def test_batch_inversion(self):
l = 8
v = ScalarVector([random_scalar() for i in range(l)])
v.append(Scalar(1))
v.append(Scalar(dumb25519.l - 1))
w = v.invert()
for i in range(len(v)):
self.assertEqual(v[i] * w[i], Scalar(1))
def prove(data,N):
clear_cache()
M = len(data)
# curve points
G = dumb25519.G
H = hash_to_point('pybullet H')
Gi = PointVector([hash_to_point('pybullet Gi ' + str(i)) for i in range(M*N)])
Hi = PointVector([hash_to_point('pybullet Hi ' + str(i)) for i in range(M*N)])
# set amount commitments
V = PointVector([])
aL = ScalarVector([])
for v,gamma in data:
V.append((H*v + G*gamma)*inv8)
mash(V[-1])
aL.extend(scalar_to_bits(v,N))
# set bit arrays
aR = ScalarVector([])
for bit in aL.scalars:
aR.append(bit-Scalar(1))
alpha = random_scalar()
A = (Gi*aL + Hi*aR + G*alpha)*inv8
sL = ScalarVector([random_scalar()]*(M*N))
sR = ScalarVector([random_scalar()]*(M*N))
rho = random_scalar()
S = (Gi*sL + Hi*sR + G*rho)*inv8
# get challenges
mash(A)
mash(S)
y = cache
y_inv = y.invert()
mash('')
z = cache
# polynomial coefficients
l0 = aL - ScalarVector([z]*(M*N))
l1 = sL
# ugly sum
zeros_twos = []
for i in range (M*N):
zeros_twos.append(Scalar(0))
for j in range(1,M+1):
temp = Scalar(0)
if i >= (j-1)*N and i < j*N:
temp = Scalar(2)**(i-(j-1)*N)
zeros_twos[-1] += temp*(z**(1+j))
# more polynomial coefficients
r0 = aR + ScalarVector([z]*(M*N))
r0 = r0*exp_scalar(y,M*N)
r0 += ScalarVector(zeros_twos)
r1 = exp_scalar(y,M*N)*sR
# build the polynomials
t0 = l0**r0
t1 = l0**r1 + l1**r0
t2 = l1**r1
tau1 = random_scalar()
tau2 = random_scalar()
T1 = (H*t1 + G*tau1)*inv8
T2 = (H*t2 + G*tau2)*inv8
mash(T1)
mash(T2)
x = cache # challenge
taux = tau1*x + tau2*(x**2)
for j in range(1,M+1):
gamma = data[j-1][1]
taux += z**(1+j)*gamma
mu = x*rho+alpha
l = l0 + l1*x
r = r0 + r1*x
t = l**r
mash(taux)
mash(mu)
mash(t)
x_ip = cache # challenge
L = PointVector([])
R = PointVector([])
# initial inner product inputs
data_ip = [Gi,PointVector([Hi[i]*(y_inv**i) for i in range(len(Hi))]),H*x_ip,l,r,None,None]
while True:
data_ip = inner_product(data_ip)
# we have reached the end of the recursion
if len(data_ip) == 2:
return [V,A,S,T1,T2,taux,mu,L,R,data_ip[0],data_ip[1],t]
# we are not done yet
L.append(data_ip[-2])
R.append(data_ip[-1])
def verify(proofs,N):
# determine the length of the longest proof
max_MN = 2**max([len(proof[7]) for proof in proofs])
# curve points
Z = dumb25519.Z
G = dumb25519.G
H = hash_to_point('pybullet H')
Gi = PointVector([hash_to_point('pybullet Gi ' + str(i)) for i in range(max_MN)])
Hi = PointVector([hash_to_point('pybullet Hi ' + str(i)) for i in range(max_MN)])
# set up weighted aggregates
y0 = Scalar(0)
y1 = Scalar(0)
z1 = Scalar(0)
z3 = Scalar(0)
z4 = [Scalar(0)]*max_MN
z5 = [Scalar(0)]*max_MN
scalars = ScalarVector([]) # for final check
points = PointVector([]) # for final check
# run through each proof
for proof in proofs:
clear_cache()
V,A,S,T1,T2,taux,mu,L,R,a,b,t = proof
# get size information
M = 2**len(L)/N
# weighting factors for batching
weight_y = random_scalar()
weight_z = random_scalar()
if weight_y == Scalar(0) or weight_z == Scalar(0):
raise ArithmeticError
# reconstruct all challenges
for v in V:
mash(v)
mash(A)
mash(S)
if cache == Scalar(0):
raise ArithmeticError
y = cache
y_inv = y.invert()
mash('')
if cache == Scalar(0):
raise ArithmeticError
z = cache
mash(T1)
mash(T2)
if cache == Scalar(0):
raise ArithmeticError
x = cache
mash(taux)
mash(mu)
mash(t)
if cache == Scalar(0):
raise ArithmeticError
x_ip = cache
y0 += taux*weight_y
k = (z-z**2)*sum_scalar(y,M*N)
for j in range(1,M+1):
k -= (z**(j+2))*sum_scalar(Scalar(2),N)
y1 += (t-k)*weight_y
for j in range(M):
scalars.append(z**(j+2)*weight_y)
points.append(V[j]*Scalar(8))
scalars.append(x*weight_y)
points.append(T1*Scalar(8))
scalars.append(x**2*weight_y)
points.append(T2*Scalar(8))
scalars.append(weight_z)
points.append(A*Scalar(8))
scalars.append(x*weight_z)
points.append(S*Scalar(8))
# inner product
W = ScalarVector([])
for i in range(len(L)):
mash(L[i])
mash(R[i])
if cache == Scalar(0):
raise ArithmeticError
W.append(cache)
W_inv = W.invert()
for i in range(M*N):
index = i
g = a
h = b*((y_inv)**i)
for j in range(len(L)-1,-1,-1):
J = len(W)-j-1
base_power = 2**j
if index/base_power == 0:
g *= W_inv[J]
h *= W[J]
else:
g *= W[J]
h *= W_inv[J]
index -= base_power
g += z
h -= (z*(y**i) + (z**(2+i/N))*(Scalar(2)**(i%N)))*((y_inv)**i)
z4[i] += g*weight_z
z5[i] += h*weight_z
z1 += mu*weight_z
for i in range(len(L)):
scalars.append(W[i]**2*weight_z)
points.append(L[i]*Scalar(8))
scalars.append(W_inv[i]**2*weight_z)
points.append(R[i]*Scalar(8))
z3 += (t-a*b)*x_ip*weight_z
# now check all proofs together
scalars.append(-y0-z1)
points.append(G)
scalars.append(-y1+z3)
points.append(H)
for i in range(max_MN):
scalars.append(-z4[i])
points.append(Gi[i])
scalars.append(-z5[i])
points.append(Hi[i])
if not dumb25519.multiexp(scalars,points) == Z:
raise ArithmeticError('Bad z check!')
return True
def verify(proofs,N):
# determine the length of the longest proof
max_MN = 2**max([len(proof.L) for proof in proofs])
# curve points
Z = dumb25519.Z
Gi = PointVector([hash_to_point('pybullet Gi ' + str(i)) for i in range(max_MN)])
Hi = PointVector([hash_to_point('pybullet Hi ' + str(i)) for i in range(max_MN)])
# set up weighted aggregates
y0 = Scalar(0)
y1 = Scalar(0)
z1 = Scalar(0)
z3 = Scalar(0)
z4 = [Scalar(0)]*max_MN
z5 = [Scalar(0)]*max_MN
scalars = ScalarVector([]) # for final check
points = PointVector([]) # for final check
# run through each proof
for proof in proofs:
tr = transcript.Transcript('Bulletproof')
V = proof.V
A = proof.A
S = proof.S
T1 = proof.T1
T2 = proof.T2
taux = proof.taux
mu = proof.mu
L = proof.L
R = proof.R
a = proof.a
b = proof.b
t = proof.t
# get size information
M = 2**len(L)/N
# weighting factors for batching
weight_y = random_scalar()
weight_z = random_scalar()
if weight_y == Scalar(0) or weight_z == Scalar(0):
raise ArithmeticError
# reconstruct challenges
for v in V:
tr.update(v)
tr.update(A)
tr.update(S)
y = tr.challenge()
if y == Scalar(0):
raise ArithmeticError
y_inv = y.invert()
z = tr.challenge()
if z == Scalar(0):
raise ArithmeticError
tr.update(T1)
tr.update(T2)
x = tr.challenge()
if x == Scalar(0):
raise ArithmeticError
tr.update(taux)
tr.update(mu)
tr.update(t)
x_ip = tr.challenge()
if x_ip == Scalar(0):
raise ArithmeticError
y0 += taux*weight_y
k = (z-z**2)*sum_scalar(y,M*N)
for j in range(1,M+1):
k -= (z**(j+2))*sum_scalar(Scalar(2),N)
y1 += (t-k)*weight_y
for j in range(M):
scalars.append(z**(j+2)*weight_y)
points.append(V[j]*Scalar(8))
scalars.append(x*weight_y)
points.append(T1*Scalar(8))
scalars.append(x**2*weight_y)
points.append(T2*Scalar(8))
scalars.append(weight_z)
points.append(A*Scalar(8))
scalars.append(x*weight_z)
points.append(S*Scalar(8))
# inner product
W = ScalarVector([])
for i in range(len(L)):
tr.update(L[i])
tr.update(R[i])
W.append(tr.challenge())
if W[i] == Scalar(0):
raise ArithmeticError
W_inv = W.invert()
for i in range(M*N):
index = i
g = a
h = b*((y_inv)**i)
for j in range(len(L)-1,-1,-1):
J = len(W)-j-1
base_power = 2**j
if index/base_power == 0:
g *= W_inv[J]
h *= W[J]
else:
g *= W[J]
h *= W_inv[J]
index -= base_power
g += z
h -= (z*(y**i) + (z**(2+i/N))*(Scalar(2)**(i%N)))*((y_inv)**i)
z4[i] += g*weight_z
z5[i] += h*weight_z
z1 += mu*weight_z
for i in range(len(L)):
scalars.append(W[i]**2*weight_z)
points.append(L[i]*Scalar(8))
scalars.append(W_inv[i]**2*weight_z)
points.append(R[i]*Scalar(8))
z3 += (t-a*b)*x_ip*weight_z
# now check all proofs together
scalars.append(-y0-z1)
points.append(Gc)
scalars.append(-y1+z3)
points.append(Hc)
for i in range(max_MN):
scalars.append(-z4[i])
points.append(Gi[i])
scalars.append(-z5[i])
points.append(Hi[i])
if not dumb25519.multiexp(scalars,points) == Z:
raise ArithmeticError('Bad verification!')
return True
def prove(data,N):
tr = transcript.Transcript('Bulletproof')
M = len(data)
# curve points
Gi = PointVector([hash_to_point('pybullet Gi ' + str(i)) for i in range(M*N)])
Hi = PointVector([hash_to_point('pybullet Hi ' + str(i)) for i in range(M*N)])
# set amount commitments
V = PointVector([])
aL = ScalarVector([])
for v,gamma in data:
V.append(com(v,gamma)*inv8)
tr.update(V[-1])
aL.extend(scalar_to_bits(v,N))
# set bit arrays
aR = ScalarVector([])
for bit in aL.scalars:
aR.append(bit-Scalar(1))
alpha = random_scalar()
A = (Gi**aL + Hi**aR + Gc*alpha)*inv8
sL = ScalarVector([random_scalar()]*(M*N))
sR = ScalarVector([random_scalar()]*(M*N))
rho = random_scalar()
S = (Gi**sL + Hi**sR + Gc*rho)*inv8
# get challenges
tr.update(A)
tr.update(S)
y = tr.challenge()
z = tr.challenge()
y_inv = y.invert()
# polynomial coefficients
l0 = aL - ScalarVector([z]*(M*N))
l1 = sL
# for polynomial coefficients
zeros_twos = []
z_cache = z**2
for j in range(M):
for i in range(N):
zeros_twos.append(z_cache*2**i)
z_cache *= z
# more polynomial coefficients
r0 = aR + ScalarVector([z]*(M*N))
r0 = r0*exp_scalar(y,M*N)
r0 += ScalarVector(zeros_twos)
r1 = exp_scalar(y,M*N)*sR
# build the polynomials
t0 = l0**r0
t1 = l0**r1 + l1**r0
t2 = l1**r1
tau1 = random_scalar()
tau2 = random_scalar()
T1 = com(t1,tau1)*inv8
T2 = com(t2,tau2)*inv8
tr.update(T1)
tr.update(T2)
x = tr.challenge()
taux = tau1*x + tau2*(x**2)
for j in range(1,M+1):
gamma = data[j-1][1]
taux += z**(1+j)*gamma
mu = x*rho+alpha
l = l0 + l1*x
r = r0 + r1*x
t = l**r
tr.update(taux)
tr.update(mu)
tr.update(t)
x_ip = tr.challenge()
# initial inner product inputs
data = InnerProductRound(Gi,PointVector([Hi[i]*(y_inv**i) for i in range(len(Hi))]),Hc*x_ip,l,r,tr)
while True:
inner_product(data)
# we have reached the end of the recursion
if data.done:
return Bulletproof(V,A,S,T1,T2,taux,mu,data.L,data.R,data.a,data.b,t)