You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
112 lines
2.3 KiB
112 lines
2.3 KiB
# R1CS-to-CCS (https://eprint.iacr.org/2023/552) Sage prototype
|
|
|
|
# utils
|
|
def matrix_vector_product(M, v):
|
|
n = M.nrows()
|
|
r = [F(0)] * n
|
|
for i in range(0, n):
|
|
for j in range(0, M.ncols()):
|
|
r[i] += M[i][j] * v[j]
|
|
return r
|
|
def hadamard_product(a, b):
|
|
n = len(a)
|
|
r = [None] * n
|
|
for i in range(0, n):
|
|
r[i] = a[i] * b[i]
|
|
return r
|
|
def vec_add(a, b):
|
|
n = len(a)
|
|
r = [None] * n
|
|
for i in range(0, n):
|
|
r[i] = a[i] + b[i]
|
|
return r
|
|
def vec_elem_mul(a, s):
|
|
r = [None] * len(a)
|
|
for i in range(0, len(a)):
|
|
r[i] = a[i] * s
|
|
return r
|
|
# end of utils
|
|
|
|
|
|
# can use any finite field, using a small one for the example
|
|
F = GF(101)
|
|
|
|
# R1CS matrices for: x^3 + x + 5 = y (example from article
|
|
# https://www.vitalik.ca/general/2016/12/10/qap.html )
|
|
A = matrix([
|
|
[F(0), 1, 0, 0, 0, 0],
|
|
[0, 0, 0, 1, 0, 0],
|
|
[0, 1, 0, 0, 1, 0],
|
|
[5, 0, 0, 0, 0, 1],
|
|
])
|
|
B = matrix([
|
|
[F(0), 1, 0, 0, 0, 0],
|
|
[0, 1, 0, 0, 0, 0],
|
|
[1, 0, 0, 0, 0, 0],
|
|
[1, 0, 0, 0, 0, 0],
|
|
])
|
|
C = matrix([
|
|
[F(0), 0, 0, 1, 0, 0],
|
|
[0, 0, 0, 0, 1, 0],
|
|
[0, 0, 0, 0, 0, 1],
|
|
[0, 0, 1, 0, 0, 0],
|
|
])
|
|
|
|
print("R1CS matrices:")
|
|
print("A:", A)
|
|
print("B:", B)
|
|
print("C:", C)
|
|
|
|
|
|
z = [F(1), 3, 35, 9, 27, 30]
|
|
print("z:", z)
|
|
|
|
assert len(z) == A.ncols()
|
|
|
|
n = A.ncols() # == len(z)
|
|
m = A.nrows()
|
|
# l = len(io) # not used for now
|
|
|
|
# check R1CS relation
|
|
Az = matrix_vector_product(A, z)
|
|
Bz = matrix_vector_product(B, z)
|
|
Cz = matrix_vector_product(C, z)
|
|
print("\nR1CS relation check (Az ∘ Bz == Cz):", hadamard_product(Az, Bz) == Cz)
|
|
assert hadamard_product(Az, Bz) == Cz
|
|
|
|
|
|
# Translate R1CS into CCS:
|
|
print("\ntranslate R1CS into CCS:")
|
|
|
|
# fixed parameters (and n, m, l are direct from R1CS)
|
|
t=3
|
|
q=2
|
|
d=2
|
|
S1=[0,1]
|
|
S2=[2]
|
|
S = [S1, S2]
|
|
c0=1
|
|
c1=-1
|
|
c = [c0, c1]
|
|
|
|
M = [A, B, C]
|
|
|
|
print("CCS values:")
|
|
print("n: %s, m: %s, t: %s, q: %s, d: %s" % (n, m, t, q, d))
|
|
print("M:", M)
|
|
print("z:", z)
|
|
print("S:", S)
|
|
print("c:", c)
|
|
|
|
# check CCS relation
|
|
r = [F(0)] * m
|
|
for i in range(0, q):
|
|
hadamard_output = [F(1)]*m
|
|
for j in S[i]:
|
|
hadamard_output = hadamard_product(hadamard_output,
|
|
matrix_vector_product(M[j], z))
|
|
|
|
r = vec_add(r, vec_elem_mul(hadamard_output, c[i]))
|
|
|
|
print("\nCCS relation check (∑ cᵢ ⋅ ◯ Mⱼ z == 0):", r == [0]*m)
|
|
assert r == [0]*m
|