def test_second_generator() -> None:
"""
important remarks on secp256-zkp prefix for
compressed encoding of the second generator:
https://github.com/garyyu/rust-secp256k1-zkp/wiki/Pedersen-Commitment
"""
H = (
0x50929B74C1A04954B78B4B6035E97A5E078A5A0F28EC96D547BFEE9ACE803AC0,
0x31D3C6863973926E049E637CB1B5F40A36DAC28AF1766968C30C2313F3A38904,
)
assert H == pedersen.second_generator(secp256k1, sha256)
H = pedersen.second_generator(secp256r1, sha256)
H = pedersen.second_generator(secp384r1, sha384)
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 test_assorted_mult():
ec = ec23_31
H = second_generator(ec)
HJ = _jac_from_aff(H)
for k1 in range(ec.n):
K1J = _mult_jac(k1, ec.GJ, ec)
for k2 in range(ec.n):
K2J = _mult_jac(k2, HJ, ec)
shamir = _double_mult(k1, ec.GJ, k2, ec.GJ, ec)
assert ec._jac_equality(shamir, _mult_jac(k1 + k2, ec.GJ, ec))
shamir = _double_mult(k1, INFJ, k2, HJ, ec)
assert ec._jac_equality(shamir, K2J)
shamir = _double_mult(k1, ec.GJ, k2, INFJ, ec)
assert ec._jac_equality(shamir, K1J)
shamir = _double_mult(k1, ec.GJ, k2, HJ, ec)
K1JK2J = ec._add_jac(K1J, K2J)
assert ec._jac_equality(K1JK2J, shamir)
k3 = 1 + secrets.randbelow(ec.n - 1)
K3J = _mult_jac(k3, ec.GJ, ec)
K1JK2JK3J = ec._add_jac(K1JK2J, K3J)
boscoster = _multi_mult([k1, k2, k3], [ec.GJ, HJ, ec.GJ], ec)
assert ec._jac_equality(K1JK2JK3J, boscoster)
k4 = 1 + secrets.randbelow(ec.n - 1)
K4J = _mult_jac(k4, HJ, ec)
K1JK2JK3JK4J = ec._add_jac(K1JK2JK3J, K4J)
points = [ec.GJ, HJ, ec.GJ, HJ]
boscoster = _multi_mult([k1, k2, k3, k4], points, ec)
assert ec._jac_equality(K1JK2JK3JK4J, boscoster)
assert ec._jac_equality(K1JK2JK3J, _multi_mult([k1, k2, k3, 0], points, ec))
assert ec._jac_equality(K1JK2J, _multi_mult([k1, k2, 0, 0], points, ec))
assert ec._jac_equality(K1J, _multi_mult([k1, 0, 0, 0], points, ec))
assert ec._jac_equality(INFJ, _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.GJ, HJ, ec.GJ], ec)
with pytest.raises(ValueError, match="negative first coefficient: "):
_double_mult(-5, HJ, 1, ec.GJ, ec)
with pytest.raises(ValueError, match="negative second coefficient: "):
_double_mult(1, HJ, -5, ec.GJ, ec)
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(
'0250929b74c1a04954b78b4b6035e97a5e078a5a0f28ec96d547bfee9ace803ac0',
ec))
# 0*G + 1*H
T = double_mult(1, H, 0, ec.G, ec)
self.assertEqual(
T,
point_from_octets(
'0250929b74c1a04954b78b4b6035e97a5e078a5a0f28ec96d547bfee9ace803ac0',
ec))
# 0*G + 2*H
T = double_mult(2, H, 0, ec.G, ec)
self.assertEqual(
T,
point_from_octets(
'03fad265e0a0178418d006e247204bcf42edb6b92188074c9134704c8686eed37a',
ec))
T = mult(2, H, ec)
self.assertEqual(
T,
point_from_octets(
'03fad265e0a0178418d006e247204bcf42edb6b92188074c9134704c8686eed37a',
ec))
# 0*G + 3*H
T = double_mult(3, H, 0, ec.G, ec)
self.assertEqual(
T,
point_from_octets(
'025ef47fcde840a435e831bbb711d466fc1ee160da3e15437c6c469a3a40daacaa',
ec))
T = mult(3, H, ec)
self.assertEqual(
T,
point_from_octets(
'025ef47fcde840a435e831bbb711d466fc1ee160da3e15437c6c469a3a40daacaa',
ec))
# 1*G+0*H
T = double_mult(0, H, 1, ec.G, ec)
self.assertEqual(
T,
point_from_octets(
'0279be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798',
ec))
T = ec.G
self.assertEqual(
T,
point_from_octets(
'0279be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798',
ec))
# 2*G+0*H
T = double_mult(0, H, 2, ec.G, ec)
self.assertEqual(
T,
point_from_octets(
'02c6047f9441ed7d6d3045406e95c07cd85c778e4b8cef3ca7abac09b95c709ee5',
ec))
T = mult(2, ec.G, ec)
self.assertEqual(
T,
point_from_octets(
'02c6047f9441ed7d6d3045406e95c07cd85c778e4b8cef3ca7abac09b95c709ee5',
ec))
# 3*G+0*H
T = double_mult(0, H, 3, ec.G, ec)
self.assertEqual(
T,
point_from_octets(
'02f9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9',
ec))
T = mult(3, ec.G, ec)
self.assertEqual(
T,
point_from_octets(
'02f9308a019258c31049344f85f89d5229b531c845836f99b08601f113bce036f9',
ec))
# 0*G+5*H
T = double_mult(5, H, 0, ec.G, ec)
self.assertEqual(
T,
point_from_octets(
'039e431be0851721f9ce35cc0f718fce7d6d970e3ddd796643d71294d7a09b554e',
ec))
T = mult(5, H, ec)
self.assertEqual(
T,
point_from_octets(
'039e431be0851721f9ce35cc0f718fce7d6d970e3ddd796643d71294d7a09b554e',
ec))
# 0*G-5*H
T = double_mult(-5, H, 0, ec.G, ec)
self.assertEqual(
T,
point_from_octets(
'029e431be0851721f9ce35cc0f718fce7d6d970e3ddd796643d71294d7a09b554e',
ec))
T = mult(-5, H, ec)
self.assertEqual(
T,
point_from_octets(
'029e431be0851721f9ce35cc0f718fce7d6d970e3ddd796643d71294d7a09b554e',
ec))
# 1*G-5*H
U = double_mult(-5, H, 1, ec.G, ec)
self.assertEqual(
U,
point_from_octets(
'02b218ddacb34d827c71760e601b41d309bc888cf7e3ab7cc09ec082b645f77e5a',
ec))
U = ec.add(ec.G, T) # reusing previous T value
self.assertEqual(
U,
point_from_octets(
'02b218ddacb34d827c71760e601b41d309bc888cf7e3ab7cc09ec082b645f77e5a',
ec))
H = pedersen.second_generator(secp256r1, hf)
H = pedersen.second_generator(secp384r1, sha384)
def test_mult_double_mult():
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_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_threshold(self):
"""testing 2-of-3 threshold signature (Pedersen secret sharing)"""
ec = secp256k1
hf = sha256
# parameters
t = 2
H = second_generator(ec, hf)
msg = 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 = 1, 0
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 = 1, 0
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 = 1, 0
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 = 1, 0
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 = 1, 0
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 = 1, 0
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 = 1, 0
RHS3 = 1, 0
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 = 1, 0
RHS3 = 1, 0
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 = 1, 0
RHS2 = 1, 0
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 = 1, 0
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 = 1, 0
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 = 1, 0
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 = 1, 0
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 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(Q, True, ec)
ebytes += msg
e = int_from_bits(hf(ebytes).digest(), ec)
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(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:
assert legendre_symbol(K[1], ec._p) != 1
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 legendre_symbol(K[1], ec._p) == 1:
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:
assert legendre_symbol(K[1], ec._p) != 1
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(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)
