
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if u1 == u2:
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if (s1 != s2):
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# p1 and p2 are inverses. Return the point at infinity.
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return (0, 1, 0)
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# p1 == p2. The formulas below fail when the two points are equal.
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return self.double(p1)
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h = u2 - u1
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r = s2 - s1
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h_2 = (h**2) % self.p
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h_3 = (h_2 * h) % self.p
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u1_h_2 = (u1 * h_2) % self.p
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x3 = (r**2 - h_3 - 2*u1_h_2) % self.p
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y3 = (r*(u1_h_2 - x3) - s1*h_3) % self.p
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z3 = (h*z1*z2) % self.p
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return (x3, y3, z3)
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def mul(self, ps):
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"""Compute a (multi) point multiplication
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ps is a list of (Jacobian tuple, scalar) pairs.
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"""
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r = (0, 1, 0)
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for i in range(255, -1, -1):
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r = self.double(r)
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for (p, n) in ps:
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if ((n >> i) & 1):
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r = self.add(r, p)
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return r
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SECP256K1 = EllipticCurve(2**256 - 2**32 - 977, 0, 7)
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SECP256K1_G = (0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798, 0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8, 1)
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SECP256K1_ORDER = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141
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SECP256K1_ORDER_HALF = SECP256K1_ORDER // 2
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class ECPubKey():
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"""A secp256k1 public key"""
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def __init__(self):
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"""Construct an uninitialized public key"""
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self.valid = False
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def set(self, data):
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"""Construct a public key from a serialization in compressed or uncompressed format"""
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if (len(data) == 65 and data[0] == 0x04):
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p = (int.from_bytes(data[1:33], 'big'), int.from_bytes(data[33:65], 'big'), 1)
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self.valid = SECP256K1.on_curve(p)
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if self.valid:
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self.p = p
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self.compressed = False
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elif (len(data) == 33 and (data[0] == 0x02 or data[0] == 0x03)):
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x = int.from_bytes(data[1:33], 'big')
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if SECP256K1.is_x_coord(x):
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p = SECP256K1.lift_x(x)
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# if the oddness of the y co-ord isn't correct, find the other
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# valid y
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if (p[1] & 1) != (data[0] & 1):
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p = SECP256K1.negate(p)
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self.p = p
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self.valid = True
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self.compressed = True
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else:
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self.valid = False
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else:
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self.valid = False
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@property
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def is_compressed(self):
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return self.compressed
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@property
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def is_valid(self):
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return self.valid
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def get_bytes(self):
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assert(self.valid)
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p = SECP256K1.affine(self.p)
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if p is None:
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return None
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if self.compressed:
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return bytes([0x02 + (p[1] & 1)]) + p[0].to_bytes(32, 'big')
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else:
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return bytes([0x04]) + p[0].to_bytes(32, 'big') + p[1].to_bytes(32, 'big')
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def verify_ecdsa(self, sig, msg, low_s=True):
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"""Verify a strictly DER-encoded ECDSA signature against this pubkey.
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See https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm for the
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ECDSA verifier algorithm"""
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assert(self.valid)
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# Extract r and s from the DER formatted signature. Return false for
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# any DER encoding errors.
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if (sig[1] + 2 != len(sig)):
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return False
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if (len(sig) < 4):
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return False
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if (sig[0] != 0x30):
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return False
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if (sig[2] != 0x02):
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return False
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rlen = sig[3]
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if (len(sig) < 6 + rlen):
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return False
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if rlen < 1 or rlen > 33:
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return False
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if sig[4] >= 0x80:
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return False
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if (rlen > 1 and (sig[4] == 0) and not (sig[5] & 0x80)):
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return False
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r = int.from_bytes(sig[4:4+rlen], 'big')
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if (sig[4+rlen] != 0x02):
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return False
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slen = sig[5+rlen]
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if slen < 1 or slen > 33:
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return False
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if (len(sig) != 6 + rlen + slen):
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return False
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if sig[6+rlen] >= 0x80:
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return False
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if (slen > 1 and (sig[6+rlen] == 0) and not (sig[7+rlen] & 0x80)):
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return False
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s = int.from_bytes(sig[6+rlen:6+rlen+slen], 'big')
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# Verify that r and s are within the group order
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if r < 1 or s < 1 or r >= SECP256K1_ORDER or s >= SECP256K1_ORDER:
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return False
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if low_s and s >= SECP256K1_ORDER_HALF:
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return False
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z = int.from_bytes(msg, 'big')
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# Run verifier algorithm on r, s
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w = modinv(s, SECP256K1_ORDER)
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u1 = z*w % SECP256K1_ORDER
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u2 = r*w % SECP256K1_ORDER
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R = SECP256K1.affine(SECP256K1.mul([(SECP256K1_G, u1), (self.p, u2)]))
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if R is None or R[0] != r:
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return False
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return True
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class ECKey():
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"""A secp256k1 private key"""
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def __init__(self):
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self.valid = False
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def set(self, secret, compressed):
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"""Construct a private key object with given 32-byte secret and compressed flag."""
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assert(len(secret) == 32)
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secret = int.from_bytes(secret, 'big')
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self.valid = (secret > 0 and secret < SECP256K1_ORDER)
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if self.valid:
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self.secret = secret
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self.compressed = compressed
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def generate(self, compressed=True):
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"""Generate a random private key (compressed or uncompressed)."""
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self.set(random.randrange(1, SECP256K1_ORDER).to_bytes(32, 'big'), compressed)
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def get_bytes(self):
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"""Retrieve the 32-byte representation of this key."""
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assert(self.valid)
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return self.secret.to_bytes(32, 'big')
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@property
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def is_valid(self):
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return self.valid
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@property
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def is_compressed(self):
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return self.compressed
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def get_pubkey(self):
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"""Compute an ECPubKey object for this secret key."""
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assert(self.valid)
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ret = ECPubKey()
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p = SECP256K1.mul([(SECP256K1_G, self.secret)])
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ret.p = p
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ret.valid = True
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ret.compressed = self.compressed
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return ret
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def sign_ecdsa(self, msg, low_s=True):
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"""Construct a DER-encoded ECDSA signature with this key.
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See https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm for the
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ECDSA signer algorithm."""
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assert(self.valid)
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z = int.from_bytes(msg, 'big')
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# Note: no RFC6979, but a simple random nonce (some tests rely on distinct transactions for the same operation)
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k = random.randrange(1, SECP256K1_ORDER)
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R = SECP256K1.affine(SECP256K1.mul([(SECP256K1_G, k)]))
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r = R[0] % SECP256K1_ORDER
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s = (modinv(k, SECP256K1_ORDER) * (z + self.secret * r)) % SECP256K1_ORDER
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if low_s and s > SECP256K1_ORDER_HALF:
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s = SECP256K1_ORDER - s
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# Represent in DER format. The byte representations of r and s have
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# length rounded up (255 bits becomes 32 bytes and 256 bits becomes 33
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# bytes).
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rb = r.to_bytes((r.bit_length() + 8) // 8, 'big')
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sb = s.to_bytes((s.bit_length() + 8) // 8, 'big')
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return b'\x30' + bytes([4 + len(rb) + len(sb), 2, len(rb)]) + rb + bytes([2, len(sb)]) + sb
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