def test_forge_hash_sig(self):
"""forging valid hash signatures"""
ec = secp256k1
# see https://twitter.com/pwuille/status/1063582706288586752
# Satoshi's key
P = point_from_octets(
secp256k1,
"0311db93e1dcdb8a016b49840f8c53bc1eb68a382e97b1482ecad7b148a6909a5c"
)
u1 = 1
u2 = 2 # pick them at will
R = double_mult(ec, u1, ec.G, u2, P)
r = R[0] % ec.n
u2inv = mod_inv(u2, ec.n)
s = r * u2inv % ec.n
sig = r, s
e = s * u1 % ec.n
dsa._verhlp(ec, e, P, sig)
u1 = 1234567890
u2 = 987654321 # pick them at will
R = double_mult(ec, u1, ec.G, u2, P)
r = R[0] % ec.n
u2inv = mod_inv(u2, ec.n)
s = r * u2inv % ec.n
sig = r, s
e = s * u1 % ec.n
dsa._verhlp(ec, e, P, sig)
def _pubkey_recovery(ec: Curve, c: int, sig: ECDS) -> Sequence[Point]:
"""Private function provided for testing purposes only."""
# ECDSA public key recovery operation according to SEC 1
# http://www.secg.org/sec1-v2.pdf
# See SEC 1 v.2 section 4.1.6
r, s = _to_sig(ec, sig)
# precomputations
r1 = mod_inv(r, ec.n)
r1s = r1 * s
r1e = -r1 * c
keys: Sequence[Point] = list()
for j in range(ec.h): # 1
x = r + j * ec.n # 1.1
try: #TODO: check test reporting 1, 2, 3, or 4 keys
x %= ec._p
R = x, ec.y_odd(x, 1) # 1.2, 1.3, and 1.4
# skip 1.5: in this function, c is an input
Q = double_mult(ec, r1s, R, r1e, ec.G) # 1.6.1
if Q[1] != 0 and _verhlp(ec, c, Q, sig): # 1.6.2
keys.append(Q)
R = ec.opposite(R) # 1.6.3
Q = double_mult(ec, r1s, R, r1e, ec.G)
if Q[1] != 0 and _verhlp(ec, c, Q, sig): # 1.6.2
keys.append(Q) # 1.6.2
except Exception: # R is not a curve point
pass
return keys
def test_forge_hash_sig() -> None:
"""forging valid hash signatures"""
ec = CURVES["secp256k1"]
# see https://twitter.com/pwuille/status/1063582706288586752
# Satoshi's key
key = "03 11db93e1dcdb8a016b49840f8c53bc1eb68a382e97b1482ecad7b148a6909a5c"
Q = point_from_octets(key, ec)
# pick u1 and u2 at will
u1 = 1
u2 = 2
R = double_mult(u2, Q, u1, ec.G, ec)
r = R[0] % ec.n
u2inv = mod_inv(u2, ec.n)
s = r * u2inv % ec.n
e = s * u1 % ec.n
dsa.__assert_as_valid(e, (Q[0], Q[1], 1), r, s, ec)
# pick u1 and u2 at will
u1 = 1234567890
u2 = 987654321
R = double_mult(u2, Q, u1, ec.G, ec)
r = R[0] % ec.n
u2inv = mod_inv(u2, ec.n)
s = r * u2inv % ec.n
e = s * u1 % ec.n
dsa.__assert_as_valid(e, (Q[0], Q[1], 1), r, s, ec)
def sign(msg: bytes,
k: List[int],
sign_key_idx: List[int],
sign_keys: List[int],
pubk_rings: Dict[int, List[Point]]) -> Tuple[bytes, Dict[int, List[int]]]:
""" Borromean ring signature - signing algorithm
https://github.com/ElementsProject/borromean-signatures-writeup
https://github.com/Blockstream/borromean_paper/blob/master/borromean_draft_0.01_9ade1e49.pdf
inputs:
- msg: msg to be signed (bytes)
- sign_key_idx: list of indexes representing each signing key per ring
- sign_keys: list containing the whole set of signing keys (one per ring)
- pubk_rings: dictionary of lists where internal lists represent single rings of pubkeys
"""
s: Dict[int, List[int]] = defaultdict(list)
e: Dict[int, List[int]] = defaultdict(list)
m = _get_msg_format(msg, pubk_rings)
e0bytes = m
ring_size = len(pubk_rings)
# step 1
for i in range(ring_size):
keys_size = len(pubk_rings[i])
s[i] = [0]*keys_size
e[i] = [0]*keys_size
j_star = sign_key_idx[i]
start_idx = (j_star + 1) % keys_size
R = octets_from_point(ec, mult(ec, k[i], ec.G), True)
if start_idx != 0:
for j in range(start_idx, keys_size):
s[i][j] = random.getrandbits(256)
e[i][j] = int_from_bits(ec, _hash(m, R, i, j))
assert 0 < e[i][j] < ec.n, "sign fail: how did you do that?!?"
T = double_mult(ec, s[i][j], ec.G, -e[i][j], pubk_rings[i][j])
R = octets_from_point(ec, T, True)
e0bytes += R
e0 = hf(e0bytes).digest()
# step 2
for i in range(ring_size):
e[i][0] = int_from_bits(ec, _hash(m, e0, i, 0))
assert 0 < e[i][0] < ec.n, "sign fail: how did you do that?!?"
j_star = sign_key_idx[i]
for j in range(1, j_star+1):
s[i][j-1] = random.getrandbits(256)
T = double_mult(ec, s[i][j-1], ec.G, -e[i][j-1], pubk_rings[i][j-1])
R = octets_from_point(ec, T, True)
e[i][j] = int_from_bits(ec, _hash(m, R, i, j))
assert 0 < e[i][j] < ec.n, "sign fail: how did you do that?!?"
s[i][j_star] = k[i] + sign_keys[i]*e[i][j_star]
return e0, s
def test_assorted_mult() -> None:
ec = ec23_31
H = second_generator(ec)
for k1 in range(-ec.n + 1, ec.n):
K1 = mult(k1, ec.G, ec)
for k2 in range(ec.n):
K2 = mult(k2, H, ec)
shamir = double_mult(k1, ec.G, k2, ec.G, ec)
assert shamir == mult(k1 + k2, ec.G, ec)
shamir = double_mult(k1, INF, k2, H, ec)
assert ec.is_on_curve(shamir)
assert shamir == K2
shamir = double_mult(k1, ec.G, k2, INF, ec)
assert ec.is_on_curve(shamir)
assert shamir == K1
shamir = double_mult(k1, ec.G, k2, H, ec)
assert ec.is_on_curve(shamir)
K1K2 = ec.add(K1, K2)
assert K1K2 == shamir
k3 = 1 + secrets.randbelow(ec.n - 1)
K3 = mult(k3, ec.G, ec)
K1K2K3 = ec.add(K1K2, K3)
assert ec.is_on_curve(K1K2K3)
boscoster = multi_mult([k1, k2, k3], [ec.G, H, ec.G], ec)
assert ec.is_on_curve(boscoster)
assert K1K2K3 == boscoster, k3
k4 = 1 + secrets.randbelow(ec.n - 1)
K4 = mult(k4, H, ec)
K1K2K3K4 = ec.add(K1K2K3, K4)
assert ec.is_on_curve(K1K2K3K4)
points = [ec.G, H, ec.G, H]
boscoster = multi_mult([k1, k2, k3, k4], points, ec)
assert ec.is_on_curve(boscoster)
assert K1K2K3K4 == boscoster, k4
assert K1K2K3 == multi_mult([k1, k2, k3, 0], points, ec)
assert K1K2 == multi_mult([k1, k2, 0, 0], points, ec)
assert K1 == multi_mult([k1, 0, 0, 0], points, ec)
assert INF == multi_mult([0, 0, 0, 0], points, ec)
err_msg = "mismatch between number of scalars and points: "
with pytest.raises(ValueError, match=err_msg):
multi_mult([k1, k2, k3, k4], [ec.G, H, ec.G], ec)
def _verify(msg: bytes,
e0: bytes,
s: Dict[int, List[int]],
pubk_rings: Dict[int, List[Point]]) -> bool:
ring_size = len(pubk_rings)
m = _get_msg_format(msg, pubk_rings)
e: Dict[int, List[int]] = defaultdict(list)
e0bytes = m
for i in range(ring_size):
keys_size = len(pubk_rings[i])
e[i] = [0]*keys_size
e[i][0] = int_from_bits(ec, _hash(m, e0, i, 0))
assert e[i][0] != 0, "invalid sig: how did you do that?!?"
R = b'\0x00'
for j in range(keys_size):
T = double_mult(ec, s[i][j], ec.G, -e[i][j], pubk_rings[i][j])
R = octets_from_point(ec, T, True)
if j != len(pubk_rings[i])-1:
e[i][j+1] = int_from_bits(ec, _hash(m, R, i, j+1))
assert e[i][j+1] != 0, "invalid sig: how did you do that?!?"
else:
e0bytes += R
e0_prime = hf(e0bytes).digest()
return e0_prime == e0
def test_shamir(self):
ec = ec23_31
for k1 in range(ec.n):
for k2 in range(ec.n):
shamir = double_mult(ec, k1, ec.G, k2, ec.G)
std = ec.add(mult(ec, k1, ec.G),
mult(ec, k2, ec.G))
self.assertEqual(shamir, std)
shamir = double_mult(ec, k1, Inf, k2, ec.G)
std = ec.add(mult(ec, k1, Inf),
mult(ec, k2, ec.G))
self.assertEqual(shamir, std)
shamir = double_mult(ec, k1, ec.G, k2, Inf)
std = ec.add(mult(ec, k1, ec.G),
mult(ec, k2, Inf))
self.assertEqual(shamir, std)
def _pubkey_recovery(ec: Curve, hf, e: int, sig: ECSS) -> Point:
# Private function provided for testing purposes only.
r, s = _to_sig(ec, sig)
K = r, ec.y_quadratic_residue(r, True)
# FIXME y_quadratic_residue in Jacobian coordinates?
if e == 0:
raise ValueError("invalid (zero) challenge e")
e1 = mod_inv(e, ec.n)
P = double_mult(ec, e1 * s, ec.G, -e1, K)
assert P[1] != 0, "how did you do that?!?"
return P
def test_second_generator(self):
"""
important remark on secp256-zkp prefix for compressed encoding of the second generator:
https://github.com/garyyu/rust-secp256k1-zkp/wiki/Pedersen-Commitment
"""
ec = secp256k1
hf = sha256
H = pedersen.second_generator(ec, hf)
self.assertEqual(H, point_from_octets(ec, '0250929b74c1a04954b78b4b6035e97a5e078a5a0f28ec96d547bfee9ace803ac0'))
# 0*G + 1*H
T = double_mult(ec, 1, H, 0)
self.assertEqual(T, point_from_octets(ec, '0250929b74c1a04954b78b4b6035e97a5e078a5a0f28ec96d547bfee9ace803ac0'))
# 0*G + 2*H
T = double_mult(ec, 2, H, 0)
self.assertEqual(T, point_from_octets(ec, '03fad265e0a0178418d006e247204bcf42edb6b92188074c9134704c8686eed37a'))
T = mult(ec, 2, H)
self.assertEqual(T, point_from_octets(ec, '03fad265e0a0178418d006e247204bcf42edb6b92188074c9134704c8686eed37a'))
# 0*G + 3*H
T = double_mult(ec, 3, H, 0)
self.assertEqual(T, point_from_octets(ec, '025ef47fcde840a435e831bbb711d466fc1ee160da3e15437c6c469a3a40daacaa'))
T = mult(ec, 3, H)
self.assertEqual(T, point_from_octets(ec, '025ef47fcde840a435e831bbb711d466fc1ee160da3e15437c6c469a3a40daacaa'))
# 1*G+0*H
T = double_mult(ec, 0, H, 1)
self.assertEqual(T, point_from_octets(ec, '0279be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798'))
T = mult(ec, 1)
self.assertEqual(T, point_from_octets(ec, '0279be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798'))
# 2*G+0*H
T = double_mult(ec, 0, H, 2)
self.assertEqual(T, point_from_octets(ec, '02c6047f9441ed7d6d3045406e95c07cd85c778e4b8cef3ca7abac09b95c709ee5'))
T = mult(ec, 2)
self.assertEqual(T, point_from_octets(ec, '02c6047f9441ed7d6d3045406e95c07cd85c778e4b8cef3ca7abac09b95c709ee5'))
# 3*G+0*H
T = double_mult(ec, 0, H, 3)
self.assertEqual(T, point_from_octets(ec, '02f9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9'))
T = mult(ec, 3)
self.assertEqual(T, point_from_octets(ec, '02f9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9'))
# 0*G+5*H
T = double_mult(ec, 5, H, 0)
self.assertEqual(T, point_from_octets(ec, '039e431be0851721f9ce35cc0f718fce7d6d970e3ddd796643d71294d7a09b554e'))
T = mult(ec, 5, H)
self.assertEqual(T, point_from_octets(ec, '039e431be0851721f9ce35cc0f718fce7d6d970e3ddd796643d71294d7a09b554e'))
# 0*G-5*H
T = double_mult(ec, -5, H, 0)
self.assertEqual(T, point_from_octets(ec, '029e431be0851721f9ce35cc0f718fce7d6d970e3ddd796643d71294d7a09b554e'))
T = mult(ec, -5, H)
self.assertEqual(T, point_from_octets(ec, '029e431be0851721f9ce35cc0f718fce7d6d970e3ddd796643d71294d7a09b554e'))
# 1*G-5*H
U = double_mult(ec, -5, H, 1)
self.assertEqual(U, point_from_octets(ec, '02b218ddacb34d827c71760e601b41d309bc888cf7e3ab7cc09ec082b645f77e5a'))
U = ec.add(ec.G, T) # reusing previous T value
self.assertEqual(U, point_from_octets(ec, '02b218ddacb34d827c71760e601b41d309bc888cf7e3ab7cc09ec082b645f77e5a'))
H = pedersen.second_generator(secp256r1, hf)
H = pedersen.second_generator(secp384r1, sha384)
def commit(r: int, v: int, ec: Curve, hf) -> Point:
"""Return rG + vH, with H being second (NUMS) generator of the curve"""
H = second_generator(ec, hf)
Q = double_mult(ec, r, ec.G, v, H)
assert Q[1] != 0, "how did you do that?!?"
return Q
def test_threshold() -> None:
"testing 2-of-3 threshold signature (Pedersen secret sharing)"
ec = CURVES["secp256k1"]
# parameters
m = 2
H = second_generator(ec)
# FIRST PHASE: key pair generation ###################################
# 1.1 signer one acting as the dealer
commits1: List[Point] = []
q1, _ = ssa.gen_keys()
q1_prime, _ = ssa.gen_keys()
commits1.append(double_mult(q1_prime, H, q1, ec.G))
# sharing polynomials
f1 = [q1]
f1_prime = [q1_prime]
for i in range(1, m):
f1.append(ssa.gen_keys()[0])
f1_prime.append(ssa.gen_keys()[0])
commits1.append(double_mult(f1_prime[i], H, f1[i], ec.G))
# shares of the secret
alpha12 = 0 # share of q1 belonging to signer two
alpha12_prime = 0
alpha13 = 0 # share of q1 belonging to signer three
alpha13_prime = 0
for i in range(m):
alpha12 += (f1[i] * pow(2, i)) % ec.n
alpha12_prime += (f1_prime[i] * pow(2, i)) % ec.n
alpha13 += (f1[i] * pow(3, i)) % ec.n
alpha13_prime += (f1_prime[i] * pow(3, i)) % ec.n
# signer two verifies consistency of his share
RHS = INF
for i in range(m):
RHS = ec.add(RHS, mult(pow(2, i), commits1[i]))
t = double_mult(alpha12_prime, H, alpha12, ec.G)
assert t == RHS, "signer one is cheating"
# signer three verifies consistency of his share
RHS = INF
for i in range(m):
RHS = ec.add(RHS, mult(pow(3, i), commits1[i]))
t = double_mult(alpha13_prime, H, alpha13, ec.G)
assert t == RHS, "signer one is cheating"
# 1.2 signer two acting as the dealer
commits2: List[Point] = []
q2, _ = ssa.gen_keys()
q2_prime, _ = ssa.gen_keys()
commits2.append(double_mult(q2_prime, H, q2, ec.G))
# sharing polynomials
f2 = [q2]
f2_prime = [q2_prime]
for i in range(1, m):
f2.append(ssa.gen_keys()[0])
f2_prime.append(ssa.gen_keys()[0])
commits2.append(double_mult(f2_prime[i], H, f2[i], ec.G))
# shares of the secret
alpha21 = 0 # share of q2 belonging to signer one
alpha21_prime = 0
alpha23 = 0 # share of q2 belonging to signer three
alpha23_prime = 0
for i in range(m):
alpha21 += (f2[i] * pow(1, i)) % ec.n
alpha21_prime += (f2_prime[i] * pow(1, i)) % ec.n
alpha23 += (f2[i] * pow(3, i)) % ec.n
alpha23_prime += (f2_prime[i] * pow(3, i)) % ec.n
# signer one verifies consistency of his share
RHS = INF
for i in range(m):
RHS = ec.add(RHS, mult(pow(1, i), commits2[i]))
t = double_mult(alpha21_prime, H, alpha21, ec.G)
assert t == RHS, "signer two is cheating"
# signer three verifies consistency of his share
RHS = INF
for i in range(m):
RHS = ec.add(RHS, mult(pow(3, i), commits2[i]))
t = double_mult(alpha23_prime, H, alpha23, ec.G)
assert t == RHS, "signer two is cheating"
# 1.3 signer three acting as the dealer
commits3: List[Point] = []
q3, _ = ssa.gen_keys()
q3_prime, _ = ssa.gen_keys()
commits3.append(double_mult(q3_prime, H, q3, ec.G))
# sharing polynomials
f3 = [q3]
f3_prime = [q3_prime]
for i in range(1, m):
f3.append(ssa.gen_keys()[0])
f3_prime.append(ssa.gen_keys()[0])
commits3.append(double_mult(f3_prime[i], H, f3[i], ec.G))
# shares of the secret
alpha31 = 0 # share of q3 belonging to signer one
alpha31_prime = 0
alpha32 = 0 # share of q3 belonging to signer two
alpha32_prime = 0
for i in range(m):
alpha31 += (f3[i] * pow(1, i)) % ec.n
alpha31_prime += (f3_prime[i] * pow(1, i)) % ec.n
alpha32 += (f3[i] * pow(2, i)) % ec.n
alpha32_prime += (f3_prime[i] * pow(2, i)) % ec.n
# signer one verifies consistency of his share
RHS = INF
for i in range(m):
RHS = ec.add(RHS, mult(pow(1, i), commits3[i]))
t = double_mult(alpha31_prime, H, alpha31, ec.G)
assert t == RHS, "signer three is cheating"
# signer two verifies consistency of his share
RHS = INF
for i in range(m):
RHS = ec.add(RHS, mult(pow(2, i), commits3[i]))
t = double_mult(alpha32_prime, H, alpha32, ec.G)
assert t == RHS, "signer three is cheating"
# shares of the secret key q = q1 + q2 + q3
alpha1 = (alpha21 + alpha31) % ec.n
alpha2 = (alpha12 + alpha32) % ec.n
alpha3 = (alpha13 + alpha23) % ec.n
for i in range(m):
alpha1 += (f1[i] * pow(1, i)) % ec.n
alpha2 += (f2[i] * pow(2, i)) % ec.n
alpha3 += (f3[i] * pow(3, i)) % ec.n
# 1.4 it's time to recover the public key
# each participant i = 1, 2, 3 shares Qi as follows
# Q = Q1 + Q2 + Q3 = (q1 + q2 + q3) G
A1: List[Point] = []
A2: List[Point] = []
A3: List[Point] = []
for i in range(m):
A1.append(mult(f1[i]))
A2.append(mult(f2[i]))
A3.append(mult(f3[i]))
# signer one checks others' values
RHS2 = INF
RHS3 = INF
for i in range(m):
RHS2 = ec.add(RHS2, mult(pow(1, i), A2[i]))
RHS3 = ec.add(RHS3, mult(pow(1, i), A3[i]))
assert mult(alpha21) == RHS2, "signer two is cheating"
assert mult(alpha31) == RHS3, "signer three is cheating"
# signer two checks others' values
RHS1 = INF
RHS3 = INF
for i in range(m):
RHS1 = ec.add(RHS1, mult(pow(2, i), A1[i]))
RHS3 = ec.add(RHS3, mult(pow(2, i), A3[i]))
assert mult(alpha12) == RHS1, "signer one is cheating"
assert mult(alpha32) == RHS3, "signer three is cheating"
# signer three checks others' values
RHS1 = INF
RHS2 = INF
for i in range(m):
RHS1 = ec.add(RHS1, mult(pow(3, i), A1[i]))
RHS2 = ec.add(RHS2, mult(pow(3, i), A2[i]))
assert mult(alpha13) == RHS1, "signer one is cheating"
assert mult(alpha23) == RHS2, "signer two is cheating"
# commitment at the global sharing polynomial
A: List[Point] = []
for i in range(m):
A.append(ec.add(A1[i], ec.add(A2[i], A3[i])))
# aggregated public key
Q = A[0]
if Q[1] % 2:
# print('Q has been negated')
A[1] = ec.negate(A[1]) # pragma: no cover
alpha1 = ec.n - alpha1 # pragma: no cover
alpha2 = ec.n - alpha2 # pragma: no cover
alpha3 = ec.n - alpha3 # pragma: no cover
Q = ec.negate(Q) # pragma: no cover
# SECOND PHASE: generation of the nonces' pair ######################
# Assume signer one and three want to sign
msg = "message to sign"
# 2.1 signer one acting as the dealer
commits1 = []
k1 = ssa.det_nonce(msg, q1, ec, hf)
k1_prime = ssa.det_nonce(msg, q1_prime, ec, hf)
commits1.append(double_mult(k1_prime, H, k1, ec.G))
# sharing polynomials
f1 = [k1]
f1_prime = [k1_prime]
for i in range(1, m):
f1.append(ssa.gen_keys()[0])
f1_prime.append(ssa.gen_keys()[0])
commits1.append(double_mult(f1_prime[i], H, f1[i], ec.G))
# shares of the secret
beta13 = 0 # share of k1 belonging to signer three
beta13_prime = 0
for i in range(m):
beta13 += (f1[i] * pow(3, i)) % ec.n
beta13_prime += (f1_prime[i] * pow(3, i)) % ec.n
# signer three verifies consistency of his share
RHS = INF
for i in range(m):
RHS = ec.add(RHS, mult(pow(3, i), commits1[i]))
t = double_mult(beta13_prime, H, beta13, ec.G)
assert t == RHS, "signer one is cheating"
# 2.2 signer three acting as the dealer
commits3 = []
k3 = ssa.det_nonce(msg, q3, ec, hf)
k3_prime = ssa.det_nonce(msg, q3_prime, ec, hf)
commits3.append(double_mult(k3_prime, H, k3, ec.G))
# sharing polynomials
f3 = [k3]
f3_prime = [k3_prime]
for i in range(1, m):
f3.append(ssa.gen_keys()[0])
f3_prime.append(ssa.gen_keys()[0])
commits3.append(double_mult(f3_prime[i], H, f3[i], ec.G))
# shares of the secret
beta31 = 0 # share of k3 belonging to signer one
beta31_prime = 0
for i in range(m):
beta31 += (f3[i] * pow(1, i)) % ec.n
beta31_prime += (f3_prime[i] * pow(1, i)) % ec.n
# signer one verifies consistency of his share
RHS = INF
for i in range(m):
RHS = ec.add(RHS, mult(pow(1, i), commits3[i]))
t = double_mult(beta31_prime, H, beta31, ec.G)
assert t == RHS, "signer three is cheating"
# 2.3 shares of the secret nonce
beta1 = beta31 % ec.n
beta3 = beta13 % ec.n
for i in range(m):
beta1 += (f1[i] * pow(1, i)) % ec.n
beta3 += (f3[i] * pow(3, i)) % ec.n
# 2.4 it's time to recover the public nonce
# each participant i = 1, 3 shares Qi as follows
B1: List[Point] = []
B3: List[Point] = []
for i in range(m):
B1.append(mult(f1[i]))
B3.append(mult(f3[i]))
# signer one checks values from signer three
RHS3 = INF
for i in range(m):
RHS3 = ec.add(RHS3, mult(pow(1, i), B3[i]))
assert mult(beta31) == RHS3, "signer three is cheating"
# signer three checks values from signer one
RHS1 = INF
for i in range(m):
RHS1 = ec.add(RHS1, mult(pow(3, i), B1[i]))
assert mult(beta13) == RHS1, "signer one is cheating"
# commitment at the global sharing polynomial
B: List[Point] = []
for i in range(m):
B.append(ec.add(B1[i], B3[i]))
# aggregated public nonce
K = B[0]
if K[1] % 2:
# print('K has been negated')
B[1] = ec.negate(B[1]) # pragma: no cover
beta1 = ec.n - beta1 # pragma: no cover
beta3 = ec.n - beta3 # pragma: no cover
K = ec.negate(K) # pragma: no cover
# PHASE THREE: signature generation ###
# partial signatures
e = ssa.challenge(msg, Q[0], K[0], ec, hf)
gamma1 = (beta1 + e * alpha1) % ec.n
gamma3 = (beta3 + e * alpha3) % ec.n
# each participant verifies the other partial signatures
# signer one
RHS3 = ec.add(K, mult(e, Q))
for i in range(1, m):
temp = double_mult(pow(3, i), B[i], e * pow(3, i), A[i])
RHS3 = ec.add(RHS3, temp)
assert mult(gamma3) == RHS3, "signer three is cheating"
# signer three
RHS1 = ec.add(K, mult(e, Q))
for i in range(1, m):
temp = double_mult(pow(1, i), B[i], e * pow(1, i), A[i])
RHS1 = ec.add(RHS1, temp)
assert mult(gamma1) == RHS1, "signer one is cheating"
# PHASE FOUR: aggregating the signature ###
omega1 = 3 * mod_inv(3 - 1, ec.n) % ec.n
omega3 = 1 * mod_inv(1 - 3, ec.n) % ec.n
sigma = (gamma1 * omega1 + gamma3 * omega3) % ec.n
sig = K[0], sigma
assert ssa.verify(msg, Q[0], sig)
# ADDITIONAL PHASE: reconstruction of the private key ###
secret = (omega1 * alpha1 + omega3 * alpha3) % ec.n
assert (q1 + q2 + q3) % ec.n in (secret, ec.n - secret)
def test_musig() -> None:
"""testing 3-of-3 MuSig.
https://github.com/ElementsProject/secp256k1-zkp/blob/secp256k1-zkp/src/modules/musig/musig.md
https://blockstream.com/2019/02/18/musig-a-new-multisignature-standard/
https://eprint.iacr.org/2018/068
https://blockstream.com/2018/01/23/musig-key-aggregation-schnorr-signatures.html
https://medium.com/@snigirev.stepan/how-schnorr-signatures-may-improve-bitcoin-91655bcb4744
"""
ec = CURVES["secp256k1"]
m = hf(b"message to sign").digest()
# the signers private and public keys,
# including both the curve Point and the BIP340-Schnorr public key
q1, x_Q1_int = ssa.gen_keys()
x_Q1 = x_Q1_int.to_bytes(ec.psize, "big")
q2, x_Q2_int = ssa.gen_keys()
x_Q2 = x_Q2_int.to_bytes(ec.psize, "big")
q3, x_Q3_int = ssa.gen_keys()
x_Q3 = x_Q3_int.to_bytes(ec.psize, "big")
# (non interactive) key setup
# this is MuSig core: the rest is just Schnorr signature additivity
# 1. lexicographic sorting of public keys
keys: List[bytes] = []
keys.append(x_Q1)
keys.append(x_Q2)
keys.append(x_Q3)
keys.sort()
# 2. coefficients
prefix = b"".join(keys)
a1 = int_from_bits(hf(prefix + x_Q1).digest(), ec.nlen) % ec.n
a2 = int_from_bits(hf(prefix + x_Q2).digest(), ec.nlen) % ec.n
a3 = int_from_bits(hf(prefix + x_Q3).digest(), ec.nlen) % ec.n
# 3. aggregated public key
Q1 = mult(q1)
Q2 = mult(q2)
Q3 = mult(q3)
Q = ec.add(double_mult(a1, Q1, a2, Q2), mult(a3, Q3))
if Q[1] % 2:
# print("Q has been negated")
a1 = ec.n - a1 # pragma: no cover
a2 = ec.n - a2 # pragma: no cover
a3 = ec.n - a3 # pragma: no cover
# ready to sign: nonces and nonce commitments
k1, _ = ssa.gen_keys()
K1 = mult(k1)
k2, _ = ssa.gen_keys()
K2 = mult(k2)
k3, _ = ssa.gen_keys()
K3 = mult(k3)
# exchange {K_i} (interactive)
# computes s_i (non interactive)
# WARNING: signers must exchange the nonces commitments {K_i}
# before sharing {s_i}
# same for all signers
K = ec.add(ec.add(K1, K2), K3)
if K[1] % 2:
k1 = ec.n - k1 # pragma: no cover
k2 = ec.n - k2 # pragma: no cover
k3 = ec.n - k3 # pragma: no cover
r = K[0]
e = ssa._challenge(m, Q[0], r, ec, hf)
s_1 = (k1 + e * a1 * q1) % ec.n
s_2 = (k2 + e * a2 * q2) % ec.n
s3 = (k3 + e * a3 * q3) % ec.n
# exchange s_i (interactive)
# finalize signature (non interactive)
s = (s_1 + s_2 + s3) % ec.n
sig = r, s
# check signature is valid
ssa._assert_as_valid(m, Q[0], sig, ec, hf)
k1 = int_from_bits(k1_bytes, ec.nlen) % ec.n
assert 0 < k1 < ec.n, "Invalid ephemeral key"
print(f"eph k: {hex(k1).upper()}")
K1 = mult(k1, ec.G)
c1 = _challenge(msg1, Q[0], K1[0], ec, sha256)
print(f" c1: {hex(c1).upper()}")
print("2. Sign message")
r1 = K1[0]
s1 = (k1 + c1 * q) % ec.n
print(f" r1: {hex(r1).upper()}")
print(f" s1: {hex(s1).upper()}")
print("3. Verify signature")
K = double_mult(-c1, Q, s1, ec.G)
print(K[0] == r1)
print("\n0. Another message to sign")
orig_msg2 = "and Paolo is right to be afraid"
msg2 = sha256(orig_msg2.encode()).digest()
print(msg2.hex().upper())
print("\n*** Ephemeral key and challenge")
# ephemeral key k must be kept secret and never reused !!!!!
k2 = k1
print(f"eph k: {hex(k2).upper()}")
K2 = mult(k2, ec.G)
c2 = _challenge(msg2, Q[0], K2[0], ec, sha256)
print(f" c2: {hex(c2).upper()}")
import time
from btclib.curve import mult, double_mult
from btclib.curves import secp256k1
random.seed(42)
ec = secp256k1
# setup
k1 = []
k2 = []
Q = []
for _ in range(50):
k1.append(random.getrandbits(ec.nlen) % ec.n)
k2.append(random.getrandbits(ec.nlen) % ec.n)
q = random.getrandbits(ec.nlen) % ec.n
Q.append(mult(ec, q, ec.G))
start = time.time()
for i in range(len(Q)):
ec.add(mult(ec, k1[i], ec.G), mult(ec, k2[i], Q[i]))
elapsed1 = time.time() - start
start = time.time()
for i in range(len(Q)):
double_mult(ec, k1[i], ec.G, k2[i], Q[i])
elapsed2 = time.time() - start
print(elapsed2 / elapsed1)
def test_musig(self):
""" testing 3-of-3 MuSig
https://github.com/ElementsProject/secp256k1-zkp/blob/secp256k1-zkp/src/modules/musig/musig.md
https://blockstream.com/2019/02/18/musig-a-new-multisignature-standard/
https://eprint.iacr.org/2018/068
https://blockstream.com/2018/01/23/musig-key-aggregation-schnorr-signatures.html
https://medium.com/@snigirev.stepan/how-schnorr-signatures-may-improve-bitcoin-91655bcb4744
"""
ec = secp256k1
M = hf('message to sign'.encode()).digest()
# key setup is not interactive
# first signer
q1 = int_from_octets(
'0c28fca386c7a227600b2fe50b7cae11ec86d3bf1fbe471be89827e19d92ad1d')
Q1 = mult(ec, q1, ec.G)
k1 = rfc6979(ec, hf, M, q1)
K1 = mult(ec, k1, ec.G)
# second signer
q2 = int_from_octets(
'0c28fca386c7a227600b2fe50b7cae11ec86d3bf1fbe471be89827e19d72aa1d')
Q2 = mult(ec, q2, ec.G)
k2 = rfc6979(ec, hf, M, q2)
K2 = mult(ec, k2, ec.G)
# third signer
q3 = int_from_octets(
'0c28fca386c7aff7600b2fe50b7cae11ec86d3bf1fbe471be89827e19d72aa1d')
Q3 = mult(ec, q3, ec.G)
k3 = rfc6979(ec, hf, M, q3)
K3 = mult(ec, k3, ec.G)
# this is MuSig core: the rest is just Schnorr signature additivity
L: List[Point] = list() # multiset of public keys
L.append(octets_from_point(ec, Q1, False))
L.append(octets_from_point(ec, Q2, False))
L.append(octets_from_point(ec, Q3, False))
L.sort() # using lexicographic ordering
L_brackets = b''
for i in range(len(L)):
L_brackets += L[i]
h1 = hf(L_brackets + octets_from_point(ec, Q1, False)).digest()
a1 = int_from_bits(ec, h1)
h2 = hf(L_brackets + octets_from_point(ec, Q2, False)).digest()
a2 = int_from_bits(ec, h2)
h3 = hf(L_brackets + octets_from_point(ec, Q3, False)).digest()
a3 = int_from_bits(ec, h3)
# aggregated public key
Q = ec.add(double_mult(ec, a1, Q1, a2, Q2), mult(ec, a3, Q3))
Q_bytes = octets_from_point(ec, Q, True)
########################
# interactive signature: exchange K, compute s
# WARNING: the signers should exchange commitments to the public
# nonces before sending the nonces themselves
# first signer
# K, r_bytes, and e as calculated by any signer
# are the same as the ones by the other signers
K = ec.add(ec.add(K1, K2), K3)
r_bytes = K[0].to_bytes(32, byteorder="big")
e = int_from_bits(ec, hf(r_bytes + Q_bytes + M).digest())
if legendre_symbol(K[1], ec._p) != 1:
# no need to actually change K[1], as it is not used anymore
# let's fix k1 instead, as it is used later
# note that all other signers will change their k too
k1 = ec.n - k1
s1 = (k1 + e * a1 * q1) % ec.n
# second signer
# K, r_bytes, and e as calculated by any signer
# are the same as the ones by the other signers
if legendre_symbol(K[1], ec._p) != 1:
# no need to actually change K[1], as it is not used anymore
# let's fix k2 instead, as it is used later
# note that all other signers will change their k too
k2 = ec.n - k2
s2 = (k2 + e * a2 * q2) % ec.n
# third signer
# K, r_bytes, and e as calculated by any signer
# are the same as the ones by the other signers
if legendre_symbol(K[1], ec._p) != 1:
# no need to actually change K[1], as it is not used anymore
# let's fix k3 instead, as it is used later
# note that all other signers will change their k too
k3 = ec.n - k3
s3 = (k3 + e * a3 * q3) % ec.n
############################################
# interactive signature: exchange signatures
# combine all (K[0], s) signatures into a single signature
# anyone can do the following
sig = K[0], (s1 + s2 + s3) % ec.n
self.assertTrue(ssa.verify(ec, hf, M, Q, sig))
def test_threshold(self):
"""testing 2-of-3 threshold signature (Pedersen secret sharing)"""
ec = secp256k1
# parameters
t = 2
H = second_generator(ec, hf)
msg = hf('message to sign'.encode()).digest()
### FIRST PHASE: key pair generation ###
# signer one acting as the dealer
commits1: List[Point] = list()
q1 = (1 + random.getrandbits(ec.nlen)) % ec.n
q1_prime = (1 + random.getrandbits(ec.nlen)) % ec.n
commits1.append(double_mult(ec, q1, ec.G, q1_prime, H))
# sharing polynomials
f1: List[int] = list()
f1.append(q1)
f1_prime: List[int] = list()
f1_prime.append(q1_prime)
for i in range(1, t):
temp = (1 + random.getrandbits(ec.nlen)) % ec.n
f1.append(temp)
temp = (1 + random.getrandbits(ec.nlen)) % ec.n
f1_prime.append(temp)
commits1.append(double_mult(ec, f1[i], ec.G, f1_prime[i], H))
# shares of the secret
alpha12 = 0 # share of q1 belonging to P2
alpha12_prime = 0
alpha13 = 0 # share of q1 belonging to P3
alpha13_prime = 0
for i in range(t):
alpha12 += (f1[i] * pow(2, i)) % ec.n
alpha12_prime += (f1_prime[i] * pow(2, i)) % ec.n
alpha13 += (f1[i] * pow(3, i)) % ec.n
alpha13_prime += (f1_prime[i] * pow(3, i)) % ec.n
# player two verifies consistency of his share
RHS = 1, 0
for i in range(t):
RHS = ec.add(RHS, mult(ec, pow(2, i), commits1[i]))
assert double_mult(ec, alpha12, ec.G, alpha12_prime,
H) == RHS, 'player one is cheating'
# player three verifies consistency of his share
RHS = 1, 0
for i in range(t):
RHS = ec.add(RHS, mult(ec, pow(3, i), commits1[i]))
assert double_mult(ec, alpha13, ec.G, alpha13_prime,
H) == RHS, 'player one is cheating'
# signer two acting as the dealer
commits2: List[Point] = list()
q2 = (1 + random.getrandbits(ec.nlen)) % ec.n
q2_prime = (1 + random.getrandbits(ec.nlen)) % ec.n
commits2.append(double_mult(ec, q2, ec.G, q2_prime, H))
# sharing polynomials
f2: List[int] = list()
f2.append(q2)
f2_prime: List[int] = list()
f2_prime.append(q2_prime)
for i in range(1, t):
temp = (1 + random.getrandbits(ec.nlen)) % ec.n
f2.append(temp)
temp = (1 + random.getrandbits(ec.nlen)) % ec.n
f2_prime.append(temp)
commits2.append(double_mult(ec, f2[i], ec.G, f2_prime[i], H))
# shares of the secret
alpha21 = 0 # share of q2 belonging to P1
alpha21_prime = 0
alpha23 = 0 # share of q2 belonging to P3
alpha23_prime = 0
for i in range(t):
alpha21 += (f2[i] * pow(1, i)) % ec.n
alpha21_prime += (f2_prime[i] * pow(1, i)) % ec.n
alpha23 += (f2[i] * pow(3, i)) % ec.n
alpha23_prime += (f2_prime[i] * pow(3, i)) % ec.n
# player one verifies consistency of his share
RHS = 1, 0
for i in range(t):
RHS = ec.add(RHS, mult(ec, pow(1, i), commits2[i]))
assert double_mult(ec, alpha21, ec.G, alpha21_prime,
H) == RHS, 'player two is cheating'
# player three verifies consistency of his share
RHS = 1, 0
for i in range(t):
RHS = ec.add(RHS, mult(ec, pow(3, i), commits2[i]))
assert double_mult(ec, alpha23, ec.G, alpha23_prime,
H) == RHS, 'player two is cheating'
# signer three acting as the dealer
commits3: List[Point] = list()
q3 = (1 + random.getrandbits(ec.nlen)) % ec.n
q3_prime = (1 + random.getrandbits(ec.nlen)) % ec.n
commits3.append(double_mult(ec, q3, ec.G, q3_prime, H))
# sharing polynomials
f3: List[int] = list()
f3.append(q3)
f3_prime: List[int] = list()
f3_prime.append(q3_prime)
for i in range(1, t):
temp = (1 + random.getrandbits(ec.nlen)) % ec.n
f3.append(temp)
temp = (1 + random.getrandbits(ec.nlen)) % ec.n
f3_prime.append(temp)
commits3.append(double_mult(ec, f3[i], ec.G, f3_prime[i], H))
# shares of the secret
alpha31 = 0 # share of q3 belonging to P1
alpha31_prime = 0
alpha32 = 0 # share of q3 belonging to P2
alpha32_prime = 0
for i in range(t):
alpha31 += (f3[i] * pow(1, i)) % ec.n
alpha31_prime += (f3_prime[i] * pow(1, i)) % ec.n
alpha32 += (f3[i] * pow(2, i)) % ec.n
alpha32_prime += (f3_prime[i] * pow(2, i)) % ec.n
# player one verifies consistency of his share
RHS = 1, 0
for i in range(t):
RHS = ec.add(RHS, mult(ec, pow(1, i), commits3[i]))
assert double_mult(ec, alpha31, ec.G, alpha31_prime,
H) == RHS, 'player three is cheating'
# player two verifies consistency of his share
RHS = 1, 0
for i in range(t):
RHS = ec.add(RHS, mult(ec, pow(2, i), commits3[i]))
assert double_mult(ec, alpha32, ec.G, alpha32_prime,
H) == RHS, 'player two is cheating'
# shares of the secret key q = q1 + q2 + q3
alpha1 = (alpha21 + alpha31) % ec.n
alpha2 = (alpha12 + alpha32) % ec.n
alpha3 = (alpha13 + alpha23) % ec.n
for i in range(t):
alpha1 += (f1[i] * pow(1, i)) % ec.n
alpha2 += (f2[i] * pow(2, i)) % ec.n
alpha3 += (f3[i] * pow(3, i)) % ec.n
# it's time to recover the public key Q = Q1 + Q2 + Q3 = (q1 + q2 + q3)G
A1: List[Point] = list()
A2: List[Point] = list()
A3: List[Point] = list()
# each participant i = 1, 2, 3 shares Qi as follows
# he broadcasts these values
for i in range(t):
A1.append(mult(ec, f1[i], ec.G))
A2.append(mult(ec, f2[i], ec.G))
A3.append(mult(ec, f3[i], ec.G))
# he checks the others' values
# player one
RHS2 = 1, 0
RHS3 = 1, 0
for i in range(t):
RHS2 = ec.add(RHS2, mult(ec, pow(1, i), A2[i]))
RHS3 = ec.add(RHS3, mult(ec, pow(1, i), A3[i]))
assert mult(ec, alpha21, ec.G) == RHS2, 'player two is cheating'
assert mult(ec, alpha31, ec.G) == RHS3, 'player three is cheating'
# player two
RHS1 = 1, 0
RHS3 = 1, 0
for i in range(t):
RHS1 = ec.add(RHS1, mult(ec, pow(2, i), A1[i]))
RHS3 = ec.add(RHS3, mult(ec, pow(2, i), A3[i]))
assert mult(ec, alpha12, ec.G) == RHS1, 'player one is cheating'
assert mult(ec, alpha32, ec.G) == RHS3, 'player three is cheating'
# player three
RHS1 = 1, 0
RHS2 = 1, 0
for i in range(t):
RHS1 = ec.add(RHS1, mult(ec, pow(3, i), A1[i]))
RHS2 = ec.add(RHS2, mult(ec, pow(3, i), A2[i]))
assert mult(ec, alpha13, ec.G) == RHS1, 'player one is cheating'
assert mult(ec, alpha23, ec.G) == RHS2, 'player two is cheating'
A: List[Point] = list() # commitment at the global sharing polynomial
for i in range(t):
A.append(ec.add(A1[i], ec.add(A2[i], A3[i])))
Q = A[0] # aggregated public key
### SECOND PHASE: generation of the nonces' pair ###
# This phase follows exactly the key generation procedure
# suppose that player one and three want to sign
# signer one acting as the dealer
commits1: List[Point] = list()
k1 = (1 + random.getrandbits(ec.nlen)) % ec.n
k1_prime = (1 + random.getrandbits(ec.nlen)) % ec.n
commits1.append(double_mult(ec, k1, ec.G, k1_prime, H))
# sharing polynomials
f1: List[int] = list()
f1.append(k1)
f1_prime: List[int] = list()
f1_prime.append(k1_prime)
for i in range(1, t):
temp = (1 + random.getrandbits(ec.nlen)) % ec.n
f1.append(temp)
temp = (1 + random.getrandbits(ec.nlen)) % ec.n
f1_prime.append(temp)
commits1.append(double_mult(ec, f1[i], ec.G, f1_prime[i], H))
# shares of the secret
beta13 = 0 # share of k1 belonging to P3
beta13_prime = 0
for i in range(t):
beta13 += (f1[i] * pow(3, i)) % ec.n
beta13_prime += (f1_prime[i] * pow(3, i)) % ec.n
# player three verifies consistency of his share
RHS = 1, 0
for i in range(t):
RHS = ec.add(RHS, mult(ec, pow(3, i), commits1[i]))
assert double_mult(ec, beta13, ec.G, beta13_prime,
H) == RHS, 'player one is cheating'
# signer three acting as the dealer
commits3: List[Point] = list()
k3 = (1 + random.getrandbits(ec.nlen)) % ec.n
k3_prime = (1 + random.getrandbits(ec.nlen)) % ec.n
commits3.append(double_mult(ec, k3, ec.G, k3_prime, H))
# sharing polynomials
f3: List[int] = list()
f3.append(k3)
f3_prime: List[int] = list()
f3_prime.append(k3_prime)
for i in range(1, t):
temp = (1 + random.getrandbits(ec.nlen)) % ec.n
f3.append(temp)
temp = (1 + random.getrandbits(ec.nlen)) % ec.n
f3_prime.append(temp)
commits3.append(double_mult(ec, f3[i], ec.G, f3_prime[i], H))
# shares of the secret
beta31 = 0 # share of k3 belonging to P1
beta31_prime = 0
for i in range(t):
beta31 += (f3[i] * pow(1, i)) % ec.n
beta31_prime += (f3_prime[i] * pow(1, i)) % ec.n
# player one verifies consistency of his share
RHS = 1, 0
for i in range(t):
RHS = ec.add(RHS, mult(ec, pow(1, i), commits3[i]))
assert double_mult(ec, beta31, ec.G, beta31_prime,
H) == RHS, 'player three is cheating'
# shares of the secret nonce
beta1 = beta31 % ec.n
beta3 = beta13 % ec.n
for i in range(t):
beta1 += (f1[i] * pow(1, i)) % ec.n
beta3 += (f3[i] * pow(3, i)) % ec.n
# it's time to recover the public nonce
B1: List[Point] = list()
B3: List[Point] = list()
# each participant i = 1, 3 shares Qi as follows
# he broadcasts these values
for i in range(t):
B1.append(mult(ec, f1[i], ec.G))
B3.append(mult(ec, f3[i], ec.G))
# he checks the others' values
# player one
RHS3 = 1, 0
for i in range(t):
RHS3 = ec.add(RHS3, mult(ec, pow(1, i), B3[i]))
assert mult(ec, beta31, ec.G) == RHS3, 'player three is cheating'
# player three
RHS1 = 1, 0
for i in range(t):
RHS1 = ec.add(RHS1, mult(ec, pow(3, i), B1[i]))
assert mult(ec, beta13, ec.G) == RHS1, 'player one is cheating'
B: List[Point] = list() # commitment at the global sharing polynomial
for i in range(t):
B.append(ec.add(B1[i], B3[i]))
K = B[0] # aggregated public nonce
if legendre_symbol(K[1], ec._p) != 1:
beta1 = ec.n - beta1
beta3 = ec.n - beta3
### PHASE THREE: signature generation ###
# partial signatures
ebytes = K[0].to_bytes(32, byteorder="big")
ebytes += octets_from_point(ec, Q, True)
ebytes += msg
e = int_from_bits(ec, hf(ebytes).digest())
gamma1 = (beta1 + e * alpha1) % ec.n
gamma3 = (beta3 + e * alpha3) % ec.n
# each participant verifies the other partial signatures
# player one
if legendre_symbol(K[1], ec._p) == 1:
RHS3 = ec.add(K, mult(ec, e, Q))
for i in range(1, t):
temp = double_mult(ec, pow(3, i), B[i], e * pow(3, i), A[i])
RHS3 = ec.add(RHS3, temp)
else:
assert legendre_symbol(K[1], ec._p) != 1
RHS3 = ec.add(ec.opposite(K), mult(ec, e, Q))
for i in range(1, t):
temp = double_mult(ec, pow(3, i), ec.opposite(B[i]),
e * pow(3, i), A[i])
RHS3 = ec.add(RHS3, temp)
assert mult(ec, gamma3, ec.G) == RHS3, 'player three is cheating'
# player three
if legendre_symbol(K[1], ec._p) == 1:
RHS1 = ec.add(K, mult(ec, e, Q))
for i in range(1, t):
temp = double_mult(ec, pow(1, i), B[i], e * pow(1, i), A[i])
RHS1 = ec.add(RHS1, temp)
else:
assert legendre_symbol(K[1], ec._p) != 1
RHS1 = ec.add(ec.opposite(K), mult(ec, e, Q))
for i in range(1, t):
temp = double_mult(ec, pow(1, i), ec.opposite(B[i]),
e * pow(1, i), A[i])
RHS1 = ec.add(RHS1, temp)
assert mult(ec, gamma1, ec.G) == RHS1, 'player two is cheating'
### PHASE FOUR: aggregating the signature ###
omega1 = 3 * mod_inv(3 - 1, ec.n) % ec.n
omega3 = 1 * mod_inv(1 - 3, ec.n) % ec.n
sigma = (gamma1 * omega1 + gamma3 * omega3) % ec.n
sig = K[0], sigma
self.assertTrue(ssa._verify(ec, hf, msg, Q, sig))
### ADDITIONAL PHASE: reconstruction of the private key ###
secret = (omega1 * alpha1 + omega3 * alpha3) % ec.n
self.assertEqual((q1 + q2 + q3) % ec.n, secret)
def test_double_mult() -> None:
H = second_generator(secp256k1)
G = secp256k1.G
# 0*G + 1*H
T = double_mult(1, H, 0, G)
assert T == H
T = multi_mult([1, 0], [H, G])
assert T == H
# 0*G + 2*H
exp = mult(2, H)
T = double_mult(2, H, 0, G)
assert T == exp
T = multi_mult([2, 0], [H, G])
assert T == exp
# 0*G + 3*H
exp = mult(3, H)
T = double_mult(3, H, 0, G)
assert T == exp
T = multi_mult([3, 0], [H, G])
assert T == exp
# 1*G + 0*H
T = double_mult(0, H, 1, G)
assert T == G
T = multi_mult([0, 1], [H, G])
assert T == G
# 2*G + 0*H
exp = mult(2, G)
T = double_mult(0, H, 2, G)
assert T == exp
T = multi_mult([0, 2], [H, G])
assert T == exp
# 3*G + 0*H
exp = mult(3, G)
T = double_mult(0, H, 3, G)
assert T == exp
T = multi_mult([0, 3], [H, G])
assert T == exp
# 0*G + 5*H
exp = mult(5, H)
T = double_mult(5, H, 0, G)
assert T == exp
T = multi_mult([5, 0], [H, G])
assert T == exp
# 0*G - 5*H
exp = mult(-5, H)
T = double_mult(-5, H, 0, G)
assert T == exp
T = multi_mult([-5, 0], [H, G])
assert T == exp
# 1*G - 5*H
exp = secp256k1.add(G, T)
T = double_mult(-5, H, 1, G)
assert T == exp
def test_musig(self):
""" testing 3-of-3 MuSig
https://eprint.iacr.org/2018/068
https://blockstream.com/2018/01/23/musig-key-aggregation-schnorr-signatures.html
https://medium.com/@snigirev.stepan/how-schnorr-signatures-may-improve-bitcoin-91655bcb4744
"""
ec = secp256k1
L: List[Point] = list() # multiset of public keys
M = hf('message to sign'.encode()).digest()
# first signer
q1 = int_from_octets(
'0c28fca386c7a227600b2fe50b7cae11ec86d3bf1fbe471be89827e19d92ad1d')
Q1 = mult(ec, q1, ec.G)
L.append(octets_from_point(ec, Q1, False))
# ephemeral private nonce
k1 = 0x012a2a833eac4e67e06611aba01345b85cdd4f5ad44f72e369ef0dd640424dbb
K1 = mult(ec, k1, ec.G)
K1_x = K1[0]
if legendre_symbol(K1[1], ec._p) != 1:
k1 = ec.n - k1
K1 = K1_x, ec.y_quadratic_residue(K1_x, True)
#K1 = mult(ec, k1, ec.G)
# second signer
q2 = int_from_octets(
'0c28fca386c7a227600b2fe50b7cae11ec86d3bf1fbe471be89827e19d72aa1d')
Q2 = mult(ec, q2, ec.G)
L.append(octets_from_point(ec, Q2, False))
k2 = 0x01a2a0d3eac4e67e06611aba01345b85cdd4f5ad44f72e369ef0dd640424dbdb
K2 = mult(ec, k2, ec.G)
K2_x = K2[0]
if legendre_symbol(K2[1], ec._p) != 1:
k2 = ec.n - k2
K2 = K2_x, ec.y_quadratic_residue(K2_x, True)
#K2 = mult(ec, k2, ec.G)
# third signer
q3 = random.getrandbits(ec.nlen) % ec.n
Q3 = mult(ec, q3, ec.G)
while Q3 == None: # plausible only for small (test) cardinality groups
q3 = random.getrandbits(ec.nlen) % ec.n
Q3 = mult(ec, q3, ec.G)
L.append(octets_from_point(ec, Q3, False))
k3 = random.getrandbits(ec.nlen) % ec.n
K3 = mult(ec, k3, ec.G)
while K3 == None: # plausible only for small (test) cardinality groups
k3 = random.getrandbits(ec.nlen) % ec.n
K3 = mult(ec, k3, ec.G)
K3_x = K3[0]
if legendre_symbol(K3[1], ec._p) != 1:
k3 = ec.n - k3
K3 = K3_x, ec.y_quadratic_residue(K3_x, True)
#K3 = mult(ec, k3, ec.G)
L.sort() # using lexicographic ordering
L_brackets = b''
for i in range(len(L)):
L_brackets += L[i]
h1 = hf(L_brackets + octets_from_point(ec, Q1, False)).digest()
a1 = int_from_bits(ec, h1)
h2 = hf(L_brackets + octets_from_point(ec, Q2, False)).digest()
a2 = int_from_bits(ec, h2)
h3 = hf(L_brackets + octets_from_point(ec, Q3, False)).digest()
a3 = int_from_bits(ec, h3)
# aggregated public key
Q_All = double_mult(ec, a1, Q1, a2, Q2)
Q_All = ec.add(Q_All, mult(ec, a3, Q3))
Q_All_bytes = octets_from_point(ec, Q_All, True)
########################
# exchange K_x, compute s
# WARNING: the signers should exchange commitments to the public
# nonces before sending the nonces themselves
# first signer use K2_x and K3_x
y = ec.y_quadratic_residue(K2_x, True)
K2_recovered = (K2_x, y)
y = ec.y_quadratic_residue(K3_x, True)
K3_recovered = (K3_x, y)
K1_All = ec.add(ec.add(K1, K2_recovered), K3_recovered)
if legendre_symbol(K1_All[1], ec._p) != 1:
# no need to actually change K1_All[1], as it is not used anymore
# let's fix k1 instead, as it is used later
k1 = ec.n - k1
K1_All0_bytes = K1_All[0].to_bytes(32, byteorder="big")
h1 = hf(K1_All0_bytes + Q_All_bytes + M).digest()
c1 = int_from_bits(ec, h1)
assert 0 < c1 and c1 < ec.n, "sign fail"
s1 = (k1 + c1 * a1 * q1) % ec.n
# second signer use K1_x and K3_x
y = ec.y_quadratic_residue(K1_x, True)
K1_recovered = (K1_x, y)
y = ec.y_quadratic_residue(K3_x, True)
K3_recovered = (K3_x, y)
K2_All = ec.add(ec.add(K2, K1_recovered), K3_recovered)
if legendre_symbol(K2_All[1], ec._p) != 1:
# no need to actually change K2_All[1], as it is not used anymore
# let's fix k2 instead, as it is used later
k2 = ec.n - k2
K2_All0_bytes = K2_All[0].to_bytes(32, byteorder="big")
h2 = hf(K2_All0_bytes + Q_All_bytes + M).digest()
c2 = int_from_bits(ec, h2)
assert 0 < c2 and c2 < ec.n, "sign fail"
s2 = (k2 + c2 * a2 * q2) % ec.n
# third signer use K1_x and K2_x
y = ec.y_quadratic_residue(K1_x, True)
K1_recovered = (K1_x, y)
y = ec.y_quadratic_residue(K2_x, True)
K2_recovered = (K2_x, y)
K3_All = ec.add(ec.add(K1_recovered, K2_recovered), K3)
if legendre_symbol(K3_All[1], ec._p) != 1:
# no need to actually change K3_All[1], as it is not used anymore
# let's fix k3 instead, as it is used later
k3 = ec.n - k3
K3_All0_bytes = K3_All[0].to_bytes(32, byteorder="big")
h3 = hf(K3_All0_bytes + Q_All_bytes + M).digest()
c3 = int_from_bits(ec, h3)
assert 0 < c3 and c3 < ec.n, "sign fail"
s3 = (k3 + c3 * a3 * q3) % ec.n
############################################
# combine signatures into a single signature
# anyone can do the following
assert K1_All[0] == K2_All[0], "sign fail"
assert K2_All[0] == K3_All[0], "sign fail"
s_All = (s1 + s2 + s3) % ec.n
sig = (K1_All[0], s_All)
self.assertTrue(ssa.verify(ec, hf, M, Q_All, sig))
