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partial bitcoin#26222: Introduce secp256k1 module with field and group classes to test framework
notes: - excludes changes to test/functional/feature_taproot.py
This commit is contained in:
Kittywhiskers Van Gogh
pasta
parent
83b1c378a0
commit
ca96231181
@ -1,7 +1,7 @@
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# Copyright (c) 2019-2020 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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"""Test-only secp256k1 elliptic curve protocols 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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@ -13,239 +13,29 @@ import os
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import random
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import unittest
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from test_framework import secp256k1
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# Order of the secp256k1 curve
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ORDER = secp256k1.GE.ORDER
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def TaggedHash(tag, data):
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ss = hashlib.sha256(tag.encode('utf-8')).digest()
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ss += ss
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ss += data
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return hashlib.sha256(ss).digest()
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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 https://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 = pow(z1, -1, 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 has_even_y(self, p1):
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"""Whether the point p1 has an even Y coordinate when expressed in affine coordinates."""
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return not (p1[2] == 0 or self.affine(p1)[1] & 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 for which the Y coordinate is even."""
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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, self.p - y if y & 1 else 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_FIELD_SIZE = 2**256 - 2**32 - 977
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SECP256K1 = EllipticCurve(SECP256K1_FIELD_SIZE, 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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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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self.p = None
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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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# Make the Y coordinate odd if required (lift_x always produces
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# a point with an even Y coordinate).
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if 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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self.p = secp256k1.GE.from_bytes(data)
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self.compressed = len(data) == 33
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@property
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def is_compressed(self):
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@ -253,24 +43,21 @@ class ECPubKey():
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@property
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def is_valid(self):
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return self.valid
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return self.p is not None
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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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assert self.is_valid
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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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return self.p.to_bytes_compressed()
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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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return self.p.to_bytes_uncompressed()
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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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assert self.is_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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@ -306,24 +93,22 @@ class ECPubKey():
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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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if r < 1 or s < 1 or r >= ORDER or s >= ORDER:
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return False
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if low_s and s >= SECP256K1_ORDER_HALF:
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if low_s and s >= secp256k1.GE.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 = pow(s, -1, 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] % SECP256K1_ORDER) != r:
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w = pow(s, -1, ORDER)
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R = secp256k1.GE.mul((z * w, secp256k1.G), (r * w, self.p))
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if R.infinity or (int(R.x) % ORDER) != r:
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return False
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return True
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def generate_privkey():
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"""Generate a valid random 32-byte private key."""
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return random.randrange(1, SECP256K1_ORDER).to_bytes(32, 'big')
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return random.randrange(1, ORDER).to_bytes(32, 'big')
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def rfc6979_nonce(key):
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"""Compute signing nonce using RFC6979."""
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@ -335,7 +120,7 @@ def rfc6979_nonce(key):
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v = hmac.new(k, v, 'sha256').digest()
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return hmac.new(k, v, 'sha256').digest()
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class ECKey():
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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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@ -345,7 +130,7 @@ class ECKey():
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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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self.valid = (secret > 0 and secret < 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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@ -371,9 +156,7 @@ class ECKey():
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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.p = self.secret * secp256k1.G
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ret.compressed = self.compressed
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return ret
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@ -388,12 +171,12 @@ class ECKey():
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if rfc6979:
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k = int.from_bytes(rfc6979_nonce(self.secret.to_bytes(32, 'big') + msg), 'big')
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else:
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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 = (pow(k, -1, 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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k = random.randrange(1, ORDER)
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R = k * secp256k1.G
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r = int(R.x) % ORDER
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s = (pow(k, -1, ORDER) * (z + self.secret * r)) % ORDER
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if low_s and s > secp256k1.GE.ORDER_HALF:
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s = 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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@ -409,10 +192,10 @@ def compute_xonly_pubkey(key):
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assert len(key) == 32
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x = int.from_bytes(key, 'big')
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if x == 0 or x >= SECP256K1_ORDER:
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if x == 0 or x >= ORDER:
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return (None, None)
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P = SECP256K1.affine(SECP256K1.mul([(SECP256K1_G, x)]))
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return (P[0].to_bytes(32, 'big'), not SECP256K1.has_even_y(P))
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P = x * secp256k1.G
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return (P.to_bytes_xonly(), not P.y.is_even())
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def tweak_add_privkey(key, tweak):
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"""Tweak a private key (after negating it if needed)."""
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@ -421,14 +204,14 @@ def tweak_add_privkey(key, tweak):
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assert len(tweak) == 32
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x = int.from_bytes(key, 'big')
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if x == 0 or x >= SECP256K1_ORDER:
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if x == 0 or x >= ORDER:
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return None
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if not SECP256K1.has_even_y(SECP256K1.mul([(SECP256K1_G, x)])):
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x = SECP256K1_ORDER - x
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if not (x * secp256k1.G).y.is_even():
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x = ORDER - x
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t = int.from_bytes(tweak, 'big')
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if t >= SECP256K1_ORDER:
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if t >= ORDER:
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return None
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x = (x + t) % SECP256K1_ORDER
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x = (x + t) % ORDER
|
||||
if x == 0:
|
||||
return None
|
||||
return x.to_bytes(32, 'big')
|
||||
@ -439,19 +222,16 @@ def tweak_add_pubkey(key, tweak):
|
||||
assert len(key) == 32
|
||||
assert len(tweak) == 32
|
||||
|
||||
x_coord = int.from_bytes(key, 'big')
|
||||
if x_coord >= SECP256K1_FIELD_SIZE:
|
||||
return None
|
||||
P = SECP256K1.lift_x(x_coord)
|
||||
P = secp256k1.GE.from_bytes_xonly(key)
|
||||
if P is None:
|
||||
return None
|
||||
t = int.from_bytes(tweak, 'big')
|
||||
if t >= SECP256K1_ORDER:
|
||||
if t >= ORDER:
|
||||
return None
|
||||
Q = SECP256K1.affine(SECP256K1.mul([(SECP256K1_G, t), (P, 1)]))
|
||||
if Q is None:
|
||||
Q = t * secp256k1.G + P
|
||||
if Q.infinity:
|
||||
return None
|
||||
return (Q[0].to_bytes(32, 'big'), not SECP256K1.has_even_y(Q))
|
||||
return (Q.to_bytes_xonly(), not Q.y.is_even())
|
||||
|
||||
def verify_schnorr(key, sig, msg):
|
||||
"""Verify a Schnorr signature (see BIP 340).
|
||||
@ -464,23 +244,20 @@ def verify_schnorr(key, sig, msg):
|
||||
assert len(msg) == 32
|
||||
assert len(sig) == 64
|
||||
|
||||
x_coord = int.from_bytes(key, 'big')
|
||||
if x_coord == 0 or x_coord >= SECP256K1_FIELD_SIZE:
|
||||
return False
|
||||
P = SECP256K1.lift_x(x_coord)
|
||||
P = secp256k1.GE.from_bytes_xonly(key)
|
||||
if P is None:
|
||||
return False
|
||||
r = int.from_bytes(sig[0:32], 'big')
|
||||
if r >= SECP256K1_FIELD_SIZE:
|
||||
if r >= secp256k1.FE.SIZE:
|
||||
return False
|
||||
s = int.from_bytes(sig[32:64], 'big')
|
||||
if s >= SECP256K1_ORDER:
|
||||
if s >= ORDER:
|
||||
return False
|
||||
e = int.from_bytes(TaggedHash("BIP0340/challenge", sig[0:32] + key + msg), 'big') % SECP256K1_ORDER
|
||||
R = SECP256K1.mul([(SECP256K1_G, s), (P, SECP256K1_ORDER - e)])
|
||||
if not SECP256K1.has_even_y(R):
|
||||
e = int.from_bytes(TaggedHash("BIP0340/challenge", sig[0:32] + key + msg), 'big') % ORDER
|
||||
R = secp256k1.GE.mul((s, secp256k1.G), (-e, P))
|
||||
if R.infinity or not R.y.is_even():
|
||||
return False
|
||||
if ((r * R[2] * R[2]) % SECP256K1_FIELD_SIZE) != R[0]:
|
||||
if r != R.x:
|
||||
return False
|
||||
return True
|
||||
|
||||
@ -495,23 +272,24 @@ def sign_schnorr(key, msg, aux=None, flip_p=False, flip_r=False):
|
||||
assert len(aux) == 32
|
||||
|
||||
sec = int.from_bytes(key, 'big')
|
||||
if sec == 0 or sec >= SECP256K1_ORDER:
|
||||
if sec == 0 or sec >= ORDER:
|
||||
return None
|
||||
P = SECP256K1.affine(SECP256K1.mul([(SECP256K1_G, sec)]))
|
||||
if SECP256K1.has_even_y(P) == flip_p:
|
||||
sec = SECP256K1_ORDER - sec
|
||||
P = sec * secp256k1.G
|
||||
if P.y.is_even() == flip_p:
|
||||
sec = ORDER - sec
|
||||
t = (sec ^ int.from_bytes(TaggedHash("BIP0340/aux", aux), 'big')).to_bytes(32, 'big')
|
||||
kp = int.from_bytes(TaggedHash("BIP0340/nonce", t + P[0].to_bytes(32, 'big') + msg), 'big') % SECP256K1_ORDER
|
||||
kp = int.from_bytes(TaggedHash("BIP0340/nonce", t + P.to_bytes_xonly() + msg), 'big') % ORDER
|
||||
assert kp != 0
|
||||
R = SECP256K1.affine(SECP256K1.mul([(SECP256K1_G, kp)]))
|
||||
k = kp if SECP256K1.has_even_y(R) != flip_r else SECP256K1_ORDER - kp
|
||||
e = int.from_bytes(TaggedHash("BIP0340/challenge", R[0].to_bytes(32, 'big') + P[0].to_bytes(32, 'big') + msg), 'big') % SECP256K1_ORDER
|
||||
return R[0].to_bytes(32, 'big') + ((k + e * sec) % SECP256K1_ORDER).to_bytes(32, 'big')
|
||||
R = kp * secp256k1.G
|
||||
k = kp if R.y.is_even() != flip_r else ORDER - kp
|
||||
e = int.from_bytes(TaggedHash("BIP0340/challenge", R.to_bytes_xonly() + P.to_bytes_xonly() + msg), 'big') % ORDER
|
||||
return R.to_bytes_xonly() + ((k + e * sec) % ORDER).to_bytes(32, 'big')
|
||||
|
||||
|
||||
class TestFrameworkKey(unittest.TestCase):
|
||||
def test_schnorr(self):
|
||||
"""Test the Python Schnorr implementation."""
|
||||
byte_arrays = [generate_privkey() for _ in range(3)] + [v.to_bytes(32, 'big') for v in [0, SECP256K1_ORDER - 1, SECP256K1_ORDER, 2**256 - 1]]
|
||||
byte_arrays = [generate_privkey() for _ in range(3)] + [v.to_bytes(32, 'big') for v in [0, ORDER - 1, ORDER, 2**256 - 1]]
|
||||
keys = {}
|
||||
for privkey in byte_arrays: # build array of key/pubkey pairs
|
||||
pubkey, _ = compute_xonly_pubkey(privkey)
|
||||
|
||||
346
test/functional/test_framework/secp256k1.py
Normal file
346
test/functional/test_framework/secp256k1.py
Normal file
@ -0,0 +1,346 @@
|
||||
# Copyright (c) 2022-2023 The Bitcoin Core developers
|
||||
# Distributed under the MIT software license, see the accompanying
|
||||
# file COPYING or http://www.opensource.org/licenses/mit-license.php.
|
||||
|
||||
"""Test-only implementation of low-level secp256k1 field and group arithmetic
|
||||
|
||||
It is designed for ease of understanding, not performance.
|
||||
|
||||
WARNING: This code is slow and trivially vulnerable to side channel attacks. Do not use for
|
||||
anything but tests.
|
||||
|
||||
Exports:
|
||||
* FE: class for secp256k1 field elements
|
||||
* GE: class for secp256k1 group elements
|
||||
* G: the secp256k1 generator point
|
||||
"""
|
||||
|
||||
|
||||
class FE:
|
||||
"""Objects of this class represent elements of the field GF(2**256 - 2**32 - 977).
|
||||
|
||||
They are represented internally in numerator / denominator form, in order to delay inversions.
|
||||
"""
|
||||
|
||||
# The size of the field (also its modulus and characteristic).
|
||||
SIZE = 2**256 - 2**32 - 977
|
||||
|
||||
def __init__(self, a=0, b=1):
|
||||
"""Initialize a field element a/b; both a and b can be ints or field elements."""
|
||||
if isinstance(a, FE):
|
||||
num = a._num
|
||||
den = a._den
|
||||
else:
|
||||
num = a % FE.SIZE
|
||||
den = 1
|
||||
if isinstance(b, FE):
|
||||
den = (den * b._num) % FE.SIZE
|
||||
num = (num * b._den) % FE.SIZE
|
||||
else:
|
||||
den = (den * b) % FE.SIZE
|
||||
assert den != 0
|
||||
if num == 0:
|
||||
den = 1
|
||||
self._num = num
|
||||
self._den = den
|
||||
|
||||
def __add__(self, a):
|
||||
"""Compute the sum of two field elements (second may be int)."""
|
||||
if isinstance(a, FE):
|
||||
return FE(self._num * a._den + self._den * a._num, self._den * a._den)
|
||||
return FE(self._num + self._den * a, self._den)
|
||||
|
||||
def __radd__(self, a):
|
||||
"""Compute the sum of an integer and a field element."""
|
||||
return FE(a) + self
|
||||
|
||||
def __sub__(self, a):
|
||||
"""Compute the difference of two field elements (second may be int)."""
|
||||
if isinstance(a, FE):
|
||||
return FE(self._num * a._den - self._den * a._num, self._den * a._den)
|
||||
return FE(self._num - self._den * a, self._den)
|
||||
|
||||
def __rsub__(self, a):
|
||||
"""Compute the difference of an integer and a field element."""
|
||||
return FE(a) - self
|
||||
|
||||
def __mul__(self, a):
|
||||
"""Compute the product of two field elements (second may be int)."""
|
||||
if isinstance(a, FE):
|
||||
return FE(self._num * a._num, self._den * a._den)
|
||||
return FE(self._num * a, self._den)
|
||||
|
||||
def __rmul__(self, a):
|
||||
"""Compute the product of an integer with a field element."""
|
||||
return FE(a) * self
|
||||
|
||||
def __truediv__(self, a):
|
||||
"""Compute the ratio of two field elements (second may be int)."""
|
||||
return FE(self, a)
|
||||
|
||||
def __pow__(self, a):
|
||||
"""Raise a field element to an integer power."""
|
||||
return FE(pow(self._num, a, FE.SIZE), pow(self._den, a, FE.SIZE))
|
||||
|
||||
def __neg__(self):
|
||||
"""Negate a field element."""
|
||||
return FE(-self._num, self._den)
|
||||
|
||||
def __int__(self):
|
||||
"""Convert a field element to an integer in range 0..p-1. The result is cached."""
|
||||
if self._den != 1:
|
||||
self._num = (self._num * pow(self._den, -1, FE.SIZE)) % FE.SIZE
|
||||
self._den = 1
|
||||
return self._num
|
||||
|
||||
def sqrt(self):
|
||||
"""Compute the square root of a field element if it exists (None otherwise).
|
||||
|
||||
Due to the fact that our modulus is of the form (p % 4) == 3, the Tonelli-Shanks
|
||||
algorithm (https://en.wikipedia.org/wiki/Tonelli-Shanks_algorithm) is simply
|
||||
raising the argument to the power (p + 1) / 4.
|
||||
|
||||
To see why: (p-1) % 2 = 0, so 2 divides the order of the multiplicative group,
|
||||
and thus only half of the non-zero field elements are squares. An element a is
|
||||
a (nonzero) square when Euler's criterion, a^((p-1)/2) = 1 (mod p), holds. We're
|
||||
looking for x such that x^2 = a (mod p). Given a^((p-1)/2) = 1, that is equivalent
|
||||
to x^2 = a^(1 + (p-1)/2) mod p. As (1 + (p-1)/2) is even, this is equivalent to
|
||||
x = a^((1 + (p-1)/2)/2) mod p, or x = a^((p+1)/4) mod p."""
|
||||
v = int(self)
|
||||
s = pow(v, (FE.SIZE + 1) // 4, FE.SIZE)
|
||||
if s**2 % FE.SIZE == v:
|
||||
return FE(s)
|
||||
return None
|
||||
|
||||
def is_square(self):
|
||||
"""Determine if this field element has a square root."""
|
||||
# A more efficient algorithm is possible here (Jacobi symbol).
|
||||
return self.sqrt() is not None
|
||||
|
||||
def is_even(self):
|
||||
"""Determine whether this field element, represented as integer in 0..p-1, is even."""
|
||||
return int(self) & 1 == 0
|
||||
|
||||
def __eq__(self, a):
|
||||
"""Check whether two field elements are equal (second may be an int)."""
|
||||
if isinstance(a, FE):
|
||||
return (self._num * a._den - self._den * a._num) % FE.SIZE == 0
|
||||
return (self._num - self._den * a) % FE.SIZE == 0
|
||||
|
||||
def to_bytes(self):
|
||||
"""Convert a field element to a 32-byte array (BE byte order)."""
|
||||
return int(self).to_bytes(32, 'big')
|
||||
|
||||
@staticmethod
|
||||
def from_bytes(b):
|
||||
"""Convert a 32-byte array to a field element (BE byte order, no overflow allowed)."""
|
||||
v = int.from_bytes(b, 'big')
|
||||
if v >= FE.SIZE:
|
||||
return None
|
||||
return FE(v)
|
||||
|
||||
def __str__(self):
|
||||
"""Convert this field element to a 64 character hex string."""
|
||||
return f"{int(self):064x}"
|
||||
|
||||
def __repr__(self):
|
||||
"""Get a string representation of this field element."""
|
||||
return f"FE(0x{int(self):x})"
|
||||
|
||||
|
||||
class GE:
|
||||
"""Objects of this class represent secp256k1 group elements (curve points or infinity)
|
||||
|
||||
Normal points on the curve have fields:
|
||||
* x: the x coordinate (a field element)
|
||||
* y: the y coordinate (a field element, satisfying y^2 = x^3 + 7)
|
||||
* infinity: False
|
||||
|
||||
The point at infinity has field:
|
||||
* infinity: True
|
||||
"""
|
||||
|
||||
# Order of the group (number of points on the curve, plus 1 for infinity)
|
||||
ORDER = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141
|
||||
|
||||
# Number of valid distinct x coordinates on the curve.
|
||||
ORDER_HALF = ORDER // 2
|
||||
|
||||
def __init__(self, x=None, y=None):
|
||||
"""Initialize a group element with specified x and y coordinates, or infinity."""
|
||||
if x is None:
|
||||
# Initialize as infinity.
|
||||
assert y is None
|
||||
self.infinity = True
|
||||
else:
|
||||
# Initialize as point on the curve (and check that it is).
|
||||
fx = FE(x)
|
||||
fy = FE(y)
|
||||
assert fy**2 == fx**3 + 7
|
||||
self.infinity = False
|
||||
self.x = fx
|
||||
self.y = fy
|
||||
|
||||
def __add__(self, a):
|
||||
"""Add two group elements together."""
|
||||
# Deal with infinity: a + infinity == infinity + a == a.
|
||||
if self.infinity:
|
||||
return a
|
||||
if a.infinity:
|
||||
return self
|
||||
if self.x == a.x:
|
||||
if self.y != a.y:
|
||||
# A point added to its own negation is infinity.
|
||||
assert self.y + a.y == 0
|
||||
return GE()
|
||||
else:
|
||||
# For identical inputs, use the tangent (doubling formula).
|
||||
lam = (3 * self.x**2) / (2 * self.y)
|
||||
else:
|
||||
# For distinct inputs, use the line through both points (adding formula).
|
||||
lam = (self.y - a.y) / (self.x - a.x)
|
||||
# Determine point opposite to the intersection of that line with the curve.
|
||||
x = lam**2 - (self.x + a.x)
|
||||
y = lam * (self.x - x) - self.y
|
||||
return GE(x, y)
|
||||
|
||||
@staticmethod
|
||||
def mul(*aps):
|
||||
"""Compute a (batch) scalar group element multiplication.
|
||||
|
||||
GE.mul((a1, p1), (a2, p2), (a3, p3)) is identical to a1*p1 + a2*p2 + a3*p3,
|
||||
but more efficient."""
|
||||
# Reduce all the scalars modulo order first (so we can deal with negatives etc).
|
||||
naps = [(a % GE.ORDER, p) for a, p in aps]
|
||||
# Start with point at infinity.
|
||||
r = GE()
|
||||
# Iterate over all bit positions, from high to low.
|
||||
for i in range(255, -1, -1):
|
||||
# Double what we have so far.
|
||||
r = r + r
|
||||
# Add then add the points for which the corresponding scalar bit is set.
|
||||
for (a, p) in naps:
|
||||
if (a >> i) & 1:
|
||||
r += p
|
||||
return r
|
||||
|
||||
def __rmul__(self, a):
|
||||
"""Multiply an integer with a group element."""
|
||||
if self == G:
|
||||
return FAST_G.mul(a)
|
||||
return GE.mul((a, self))
|
||||
|
||||
def __neg__(self):
|
||||
"""Compute the negation of a group element."""
|
||||
if self.infinity:
|
||||
return self
|
||||
return GE(self.x, -self.y)
|
||||
|
||||
def to_bytes_compressed(self):
|
||||
"""Convert a non-infinite group element to 33-byte compressed encoding."""
|
||||
assert not self.infinity
|
||||
return bytes([3 - self.y.is_even()]) + self.x.to_bytes()
|
||||
|
||||
def to_bytes_uncompressed(self):
|
||||
"""Convert a non-infinite group element to 65-byte uncompressed encoding."""
|
||||
assert not self.infinity
|
||||
return b'\x04' + self.x.to_bytes() + self.y.to_bytes()
|
||||
|
||||
def to_bytes_xonly(self):
|
||||
"""Convert (the x coordinate of) a non-infinite group element to 32-byte xonly encoding."""
|
||||
assert not self.infinity
|
||||
return self.x.to_bytes()
|
||||
|
||||
@staticmethod
|
||||
def lift_x(x):
|
||||
"""Return group element with specified field element as x coordinate (and even y)."""
|
||||
y = (FE(x)**3 + 7).sqrt()
|
||||
if y is None:
|
||||
return None
|
||||
if not y.is_even():
|
||||
y = -y
|
||||
return GE(x, y)
|
||||
|
||||
@staticmethod
|
||||
def from_bytes(b):
|
||||
"""Convert a compressed or uncompressed encoding to a group element."""
|
||||
assert len(b) in (33, 65)
|
||||
if len(b) == 33:
|
||||
if b[0] != 2 and b[0] != 3:
|
||||
return None
|
||||
x = FE.from_bytes(b[1:])
|
||||
if x is None:
|
||||
return None
|
||||
r = GE.lift_x(x)
|
||||
if r is None:
|
||||
return None
|
||||
if b[0] == 3:
|
||||
r = -r
|
||||
return r
|
||||
else:
|
||||
if b[0] != 4:
|
||||
return None
|
||||
x = FE.from_bytes(b[1:33])
|
||||
y = FE.from_bytes(b[33:])
|
||||
if y**2 != x**3 + 7:
|
||||
return None
|
||||
return GE(x, y)
|
||||
|
||||
@staticmethod
|
||||
def from_bytes_xonly(b):
|
||||
"""Convert a point given in xonly encoding to a group element."""
|
||||
assert len(b) == 32
|
||||
x = FE.from_bytes(b)
|
||||
if x is None:
|
||||
return None
|
||||
return GE.lift_x(x)
|
||||
|
||||
@staticmethod
|
||||
def is_valid_x(x):
|
||||
"""Determine whether the provided field element is a valid X coordinate."""
|
||||
return (FE(x)**3 + 7).is_square()
|
||||
|
||||
def __str__(self):
|
||||
"""Convert this group element to a string."""
|
||||
if self.infinity:
|
||||
return "(inf)"
|
||||
return f"({self.x},{self.y})"
|
||||
|
||||
def __repr__(self):
|
||||
"""Get a string representation for this group element."""
|
||||
if self.infinity:
|
||||
return "GE()"
|
||||
return f"GE(0x{int(self.x):x},0x{int(self.y):x})"
|
||||
|
||||
# The secp256k1 generator point
|
||||
G = GE.lift_x(0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798)
|
||||
|
||||
|
||||
class FastGEMul:
|
||||
"""Table for fast multiplication with a constant group element.
|
||||
|
||||
Speed up scalar multiplication with a fixed point P by using a precomputed lookup table with
|
||||
its powers of 2:
|
||||
|
||||
table = [P, 2*P, 4*P, (2^3)*P, (2^4)*P, ..., (2^255)*P]
|
||||
|
||||
During multiplication, the points corresponding to each bit set in the scalar are added up,
|
||||
i.e. on average ~128 point additions take place.
|
||||
"""
|
||||
|
||||
def __init__(self, p):
|
||||
self.table = [p] # table[i] = (2^i) * p
|
||||
for _ in range(255):
|
||||
p = p + p
|
||||
self.table.append(p)
|
||||
|
||||
def mul(self, a):
|
||||
result = GE()
|
||||
a = a % GE.ORDER
|
||||
for bit in range(a.bit_length()):
|
||||
if a & (1 << bit):
|
||||
result += self.table[bit]
|
||||
return result
|
||||
|
||||
# Precomputed table with multiples of G for fast multiplication
|
||||
FAST_G = FastGEMul(G)
|
||||
Loading…
Reference in New Issue
Block a user