def test_low_cardinality(self):
"""test low-cardinality curves for all msg/key pairs."""
# ec.n has to be prime to sign
prime = [11, 13, 17, 19]
# for dsa it is possible to directy iterate on all
# possible values for e;
# for ssa we have to iterate over all possible hash values
hsize = hf().digest_size
H = [i.to_bytes(hsize, 'big') for i in range(max(prime) * 4)]
# only low card curves or it would take forever
for ec in low_card_curves:
if ec._p in prime: # only few curves or it would take too long
# BIP340 Schnorr only applies to curve whose prime p = 3 %4
if not ec.pIsThreeModFour:
self.assertRaises(ValueError, ssa.sign, H[0], 1, None, ec)
continue
for q in range(1, ec.n): # all possible private keys
Q = mult(q, ec.G, ec) # public key
if not ec.has_square_y(Q):
q = ec.n - q
for h in H: # all possible hashed messages
k = ssa.k(h, q, ec, hf)
K = mult(k, ec.G, ec)
if not ec.has_square_y(K):
k = ec.n - k
x_K = K[0]
try:
c = ssa._challenge(x_K, Q[0], h, ec, hf)
except Exception:
pass
else:
s = (k + c * q) % ec.n
sig = ssa.sign(h, q, None, ec)
self.assertEqual((x_K, s), sig)
# valid signature must validate
self.assertIsNone(ssa._verify(h, Q, sig, ec, hf))
if c != 0: # FIXME
x_Q = ssa._recover_pubkeys(c, x_K, s, ec)
self.assertEqual(Q[0], x_Q)
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
"""
mhd = 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 = ssa.gen_keys()
x_Q1 = x_Q1.to_bytes(ec.psize, 'big')
q2, x_Q2 = ssa.gen_keys()
x_Q2 = x_Q2.to_bytes(ec.psize, 'big')
q3, x_Q3 = ssa.gen_keys()
x_Q3 = x_Q3.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] = list()
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 not ec.has_square_y(Q):
# 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 not ec.has_square_y(K):
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(r, Q[0], mhd, ec, hf)
s1 = (k1 + e * a1 * q1) % ec.n
s2 = (k2 + e * a2 * q2) % ec.n
s3 = (k3 + e * a3 * q3) % ec.n
# exchange s_i (interactive)
# finalize signature (non interactive)
s = (s1 + s2 + s3) % ec.n
sig = r, s
# check signature is valid
ssa._verify(mhd, Q, sig, ec, hf)
self.assertTrue(ssa.verify(mhd, Q, sig))
def test_threshold(self):
"""testing 2-of-3 threshold signature (Pedersen secret sharing)"""
# parameters
m = 2
H = second_generator(ec, hf)
mhd = hf(b'message to sign').digest()
# FIRST PHASE: key pair generation ###################################
# 1.1 signer one acting as the dealer
commits1: List[Point] = list()
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] = list()
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] = list()
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] = list()
A2: List[Point] = list()
A3: List[Point] = list()
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] = list()
for i in range(m):
A.append(ec.add(A1[i], ec.add(A2[i], A3[i])))
# aggregated public key
Q = A[0]
if not ec.has_square_y(Q):
# 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
# 2.1 signer one acting as the dealer
commits1: List[Point] = list()
k1 = ssa.k(mhd, q1, ec, hf)
k1_prime = ssa.k(mhd, 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: List[Point] = list()
k3 = ssa.k(mhd, q3, ec, hf)
k3_prime = ssa.k(mhd, 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] = list()
B3: List[Point] = list()
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] = list()
for i in range(m):
B.append(ec.add(B1[i], B3[i]))
# aggregated public nonce
K = B[0]
if not ec.has_square_y(K):
# 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(K[0], Q[0], mhd, 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
self.assertTrue(ssa.verify(mhd, Q, sig))
# ADDITIONAL PHASE: reconstruction of the private key ###
secret = (omega1 * alpha1 + omega3 * alpha3) % ec.n
self.assertIn((q1 + q2 + q3) % ec.n, (secret, ec.n - secret))
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
"""
mhd = hf(b'message to sign').digest()
# the signers private and public keys,
# including both the curve Point and the BIP340-Schnorr public key
q1 = 0x010101
Q1 = mult(q1)
Q1_x = Q1[0].to_bytes(ec.psize, 'big')
q2 = 0x020202
Q2 = mult(q2)
Q2_x = Q2[0].to_bytes(ec.psize, 'big')
q3 = 0x030303
Q3 = mult(q3)
Q3_x = Q3[0].to_bytes(ec.psize, 'big')
# ready to sign: nonces and nonce commitments
k1 = 1 + secrets.randbelow(ec.n - 1)
K1 = mult(k1)
k2 = 1 + secrets.randbelow(ec.n - 1)
K2 = mult(k2)
k3 = 1 + secrets.randbelow(ec.n - 1)
K3 = mult(k3)
#### (non interactive) key setup
# this is MuSig core: the rest is just Schnorr signature additivity
# 1. lexicographic sorting of public keys
keys: List[bytes] = list()
keys.append(Q1_x)
keys.append(Q2_x)
keys.append(Q3_x)
keys.sort()
# 2. coefficients
prefix = b''.join(keys)
a1 = int_from_bits(hf(prefix + Q1_x).digest(), ec.nlen) % ec.n
a2 = int_from_bits(hf(prefix + Q2_x).digest(), ec.nlen) % ec.n
a3 = int_from_bits(hf(prefix + Q3_x).digest(), ec.nlen) % ec.n
# 3. aggregated public key
Q = ec.add(double_mult(a1, Q1, a2, Q2), mult(a3, Q3))
#### exchange {K_i} (interactive)
#### computes s_i (non interactive)
# WARNING:
# the signers must exchange the nonces
# commitments {K_i} before sharing {s_i}
# same for all signers
K = ec.add(ec.add(K1, K2), K3)
r = K[0]
e = ssa._challenge(r, Q[0], mhd, ec, hf)
# first signer
if not ec.has_square_y(K):
k1 = ec.n - k1
s1 = (k1 + e * a1 * q1) % ec.n
# second signer
if not ec.has_square_y(K):
k2 = ec.n - k2
s2 = (k2 + e * a2 * q2) % ec.n
# third signer
if not ec.has_square_y(K):
k3 = ec.n - k3
s3 = (k3 + e * a3 * q3) % ec.n
#### exchange s_i (interactive)
#### finalize signature (non interactive)
s = (s1 + s2 + s3) % ec.n
sig = r, s
# check signature is valid
self.assertTrue(ssa.verify(mhd, Q, sig))
def test_threshold(self):
"""testing 2-of-3 threshold signature (Pedersen secret sharing)"""
# parameters
t = 2
H = second_generator(ec, hf)
mhd = hf(b'message to sign').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(q1_prime, H, q1))
# 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(f1_prime[i], H, f1[i]))
# 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 = INF
for i in range(t):
RHS = ec.add(RHS, mult(pow(2, i), commits1[i]))
assert double_mult(alpha12_prime, H,
alpha12) == RHS, 'player one is cheating'
# player three verifies consistency of his share
RHS = INF
for i in range(t):
RHS = ec.add(RHS, mult(pow(3, i), commits1[i]))
assert double_mult(alpha13_prime, H,
alpha13) == 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(q2_prime, H, q2))
# 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(f2_prime[i], H, f2[i]))
# 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 = INF
for i in range(t):
RHS = ec.add(RHS, mult(pow(1, i), commits2[i]))
assert double_mult(alpha21_prime, H,
alpha21) == RHS, 'player two is cheating'
# player three verifies consistency of his share
RHS = INF
for i in range(t):
RHS = ec.add(RHS, mult(pow(3, i), commits2[i]))
assert double_mult(alpha23_prime, H,
alpha23) == 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(q3_prime, H, q3))
# 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(f3_prime[i], H, f3[i]))
# 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 = INF
for i in range(t):
RHS = ec.add(RHS, mult(pow(1, i), commits3[i]))
assert double_mult(alpha31_prime, H,
alpha31) == RHS, 'player three is cheating'
# player two verifies consistency of his share
RHS = INF
for i in range(t):
RHS = ec.add(RHS, mult(pow(2, i), commits3[i]))
assert double_mult(alpha32_prime, H,
alpha32) == 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(f1[i]))
A2.append(mult(f2[i]))
A3.append(mult(f3[i]))
# he checks the others' values
# player one
RHS2 = INF
RHS3 = INF
for i in range(t):
RHS2 = ec.add(RHS2, mult(pow(1, i), A2[i]))
RHS3 = ec.add(RHS3, mult(pow(1, i), A3[i]))
assert mult(alpha21) == RHS2, 'player two is cheating'
assert mult(alpha31) == RHS3, 'player three is cheating'
# player two
RHS1 = INF
RHS3 = INF
for i in range(t):
RHS1 = ec.add(RHS1, mult(pow(2, i), A1[i]))
RHS3 = ec.add(RHS3, mult(pow(2, i), A3[i]))
assert mult(alpha12) == RHS1, 'player one is cheating'
assert mult(alpha32) == RHS3, 'player three is cheating'
# player three
RHS1 = INF
RHS2 = INF
for i in range(t):
RHS1 = ec.add(RHS1, mult(pow(3, i), A1[i]))
RHS2 = ec.add(RHS2, mult(pow(3, i), A2[i]))
assert mult(alpha13) == RHS1, 'player one is cheating'
assert mult(alpha23) == 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(k1_prime, H, k1))
# 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(f1_prime[i], H, f1[i]))
# 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 = INF
for i in range(t):
RHS = ec.add(RHS, mult(pow(3, i), commits1[i]))
assert double_mult(beta13_prime, H,
beta13) == 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(k3_prime, H, k3))
# 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(f3_prime[i], H, f3[i]))
# 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 = INF
for i in range(t):
RHS = ec.add(RHS, mult(pow(1, i), commits3[i]))
assert double_mult(beta31_prime, H,
beta31) == 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(f1[i]))
B3.append(mult(f3[i]))
# he checks the others' values
# player one
RHS3 = INF
for i in range(t):
RHS3 = ec.add(RHS3, mult(pow(1, i), B3[i]))
assert mult(beta31) == RHS3, 'player three is cheating'
# player three
RHS1 = INF
for i in range(t):
RHS1 = ec.add(RHS1, mult(pow(3, i), B1[i]))
assert mult(beta13) == 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 not ec.has_square_y(K):
beta1 = ec.n - beta1
beta3 = ec.n - beta3
### PHASE THREE: signature generation ###
# partial signatures
e = ssa._challenge(K[0], Q[0], mhd, ec, hf)
gamma1 = (beta1 + e * alpha1) % ec.n
gamma3 = (beta3 + e * alpha3) % ec.n
# each participant verifies the other partial signatures
# player one
if ec.has_square_y(K):
RHS3 = ec.add(K, mult(e, Q))
for i in range(1, t):
temp = double_mult(pow(3, i), B[i], e * pow(3, i), A[i])
RHS3 = ec.add(RHS3, temp)
else:
RHS3 = ec.add(ec.opposite(K), mult(e, Q))
for i in range(1, t):
temp = double_mult(pow(3, i), ec.opposite(B[i]), e * pow(3, i),
A[i])
RHS3 = ec.add(RHS3, temp)
assert mult(gamma3) == RHS3, 'player three is cheating'
# player three
if ec.has_square_y(K):
RHS1 = ec.add(K, mult(e, Q))
for i in range(1, t):
temp = double_mult(pow(1, i), B[i], e * pow(1, i), A[i])
RHS1 = ec.add(RHS1, temp)
else:
RHS1 = ec.add(ec.opposite(K), mult(e, Q))
for i in range(1, t):
temp = double_mult(pow(1, i), ec.opposite(B[i]), e * pow(1, i),
A[i])
RHS1 = ec.add(RHS1, temp)
assert mult(gamma1) == 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(mhd, Q, sig))
### ADDITIONAL PHASE: reconstruction of the private key ###
secret = (omega1 * alpha1 + omega3 * alpha3) % ec.n
self.assertEqual((q1 + q2 + q3) % ec.n, secret)