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parallel_compressed.py
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parallel_compressed.py
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# Parallel compressed one-of-many Groth/Bootle-type proving system
#
# {F,{S,V},S1,V1 ; (l,s,v) | S_l - S1 = sF, V_l - V1 = vF}
from dumb25519 import *
import transcript
class ParallelCompressedParameters:
def __init__(self,F,n,m):
if not isinstance(F,Point):
raise TypeError('Bad type for parameter F!')
if not isinstance(n,int) or not n > 1:
raise TypeError('Bad type or value for parameter n!')
if not isinstance(m,int) or not m > 1:
raise TypeError('Bad type or value for parameter m!')
self.F = F
self.n = n
self.m = m
class ParallelCompressedStatement:
def __init__(self,params,S,V,S1,V1):
if not isinstance(params,ParallelCompressedParameters):
raise TypeError('Bad type for parameters!')
n = params.n
m = params.m
if not isinstance(S,PointVector) or not len(S) == n**m:
raise TypeError('Bad type or length for parallel statement input S!')
if not isinstance(V,PointVector) or not len(V) == n**m:
raise TypeError('Bad type or length for parallel statement input V!')
if not isinstance(S1,Point):
raise TypeError('Bad type for parallel statement input S1!')
if not isinstance(V1,Point):
raise TypeError('Bad type for parallel statement input V1!')
self.F = params.F
self.n = n
self.m = m
self.S = S
self.V = V
self.S1 = S1
self.V1 = V1
self.Gi = [PointVector([hash_to_point('Gi',j,i) for i in range(n)]) for j in range(m)]
class ParallelCompressedWitness:
def __init__(self,l,s,v):
if not isinstance(l,int):
raise TypeError('Bad type for parallel witness l!')
if not isinstance(s,Scalar):
raise TypeError('Bad type for parallel witness s!')
if not isinstance(v,Scalar):
raise TypeError('Bad type for parallel witness v!')
self.l = l
self.s = s
self.v = v
class ParallelCompressedProof:
def __repr__(self):
return repr(hash_to_scalar(
self.A,
self.B,
self.C,
self.D,
self.G,
self.f,
self.zA,
self.zC,
self.z
))
def __init__(self,A,B,C,D,G,f,zA,zC,z):
if not isinstance(A,Point):
raise TypeError('Bad type for parallel proof element A!')
if not isinstance(B,Point):
raise TypeError('Bad type for parallel proof element B!')
if not isinstance(C,Point):
raise TypeError('Bad type for parallel proof element C!')
if not isinstance(D,Point):
raise TypeError('Bad type for parallel proof element D!')
if not isinstance(G,PointVector):
raise TypeError('Bad type for parallel proof element G!')
if not isinstance(f,list):
raise TypeError('Bad type for parallel proof element f!')
for f_ in f:
if not isinstance(f_,ScalarVector):
raise TypeError('Bad type for parallel proof element f!')
if not isinstance(zA,Scalar):
raise TypeError('Bad type for parallel proof element zA!')
if not isinstance(zC,Scalar):
raise TypeError('Bad type for parallel proof element zC!')
if not isinstance(z,Scalar):
raise TypeError('Bad type for parallel proof element z!')
self.A = A
self.B = B
self.C = C
self.D = D
self.G = G
self.f = f
self.zA = zA
self.zC = zC
self.z = z
# Pedersen matrix commitment
def com_matrix(Gi,F,v,r):
C = r*F
for j in range(len(v)):
for i in range(len(v[0])):
C += Gi[j][i]*v[j][i]
return C
# Kronecker delta
def delta(x,y):
if x == y:
return Scalar(1)
return Scalar(0)
# Compute a convolution with a degree-one polynomial
def convolve(x,y):
if not len(y) == 2:
raise ValueError('Convolution requires a degree-one polynomial!')
r = [Scalar(0)]*(len(x)+1)
for i in range(len(x)):
for j in range(len(y)):
r[i+j] += x[i]*y[j]
return r
# Decompose a value with given base and size
def decompose(val,base,size):
r = []
for i in range(size-1,-1,-1):
slot = base**i
r.append(int(val/slot))
val -= slot*r[-1]
return list(reversed(r))
# Perform a commitment-to-zero proof
def prove(statement,witness):
if not isinstance(statement,ParallelCompressedStatement):
raise TypeError('Bad type for parallel statement!')
if not isinstance(witness,ParallelCompressedWitness):
raise TypeError('Bad type for parallel witness!')
# Check the statement validity
l = witness.l
n = statement.n
m = statement.m
N = n**m
if l < 0 or l >= N:
raise IndexError('Invalid parallel witness!')
if not statement.S[l] - statement.S1 == witness.s*statement.F:
raise ArithmeticError('Invalid parallel statement!')
if not statement.V[l] - statement.V1 == witness.v*statement.F:
raise ArithmeticError('Invalid parallel statement!')
# Begin the proof
rA = random_scalar()
rB = random_scalar()
rC = random_scalar()
rD = random_scalar()
# Commit to zero-sum blinders
a = [[random_scalar() for _ in range(n)] for _ in range(m)]
for j in range(m):
a[j][0] = Scalar(0)
for i in range(1,n):
a[j][0] -= a[j][i]
A = com_matrix(statement.Gi,statement.F,a,rA)
# Commit to decomposition bits
decomp_l = decompose(l,n,m)
sigma = [[Scalar(0) for _ in range(n)] for _ in range(m)]
for j in range(m):
for i in range(n):
sigma[j][i] = delta(decomp_l[j],i)
B = com_matrix(statement.Gi,statement.F,sigma,rB)
# Commit to a/sigma relationships
a_sigma = [[Scalar(0) for _ in range(n)] for _ in range(m)]
for j in range(m):
for i in range(n):
a_sigma[j][i] = a[j][i]*(Scalar(1) - Scalar(2)*sigma[j][i])
C = com_matrix(statement.Gi,statement.F,a_sigma,rC)
# Commit to squared a-values
a_sq = [[Scalar(0) for _ in range(n)] for _ in range(m)]
for j in range(m):
for i in range(n):
a_sq[j][i] = -a[j][i]*a[j][i]
D = com_matrix(statement.Gi,statement.F,a_sq,rD)
# Compute p coefficients
p = [[] for _ in range(N)]
for k in range(N):
decomp_k = decompose(k,n,m)
p[k] = [a[0][decomp_k[0]],delta(decomp_l[0],decomp_k[0])]
for j in range(1,m):
p[k] = convolve(p[k],[a[j][decomp_k[j]],delta(decomp_l[j],decomp_k[j])])
# Challenge
tr = transcript.Transcript('Parallel Groth/Bootle')
tr.update(statement.F)
tr.update(n)
tr.update(m)
tr.update(statement.S)
tr.update(statement.V)
tr.update(statement.S1)
tr.update(statement.V1)
tr.update(A)
tr.update(B)
tr.update(C)
tr.update(D)
mu = tr.challenge()
# Generate proof values
G = PointVector([Z for _ in range(m)])
rho = ScalarVector([random_scalar() for _ in range(m)])
for j in range(m):
for i in range(N):
G[j] += ((statement.S[i] - statement.S1) + mu*(statement.V[i] - statement.V1))*p[i][j]
G[j] += rho[j]*statement.F
# Challenge
tr.update(G)
x = tr.challenge()
f = [ScalarVector([Scalar(0) for _ in range(n-1)]) for _ in range(m)]
for j in range(m):
for i in range(1,n):
f[j][i-1] = sigma[j][i]*x + a[j][i]
zA = rB*x + rA
zC = rC*x + rD
z = (witness.s + mu*witness.v)*x**m
for j in range(m):
z -= rho[j]*x**j
return ParallelCompressedProof(A,B,C,D,G,f,zA,zC,z)
# Verify a commitment-to-zero proof
def verify(statement,proof):
# Check statement consistency
if not isinstance(statement,ParallelCompressedStatement):
raise TypeError('Bad type for parallel statement!')
if not isinstance(proof,ParallelCompressedProof):
raise TypeError('Bad type for parallel proof!')
n = statement.n
m = statement.m
N = n**m
f = [[Scalar(0) for _ in range(n)] for _ in range(m)]
# Transcript and challenges
tr = transcript.Transcript('Parallel Groth/Bootle')
tr.update(statement.F)
tr.update(n)
tr.update(m)
tr.update(statement.S)
tr.update(statement.V)
tr.update(statement.S1)
tr.update(statement.V1)
tr.update(proof.A)
tr.update(proof.B)
tr.update(proof.C)
tr.update(proof.D)
mu = tr.challenge()
tr.update(proof.G)
x = tr.challenge()
# Matrix reconstruction
for j in range(m):
f[j][0] = x
for i in range(1,n):
f[j][i] = proof.f[j][i-1]
f[j][0] -= f[j][i]
# A/B check
if not com_matrix(statement.Gi,statement.F,f,proof.zA) == proof.B*x + proof.A:
raise ArithmeticError('Failed parallel A/B check!')
# C/D check
fx = [ScalarVector([Scalar(0) for _ in range(n)]) for _ in range(m)]
for j in range(m):
for i in range(n):
fx[j][i] = f[j][i]*(x-f[j][i])
if not com_matrix(statement.Gi,statement.F,fx,proof.zC) == proof.C*x + proof.D:
raise ArithmeticError('Failed parallel C/D check!')
# Commitment check
scalars = ScalarVector([])
points = PointVector([])
scalar_S1 = Scalar(0)
scalar_V1 = Scalar(0)
for i in range(N):
s = Scalar(1)
decomp_i = decompose(i,n,m)
for j in range(m):
s *= f[j][decomp_i[j]]
scalars.append(s)
points.append(statement.S[i])
scalars.append(mu*s)
points.append(statement.V[i])
scalar_S1 -= s
scalar_V1 -= mu*s
for j in range(m):
scalars.append(-x**j)
points.append(proof.G[j])
scalars.append(scalar_S1)
points.append(statement.S1)
scalars.append(scalar_V1)
points.append(statement.V1)
if not multiexp(scalars,points) == proof.z*statement.F:
raise ArithmeticError('Failed parallel commitment check!')
return True