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