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)