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bitcoin-bips/bip-0324/reference.py
Pieter Wuille cc177ab7bc BIP324 updates
Includes:
* Simpler (but equivalent) ElligatorSwift encoding function & spec
* Improved test vectors
* Test vector generation code
* Code for converting test vectors for libsecp256k1 code.
* Code for running test vectors against SwiftEC paper authors' code.
* Miscellaneous reference code improvements (style, comments).
2023-01-11 17:39:56 -05:00

650 lines
22 KiB
Python

"""Reference implementation for the cryptographic aspects of BIP-324"""
import sys
import random
import hashlib
import hmac
### BIP-340 tagged hash
def TaggedHash(tag, data):
"""Compute BIP-340 tagged hash with specified tag string of data."""
ss = hashlib.sha256(tag.encode('utf-8')).digest()
ss += ss
ss += data
return hashlib.sha256(ss).digest()
### HKDF-SHA256
def hmac_sha256(key, data):
"""Compute HMAC-SHA256 from specified byte arrays key and data."""
return hmac.new(key, data, hashlib.sha256).digest()
def hkdf_sha256(length, ikm, salt, info):
"""Derive a key using HKDF-SHA256."""
if len(salt) == 0:
salt = bytes([0] * 32)
prk = hmac_sha256(salt, ikm)
t = b""
okm = b""
for i in range((length + 32 - 1) // 32):
t = hmac_sha256(prk, t + info + bytes([i + 1]))
okm += t
return okm[:length]
### secp256k1 field/group elements
def modinv(a, n):
"""Compute the modular inverse of a modulo n using the extended Euclidean
Algorithm. See https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#Modular_integers.
"""
a = a % n
if a == 0:
return 0
if sys.hexversion >= 0x3080000:
# More efficient version available in Python 3.8.
return pow(a, -1, n)
t1, t2 = 0, 1
r1, r2 = n, a
while r2 != 0:
q = r1 // r2
t1, t2 = t2, t1 - q * t2
r1, r2 = r2, r1 - q * r2
if r1 > 1:
return None
if t1 < 0:
t1 += n
return t1
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.
"""
SIZE = 2**256 - 2**32 - 977
def __init__(self, a=0, b=1):
"""Initialize an FE as a/b; both a and b can be ints or field elements."""
if isinstance(b, FE):
if isinstance(a, FE):
self.num = (a.num * b.den) % FE.SIZE
self.den = (a.den * b.num) % FE.SIZE
else:
self.num = (a * b.den) % FE.SIZE
self.den = b.num
else:
b = b % FE.SIZE
assert b != 0
if isinstance(a, FE):
self.num = a.num
self.den = (a.den * b) % FE.SIZE
else:
self.num = a % FE.SIZE
self.den = b
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(self.num + self.den * a, self.den)
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 between an integer and a field element."""
return FE(self.den * a - self.num, self.den)
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(self.num * a, self.den)
def __truediv__(self, a):
"""Compute the ratio of two field elements (second may be int)."""
return FE(self, a)
def __rtruediv__(self, a):
"""Compute the ratio of an integer and a field element."""
return FE(a, self)
def __pow__(self, a):
"""Raise a field element to a (positive) 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. The result is cached."""
if self.den != 1:
self.num = (self.num * modinv(self.den, FE.SIZE)) % FE.SIZE
self.den = 1
return self.num
def sqrt(self):
"""Compute the square root of a field element.
Due to the fact that our modulus p is of the form p = 3 (mod 4), 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 = 0 (mod 2), 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 (mod p), 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 sqrts(self):
"""Compute all square roots of a field element, if any."""
s = self.sqrt()
if s is None:
return []
return [FE(s), -FE(s)]
# The cube roots of 1 (mod p).
CBRT1 = [
1,
0x851695d49a83f8ef919bb86153cbcb16630fb68aed0a766a3ec693d68e6afa40,
0x7ae96a2b657c07106e64479eac3434e99cf0497512f58995c1396c28719501ee
]
def cbrts(self):
"""Compute all cube roots of a field element, if any.
Due to the fact that our modulus p is of the form p = 7 (mod 9), one cube root
can always be computed by raising to the power (p + 2) / 9. The other roots
(if any) can be found by multiplying with the two non-trivial cube roots of 1.
To see why: p-1 = 0 (mod 3), so 3 divides the order of the multiplicative group,
and thus only 1/3 of the non-zero field elements are cubes. An element a is a
(nonzero) cube when a^((p-1)/3) = 1 (mod p). We're looking for x such that
x^3 = a (mod p). Given a^((p-1)/3) = 1 (mod p), that is equivalent to
x^3 = a^(1 + (p-1)/3) (mod p). As (1 + (p-1)/3) is a multiple of 3, this is
equivalent to x = a^((1 + (p-1)/3)/3) (mod p), or x = a^((p+2)/9) (mod p)."""
v = int(self)
c = pow(v, (FE.SIZE + 2) // 9, FE.SIZE)
if pow(c, 3, FE.SIZE) == v:
return [FE(c * f) for f in FE.CBRT1]
return []
def is_square(self):
"""Determine if this field element has a square root."""
# Compute the Jacobi symbol of (self / p). Since our modulus is prime, this
# is the same as the Legendre symbol, which determines quadratic residuosity.
# See https://en.wikipedia.org/wiki/Jacobi_symbol for the algorithm.
n, k, t = (self.num * self.den) % FE.SIZE, FE.SIZE, 0
if n == 0:
return True
while n != 0:
while n & 1 == 0:
n >>= 1
r = k & 7
t ^= (r in (3, 5))
n, k = k, n
t ^= (n & k & 3 == 3)
n = n % k
assert k == 1
return not t
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 32-byte big endian encoding."""
return int(self).to_bytes(32, 'big')
@staticmethod
def from_bytes(b):
"""Convert a 32-byte big endian encoding of a field element to an FE."""
v = int.from_bytes(b, 'big')
if v >= FE.SIZE:
return None
return FE(v)
def __str__(self):
"""Convert this field element to a string."""
return f"{int(self):064x}"
def __repr__(self):
"""Get a string representation of this field element."""
return f"FE(0x{int(self):x})"
assert all(pow(c, 3, FE.SIZE) == 1 for c in FE.CBRT1)
class GE:
"""Objects of this class represent points (group elements) on the secp256k1 curve.
The point at infinity is represented as None."""
ORDER = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141
ORDER_HALF = ORDER // 2
def __init__(self, x, y):
"""Initialize a group element with specified x and y coordinates (must be on curve)."""
fx = FE(x)
fy = FE(y)
assert fy**2 == fx**3 + 7
self.x = fx
self.y = fy
def double(self):
"""Compute the double of a point."""
l = 3 * self.x**2 / (2 * self.y)
x3 = l**2 - 2 * self.x
y3 = l * (self.x - x3) - self.y
return GE(x3, y3)
def __add__(self, a):
"""Add two points, or a point and infinity, together."""
if a is None:
# Adding point at infinity
return self
if self.x != a.x:
# Adding distinct x coordinates
l = (a.y - self.y) / (a.x - self.x)
x3 = l**2 - self.x - a.x
y3 = l * (self.x - x3) - self.y
return GE(x3, y3)
if self.y == a.y:
# Adding point to itself
return self.double()
# Adding point to its negation
return None
def __radd__(self, a):
"""Add infinity to a point."""
assert a is None
return self
def __mul__(self, a):
"""Multiply a point with an integer (scalar multiplication)."""
r = None
for i in range(a.bit_length() - 1, -1, -1):
if r is not None:
r = r.double()
if (a >> i) & 1:
r += self
return r
def __rmul__(self, a):
"""Multiply an integer with a point (scalar multiplication)."""
return self * a
@staticmethod
def lift_x(x):
"""Take an FE, and return the point with that as X coordinate, and square Y."""
y = (FE(x)**3 + 7).sqrt()
if y is None:
return None
return GE(x, y)
@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."""
return f"({self.x},{self.y})"
def __repr__(self):
"""Get a string representation for this group element."""
return f"GE(0x{int(self.x)},0x{int(self.y)})"
SECP256K1_G = GE(
0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798,
0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8)
### ElligatorSwift
# Precomputed constant square root of -3 (mod p).
MINUS_3_SQRT = FE(-3).sqrt()
def xswiftec(u, t):
"""Decode field elements (u, t) to an X coordinate on the curve."""
if u == 0:
u = FE(1)
if t == 0:
t = FE(1)
if u**3 + t**2 + 7 == 0:
t = 2 * t
X = (u**3 + 7 - t**2) / (2 * t)
Y = (X + t) / (MINUS_3_SQRT * u)
for x in (u + 4 * Y**2, (-X / Y - u) / 2, (X / Y - u) / 2):
if GE.is_valid_x(x):
return x
assert False
def xswiftec_inv(x, u, case):
"""Given x and u, find t such that xswiftec(u, t) = x, or return None.
Case selects which of the up to 8 results to return."""
if case & 2 == 0:
if GE.is_valid_x(-x - u):
return None
v = x
s = -(u**3 + 7) / (u**2 + u*v + v**2)
else:
s = x - u
if s == 0:
return None
r = (-s * (4 * (u**3 + 7) + 3 * s * u**2)).sqrt()
if r is None:
return None
if case & 1 and r == 0:
return None
v = (-u + r / s) / 2
w = s.sqrt()
if w is None:
return None
if case & 5 == 0: return -w * (u * (1 - MINUS_3_SQRT) / 2 + v)
if case & 5 == 1: return w * (u * (1 + MINUS_3_SQRT) / 2 + v)
if case & 5 == 4: return w * (u * (1 - MINUS_3_SQRT) / 2 + v)
if case & 5 == 5: return -w * (u * (1 + MINUS_3_SQRT) / 2 + v)
def xelligatorswift(x):
"""Given a field element X on the curve, find (u, t) that encode them."""
while True:
u = FE(random.randrange(1, GE.ORDER))
case = random.randrange(0, 8)
t = xswiftec_inv(x, u, case)
if t is not None:
return u, t
def ellswift_create():
"""Generate a (privkey, ellswift_pubkey) pair."""
priv = random.randrange(1, GE.ORDER)
u, t = xelligatorswift((priv * SECP256K1_G).x)
return priv.to_bytes(32, 'big'), u.to_bytes() + t.to_bytes()
def ellswift_decode(ellswift):
"""Convert ellswift encoded X coordinate to 32-byte xonly format."""
u = FE(int.from_bytes(ellswift[:32], 'big'))
t = FE(int.from_bytes(ellswift[32:], 'big'))
return xswiftec(u, t).to_bytes()
def ellswift_ecdh_xonly(pubkey_theirs, privkey):
"""Compute X coordinate of shared ECDH point between elswift pubkey and privkey."""
d = int.from_bytes(privkey, 'big')
pub = ellswift_decode(pubkey_theirs)
return (d * GE.lift_x(FE.from_bytes(pub))).x.to_bytes()
### Poly1305
class Poly1305:
"""Class representing a running poly1305 computation."""
MODULUS = 2**130 - 5
def __init__(self, key):
self.r = int.from_bytes(key[:16], 'little') & 0xffffffc0ffffffc0ffffffc0fffffff
self.s = int.from_bytes(key[16:], 'little')
self.acc = 0
def add(self, msg, length=None, pad=False):
"""Add a message of any length. Input so far must be a multiple of 16 bytes."""
length = len(msg) if length is None else length
for i in range((length + 15) // 16):
chunk = msg[i * 16:i * 16 + min(16, length - i * 16)]
val = int.from_bytes(chunk, 'little') + 256**(16 if pad else len(chunk))
self.acc = (self.r * (self.acc + val)) % Poly1305.MODULUS
return self
def tag(self):
"""Compute the poly1305 tag."""
return ((self.acc + self.s) & 0xffffffffffffffffffffffffffffffff).to_bytes(16, 'little')
### ChaCha20
CHACHA20_INDICES = (
(0, 4, 8, 12), (1, 5, 9, 13), (2, 6, 10, 14), (3, 7, 11, 15),
(0, 5, 10, 15), (1, 6, 11, 12), (2, 7, 8, 13), (3, 4, 9, 14)
)
CHACHA20_CONSTANTS = (0x61707865, 0x3320646e, 0x79622d32, 0x6b206574)
def rotl32(v, bits):
"""Rotate the 32-bit value v left by bits bits."""
return ((v << bits) & 0xffffffff) | (v >> (32 - bits))
def chacha20_doubleround(s):
"""Apply a ChaCha20 double round to 16-element state array s.
See https://cr.yp.to/chacha/chacha-20080128.pdf and https://tools.ietf.org/html/rfc8439
"""
for a, b, c, d in CHACHA20_INDICES:
s[a] = (s[a] + s[b]) & 0xffffffff
s[d] = rotl32(s[d] ^ s[a], 16)
s[c] = (s[c] + s[d]) & 0xffffffff
s[b] = rotl32(s[b] ^ s[c], 12)
s[a] = (s[a] + s[b]) & 0xffffffff
s[d] = rotl32(s[d] ^ s[a], 8)
s[c] = (s[c] + s[d]) & 0xffffffff
s[b] = rotl32(s[b] ^ s[c], 7)
def chacha20_block(key, nonce, cnt):
"""Compute the 64-byte output of the ChaCha20 block function.
Takes as input a 32-byte key, 12-byte nonce, and 32-bit integer counter.
"""
# Initial state.
init = [0 for _ in range(16)]
for i in range(4):
init[i] = CHACHA20_CONSTANTS[i]
for i in range(8):
init[4 + i] = int.from_bytes(key[4 * i:4 * (i+1)], 'little')
init[12] = cnt
for i in range(3):
init[13 + i] = int.from_bytes(nonce[4 * i:4 * (i+1)], 'little')
# Perform 20 rounds.
state = list(init)
for _ in range(10):
chacha20_doubleround(state)
# Add initial values back into state.
for i in range(16):
state[i] = (state[i] + init[i]) & 0xffffffff
# Produce byte output
return b''.join(state[i].to_bytes(4, 'little') for i in range(16))
### ChaCha20Poly1305
def aead_chacha20_poly1305_encrypt(key, nonce, aad, plaintext):
"""Encrypt a plaintext using ChaCha20Poly1305."""
ret = bytearray()
msg_len = len(plaintext)
for i in range((msg_len + 63) // 64):
now = min(64, msg_len - 64 * i)
keystream = chacha20_block(key, nonce, i + 1)
for j in range(now):
ret.append(plaintext[j + 64 * i] ^ keystream[j])
poly1305 = Poly1305(chacha20_block(key, nonce, 0)[:32])
poly1305.add(aad, pad=True).add(ret, pad=True)
poly1305.add(len(aad).to_bytes(8, 'little') + msg_len.to_bytes(8, 'little'))
ret += poly1305.tag()
return bytes(ret)
def aead_chacha20_poly1305_decrypt(key, nonce, aad, ciphertext):
"""Decrypt a ChaCha20Poly1305 ciphertext."""
if len(ciphertext) < 16:
return None
msg_len = len(ciphertext) - 16
poly1305 = Poly1305(chacha20_block(key, nonce, 0)[:32])
poly1305.add(aad, pad=True)
poly1305.add(ciphertext, length=msg_len, pad=True)
poly1305.add(len(aad).to_bytes(8, 'little') + msg_len.to_bytes(8, 'little'))
if ciphertext[-16:] != poly1305.tag():
return None
ret = bytearray()
for i in range((msg_len + 63) // 64):
now = min(64, msg_len - 64 * i)
keystream = chacha20_block(key, nonce, i + 1)
for j in range(now):
ret.append(ciphertext[j + 64 * i] ^ keystream[j])
return bytes(ret)
### FSChaCha20{,Poly1305}
REKEY_INTERVAL = 224 # packets
class FSChaCha20Poly1305:
"""Rekeying wrapper AEAD around ChaCha20Poly1305."""
def __init__(self, initial_key):
self.key = initial_key
self.packet_counter = 0
def crypt(self, aad, text, is_decrypt):
"""Encrypt or decrypt the specified (plain/cipher)text."""
nonce = ((self.packet_counter % REKEY_INTERVAL).to_bytes(4, 'little') +
(self.packet_counter // REKEY_INTERVAL).to_bytes(8, 'little'))
if is_decrypt:
ret = aead_chacha20_poly1305_decrypt(self.key, nonce, aad, text)
else:
ret = aead_chacha20_poly1305_encrypt(self.key, nonce, aad, text)
if (self.packet_counter + 1) % REKEY_INTERVAL == 0:
rekey_nonce = b"\xFF\xFF\xFF\xFF" + nonce[4:]
newkey1 = aead_chacha20_poly1305_encrypt(self.key, rekey_nonce, b"", b"\x00" * 32)[:32]
newkey2 = chacha20_block(self.key, rekey_nonce, 1)[:32]
assert newkey1 == newkey2
self.key = newkey1
self.packet_counter += 1
return ret
def encrypt(self, aad, plaintext):
"""Encrypt the specified plaintext with provided AAD."""
return self.crypt(aad, plaintext, False)
def decrypt(self, aad, ciphertext):
"""Decrypt the specified ciphertext with provided AAD."""
return self.crypt(aad, ciphertext, True)
class FSChaCha20:
"""Rekeying wrapper stream cipher around ChaCha20."""
def __init__(self, initial_key):
self.key = initial_key
self.block_counter = 0
self.chunk_counter = 0
self.keystream = b''
def get_keystream_bytes(self, nbytes):
"""Generate nbytes keystream bytes."""
while len(self.keystream) < nbytes:
nonce = ((0).to_bytes(4, 'little') +
(self.chunk_counter // REKEY_INTERVAL).to_bytes(8, 'little'))
self.keystream += chacha20_block(self.key, nonce, self.block_counter)
self.block_counter += 1
ret = self.keystream[:nbytes]
self.keystream = self.keystream[nbytes:]
return ret
def crypt(self, chunk):
"""Encrypt or decypt chunk."""
ks = self.get_keystream_bytes(len(chunk))
ret = bytes([ks[i] ^ chunk[i] for i in range(len(chunk))])
if ((self.chunk_counter + 1) % REKEY_INTERVAL) == 0:
self.key = self.get_keystream_bytes(32)
self.block_counter = 0
self.chunk_counter += 1
return ret
def encrypt(self, chunk):
"""Encrypt chunk."""
return self.crypt(chunk)
def decrypt(self, chunk):
"""Decrypt chunk."""
return self.crypt(chunk)
### Shared secret computation
def v2_ecdh(priv, ellswift_theirs, ellswift_ours, initiating):
"""Compute BIP324 shared secret."""
ecdh_point_x32 = ellswift_ecdh_xonly(ellswift_theirs, priv)
if initiating:
# Initiating, place our public key encoding first.
return TaggedHash("bip324_ellswift_xonly_ecdh",
ellswift_ours + ellswift_theirs + ecdh_point_x32)
# Responding, place their public key encoding first.
return TaggedHash("bip324_ellswift_xonly_ecdh",
ellswift_theirs + ellswift_ours + ecdh_point_x32)
### Key derivation
NETWORK_MAGIC = b'\xf9\xbe\xb4\xd9'
def initialize_v2_transport(ecdh_secret, initiating):
"""Return a peer object with various BIP324 derived keys and ciphers."""
peer = {}
salt = b'bitcoin_v2_shared_secret' + NETWORK_MAGIC
for name, length in (
('initiator_L', 32), ('initiator_P', 32), ('responder_L', 32), ('responder_P', 32),
('garbage_terminators', 32), ('session_id', 32)):
peer[name] = hkdf_sha256(
salt=salt, ikm=ecdh_secret, info=name.encode('utf-8'), length=length)
peer['initiator_garbage_terminator'] = peer['garbage_terminators'][:16]
peer['responder_garbage_terminator'] = peer['garbage_terminators'][16:]
del peer['garbage_terminators']
if initiating:
peer['send_L'] = FSChaCha20(peer['initiator_L'])
peer['send_P'] = FSChaCha20Poly1305(peer['initiator_P'])
peer['send_garbage_terminator'] = peer['initiator_garbage_terminator']
peer['recv_L'] = FSChaCha20(peer['responder_L'])
peer['recv_P'] = FSChaCha20Poly1305(peer['responder_P'])
peer['recv_garbage_terminator'] = peer['responder_garbage_terminator']
else:
peer['send_L'] = FSChaCha20(peer['responder_L'])
peer['send_P'] = FSChaCha20Poly1305(peer['responder_P'])
peer['send_garbage_terminator'] = peer['responder_garbage_terminator']
peer['recv_L'] = FSChaCha20(peer['initiator_L'])
peer['recv_P'] = FSChaCha20Poly1305(peer['initiator_P'])
peer['recv_garbage_terminator'] = peer['initiator_garbage_terminator']
return peer
### Packet encryption
LENGTH_FIELD_LEN = 3
HEADER_LEN = 1
IGNORE_BIT_POS = 7
def v2_enc_packet(peer, contents, aad=b'', ignore=False):
"""Encrypt a BIP324 packet."""
assert len(contents) <= 2**24 - 1
header = (ignore << IGNORE_BIT_POS).to_bytes(HEADER_LEN, 'little')
plaintext = header + contents
aead_ciphertext = peer['send_P'].encrypt(aad, plaintext)
enc_plaintext_len = peer['send_L'].encrypt(len(contents).to_bytes(LENGTH_FIELD_LEN, 'little'))
return enc_plaintext_len + aead_ciphertext