def test_forge_hash_sig() -> None:
"""forging valid hash signatures"""
# pylint: disable=protected-access
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
s = ec.n - s if s > ec.n / 2 else s
e = s * u1 % ec.n
dsa._assert_as_valid_(e, (Q[0], Q[1], 1), r, s, lower_s=True, ec=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
s = ec.n - s if s > ec.n / 2 else s
e = s * u1 % ec.n
dsa._assert_as_valid_(e, (Q[0], Q[1], 1), r, s, lower_s=True, ec=ec)
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(BTClibValueError, match=err_msg):
multi_mult([k1, k2, k3, k4], [ec.G, H, ec.G], ec)
def assert_as_valid(msg: Octets, e0: bytes, s: SValues, pubk_rings: PubkeyRing) -> bool:
msg = bytes_from_octets(msg)
m = _get_msg_format(msg, pubk_rings)
ring_size = len(pubk_rings)
e: SValues = 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(_hash(m, e0, i, 0), ec.nlen) % ec.n
# edge case that cannot be reproduced in the test suite
if e[i][0] == 0:
err_msg = "implausibile signature failure" # pragma: no cover
raise BTClibRuntimeError(err_msg) # pragma: no cover
r = b"\0x00"
for j in range(keys_size):
t = double_mult(-e[i][j], pubk_rings[i][j], s[i][j], ec.G)
r = bytes_from_point(t, ec)
if j != len(pubk_rings[i]) - 1:
h = _hash(m, r, i, j + 1)
e[i][j + 1] = int_from_bits(h, ec.nlen) % ec.n
# edge case that cannot be reproduced in the test suite
if e[i][j + 1] == 0:
err_msg = "implausibile signature failure" # pragma: no cover
raise BTClibRuntimeError(err_msg) # pragma: no cover
else:
e0bytes += r
e0_prime = hf(e0bytes).digest()
return e0_prime == e0
def commit(r: int, v: int, ec: Curve = secp256k1, hf: HashF = sha256) -> Point:
"""Commit to r, returning rG+vH.
Commit to r, returning rG+vH. H is the second Nothing-Up-My-Sleeve
(NUMS) generator of the curve.
"""
H = second_generator(ec, hf)
Q = double_mult(v, H, r, ec.G, ec)
# edge case that cannot be reproduced in the test suite
if Q[1] == 0:
err_msg = "invalid (INF) key" # pragma: no cover
raise BTClibRuntimeError(err_msg) # pragma: no cover
return Q
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_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".encode()
msg_hash = reduce_to_hlen(msg, hf)
# 2.1 signer one acting as the dealer
commits1 = []
k1 = ssa.det_nonce_(msg_hash, q1, None, ec, hf)
k1_prime = ssa.det_nonce_(msg_hash, q1_prime, None, 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_hash, q3, None, ec, hf)
k3_prime = ssa.det_nonce_(msg_hash, q3_prime, None, 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_hash, 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 = ssa.Sig(K[0], sigma, ec)
assert ssa.verify_(msg_hash, 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"]
msg_hash = 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.p_size, byteorder="big", signed=False)
q2, x_Q2_int = ssa.gen_keys()
x_Q2 = x_Q2_int.to_bytes(ec.p_size, byteorder="big", signed=False)
q3, x_Q3_int = ssa.gen_keys()
x_Q3 = x_Q3_int.to_bytes(ec.p_size, byteorder="big", signed=False)
# (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_(msg_hash, 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 = ssa.Sig(r, s, ec)
# check signature is valid
ssa.assert_as_valid_(msg_hash, Q[0], sig, hf)
def sign(
msg: Octets,
ks: Sequence[int],
sign_key_idx: Sequence[int],
sign_keys: Sequence[int],
pubk_rings: PubkeyRing,
) -> Tuple[bytes, SValues]:
"""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: message 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 sequences representing single rings of pub_keys
"""
msg = bytes_from_octets(msg)
m = _get_msg_format(msg, pubk_rings)
e0bytes = m
s: SValues = defaultdict(list)
e: SValues = defaultdict(list)
# step 1
for i, (pubk_ring, j_star, k) in enumerate(
zip(pubk_rings.values(), sign_key_idx, ks)
):
keys_size = len(pubk_ring)
s[i] = [0] * keys_size
e[i] = [0] * keys_size
start_idx = (j_star + 1) % keys_size
r = bytes_from_point(mult(k), ec)
if start_idx != 0:
for j in range(start_idx, keys_size):
s[i][j] = secrets.randbits(256)
e[i][j] = int_from_bits(_hash(m, r, i, j), ec.nlen) % ec.n
# edge case that cannot be reproduced in the test suite
if not 0 < e[i][j] < ec.n:
err_msg = "implausibile signature failure" # pragma: no cover
raise BTClibRuntimeError(err_msg) # pragma: no cover
t = double_mult(-e[i][j], pubk_ring[j], s[i][j], ec.G)
r = bytes_from_point(t, ec)
e0bytes += r
e0 = hf(e0bytes).digest()
# step 2
for i, (j_star, k) in enumerate(zip(sign_key_idx, ks)):
e[i][0] = int_from_bits(_hash(m, e0, i, 0), ec.nlen) % ec.n
# edge case that cannot be reproduced in the test suite
if not 0 < e[i][0] < ec.n:
err_msg = "implausibile signature failure" # pragma: no cover
raise BTClibRuntimeError(err_msg) # pragma: no cover
for j in range(1, j_star + 1):
s[i][j - 1] = secrets.randbits(256)
t = double_mult(-e[i][j - 1], pubk_rings[i][j - 1], s[i][j - 1], ec.G)
r = bytes_from_point(t, ec)
e[i][j] = int_from_bits(_hash(m, r, i, j), ec.nlen) % ec.n
# edge case that cannot be reproduced in the test suite
if not 0 < e[i][j] < ec.n:
err_msg = "implausibile signature failure" # pragma: no cover
raise BTClibRuntimeError(err_msg) # pragma: no cover
s[i][j_star] = k + sign_keys[i] * e[i][j_star]
return e0, s
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("\n1. 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()}")
