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Merge bitcoin/bitcoin#26222: Introduce secp256k1 module with field and group classes to test framework
d4fb58ae8atest: EC: optimize scalar multiplication of G by using lookup table (Sebastian Falbesoner)1830dd8820test: add secp256k1 module with FE (field element) and GE (group element) classes (Pieter Wuille) Pull request description: This PR rewrites a portion of `test_framework/key.py`, in a compatible way, by introducing classes that encapsulate field element and group element logic, in an attempt to be more readable and reusable. To maximize readability, the group element logic does not use Jacobian coordinates. Instead, group elements just store (affine) X and Y coordinates directly. To compensate for the performance loss this causes, field elements are represented as fractions. This undoes most, but not all, of the performance loss, and there is a few % slowdown (as measured in `feature_taproot.py`, which heavily uses this). The upside is that the implementation for group laws (point doubling, addition, subtraction, ...) is very close to the mathematical description of elliptic curves, and this extends to potential future extensions (e.g. ElligatorSwift as needed by #27479). ACKs for top commit: achow101: ACKd4fb58ae8atheStack: re-ACKd4fb58ae8astratospher: tested ACKd4fb58a. really liked how this PR makes the secp256k1 code in the tests more intuitive and easier to follow! Tree-SHA512: 9e0d65d7de0d4fb35ad19a1c19da7f41e5e1db33631df898c6d18ea227258a8ba80c893dab862b0fa9b0fb2efd0406ad4a72229ee26d7d8d733dee1d56947f18
This commit is contained in:
Andrew Chow
commit
626d346469
3 changed files with 408 additions and 285 deletions
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@ -104,8 +104,8 @@ from test_framework.key import (
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sign_schnorr,
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tweak_add_privkey,
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ECKey,
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SECP256K1
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)
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from test_framework import secp256k1
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from test_framework.address import (
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hash160,
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program_to_witness,
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@ -695,7 +695,7 @@ def spenders_taproot_active():
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# Generate an invalid public key
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while True:
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invalid_pub = random_bytes(32)
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if not SECP256K1.is_x_coord(int.from_bytes(invalid_pub, 'big')):
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if not secp256k1.GE.is_valid_x(int.from_bytes(invalid_pub, 'big')):
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break
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# Implement a test case that detects validation logic which maps invalid public keys to the
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@ -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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@ -13,9 +13,13 @@ 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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# Point with no known discrete log.
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H_POINT = "50929b74c1a04954b78b4b6035e97a5e078a5a0f28ec96d547bfee9ace803ac0"
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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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@ -23,233 +27,18 @@ def TaggedHash(tag, data):
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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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@ -257,24 +46,21 @@ 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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@ -310,24 +96,22 @@ 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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@ -339,7 +123,7 @@ 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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@ -349,7 +133,7 @@ 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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|
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@ -375,9 +159,7 @@ 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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|
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@ -392,12 +174,12 @@ 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
|
||||
if low_s and s > SECP256K1_ORDER_HALF:
|
||||
s = SECP256K1_ORDER - s
|
||||
k = random.randrange(1, ORDER)
|
||||
R = k * secp256k1.G
|
||||
r = int(R.x) % ORDER
|
||||
s = (pow(k, -1, ORDER) * (z + self.secret * r)) % ORDER
|
||||
if low_s and s > secp256k1.GE.ORDER_HALF:
|
||||
s = ORDER - s
|
||||
# Represent in DER format. The byte representations of r and s have
|
||||
# length rounded up (255 bits becomes 32 bytes and 256 bits becomes 33
|
||||
# bytes).
|
||||
|
|
@ -413,10 +195,10 @@ def compute_xonly_pubkey(key):
|
|||
|
||||
assert len(key) == 32
|
||||
x = int.from_bytes(key, 'big')
|
||||
if x == 0 or x >= SECP256K1_ORDER:
|
||||
if x == 0 or x >= ORDER:
|
||||
return (None, None)
|
||||
P = SECP256K1.affine(SECP256K1.mul([(SECP256K1_G, x)]))
|
||||
return (P[0].to_bytes(32, 'big'), not SECP256K1.has_even_y(P))
|
||||
P = x * secp256k1.G
|
||||
return (P.to_bytes_xonly(), not P.y.is_even())
|
||||
|
||||
def tweak_add_privkey(key, tweak):
|
||||
"""Tweak a private key (after negating it if needed)."""
|
||||
|
|
@ -425,14 +207,14 @@ def tweak_add_privkey(key, tweak):
|
|||
assert len(tweak) == 32
|
||||
|
||||
x = int.from_bytes(key, 'big')
|
||||
if x == 0 or x >= SECP256K1_ORDER:
|
||||
if x == 0 or x >= ORDER:
|
||||
return None
|
||||
if not SECP256K1.has_even_y(SECP256K1.mul([(SECP256K1_G, x)])):
|
||||
x = SECP256K1_ORDER - x
|
||||
if not (x * secp256k1.G).y.is_even():
|
||||
x = ORDER - x
|
||||
t = int.from_bytes(tweak, 'big')
|
||||
if t >= SECP256K1_ORDER:
|
||||
if t >= ORDER:
|
||||
return None
|
||||
x = (x + t) % SECP256K1_ORDER
|
||||
x = (x + t) % ORDER
|
||||
if x == 0:
|
||||
return None
|
||||
return x.to_bytes(32, 'big')
|
||||
|
|
@ -443,19 +225,16 @@ 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).
|
||||
|
|
@ -468,23 +247,20 @@ 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
|
||||
|
||||
|
|
@ -499,23 +275,24 @@ 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 a new issue