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463 lines
14 KiB
463 lines
14 KiB
# This file contains two Inner Product Argument implementations:
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# - Bulletproofs version: https://eprint.iacr.org/2017/1066.pdf
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# - Halo version: https://eprint.iacr.org/2019/1021.pdf
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# IPA_bulletproofs implements the IPA version from the Bulletproofs paper: https://eprint.iacr.org/2017/1066.pdf
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# https://doc-internal.dalek.rs/bulletproofs/notes/inner_product_proof/index.html
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class IPA_bulletproofs:
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def __init__(self, F, E, g, d):
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self.g = g
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self.F = F
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self.E = E
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self.d = d
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# TODO:
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# Setup:
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self.h = E.random_element() # TMP
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self.gs = random_values(E, d)
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self.hs = random_values(E, d)
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# a: aᵢ ∈ 𝔽 coefficients of p(X)
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# r: blinding factor
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def commit(self, a, b):
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P = inner_product_point(a, self.gs) + inner_product_point(b, self.hs)
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return P
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def evaluate(self, a, x_powers):
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return inner_product_field(a, x_powers)
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def ipa(self, a_, b_, u, U):
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G = self.gs
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H = self.hs
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a = a_
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b = b_
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k = int(math.log(self.d, 2))
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L = [None] * k
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R = [None] * k
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for j in reversed(range(0, k)):
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m = len(a)/2
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a_lo = a[:m]
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a_hi = a[m:]
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b_lo = b[:m]
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b_hi = b[m:]
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H_lo = H[:m]
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H_hi = H[m:]
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G_lo = G[:m]
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G_hi = G[m:]
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# Lⱼ = <a'ₗₒ, G'ₕᵢ> + [lⱼ] H + [<a'ₗₒ, b'ₕᵢ>] U
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L[j] = inner_product_point(a_lo, G_hi) + inner_product_point(b_hi, H_lo) + int(inner_product_field(a_lo, b_hi)) * U
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# Rⱼ = <a'ₕᵢ, G'ₗₒ> + [rⱼ] H + [<a'ₕᵢ, b'ₗₒ>] U
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R[j] = inner_product_point(a_hi, G_lo) + inner_product_point(b_lo, H_hi) + int(inner_product_field(a_hi, b_lo)) * U
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# use the random challenge uⱼ ∈ 𝕀 generated by the verifier
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u_ = u[j] # uⱼ
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u_inv = u[j]^(-1) # uⱼ⁻¹
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a = vec_add(vec_scalar_mul_field(a_lo, u_), vec_scalar_mul_field(a_hi, u_inv))
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b = vec_add(vec_scalar_mul_field(b_lo, u_inv), vec_scalar_mul_field(b_hi, u_))
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G = vec_add(vec_scalar_mul_point(G_lo, u_inv), vec_scalar_mul_point(G_hi, u_))
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H = vec_add(vec_scalar_mul_point(H_lo, u_), vec_scalar_mul_point(H_hi, u_inv))
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assert len(a)==1
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assert len(b)==1
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assert len(G)==1
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assert len(H)==1
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# a, b, G have length=1
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# L, R are the "cross-terms" of the inner product
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return a[0], b[0], G[0], H[0], L, R
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def verify(self, P, a, v, x_powers, u, U, L, R, b_ipa, G_ipa, H_ipa):
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b = b_ipa
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G = G_ipa
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H = H_ipa
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# Q_0 = P' ⋅ ∑ ( [uⱼ²] Lⱼ + [uⱼ⁻²] Rⱼ)
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C = P
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for j in range(len(L)):
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u_ = u[j] # uⱼ
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u_inv = u[j]^(-1) # uⱼ⁻²
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# ∑ ( [uⱼ²] Lⱼ + [uⱼ⁻²] Rⱼ)
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C = C + int(u_^2) * L[j] + int(u_inv^2) * R[j]
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D = int(a) * G + int(b) * H + int(a * b)*U
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return C == D
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# IPA_halo implements the modified IPA from the Halo paper: https://eprint.iacr.org/2019/1021.pdf
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class IPA_halo:
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def __init__(self, F, E, g, d):
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self.g = g
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self.F = F
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self.E = E
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self.d = d
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self.h = E.random_element() # TMP
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self.gs = random_values(E, d)
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self.hs = random_values(E, d)
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# print(" h=", self.h)
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# print(" G=", self.gs)
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# print(" H=", self.hs)
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def commit(self, a, r):
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P = inner_product_point(a, self.gs) + r * self.h
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return P
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def evaluate(self, a, x_powers):
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return inner_product_field(a, x_powers)
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def ipa(self, a_, x_powers, u, U): # prove
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print(" method ipa():")
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G = self.gs
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a = a_
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b = x_powers
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k = int(math.log(self.d, 2))
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l = [None] * k
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r = [None] * k
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L = [None] * k
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R = [None] * k
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for j in reversed(range(0, k)):
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print(" j =", j)
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print(" len(a) = n =", len(a))
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print(" m = n/2 =", len(a)/2)
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m = len(a)/2
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a_lo = a[:m]
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a_hi = a[m:]
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b_lo = b[:m]
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b_hi = b[m:]
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G_lo = G[:m]
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G_hi = G[m:]
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print(" Split into a_lo,hi b_lo,hi, G_lo,hi:")
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print(" a", a)
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print(" a_lo", a_lo)
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print(" a_hi", a_hi)
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print(" b", b)
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print(" b_lo", b_lo)
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print(" b_hi", b_hi)
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print(" G", G)
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print(" G_lo", G_lo)
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print(" G_hi", G_hi)
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l[j] = self.F.random_element() # random blinding factor
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r[j] = self.F.random_element() # random blinding factor
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print(" random blinding factors:")
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print(" l[j]", l[j])
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print(" r[j]", r[j])
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# Lⱼ = <a'ₗₒ, G'ₕᵢ> + [lⱼ] H + [<a'ₗₒ, b'ₕᵢ>] U
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L[j] = inner_product_point(a_lo, G_hi) + int(l[j]) * self.h + int(inner_product_field(a_lo, b_hi)) * U
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# Rⱼ = <a'ₕᵢ, G'ₗₒ> + [rⱼ] H + [<a'ₕᵢ, b'ₗₒ>] U
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R[j] = inner_product_point(a_hi, G_lo) + int(r[j]) * self.h + int(inner_product_field(a_hi, b_lo)) * U
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print(" Compute Lⱼ = <a'ₗₒ, G'ₕᵢ> + [lⱼ] H + [<a'ₗₒ, b'ₕᵢ>] U")
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print(" L[j]", L[j])
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print(" Compute Rⱼ = <a'ₕᵢ, G'ₗₒ> + [rⱼ] H + [<a'ₕᵢ, b'ₗₒ>] U")
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print(" R[j]", R[j])
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# use the random challenge uⱼ ∈ 𝕀 generated by the verifier
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u_ = u[j] # uⱼ
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u_inv = self.F(u[j])^(-1) # uⱼ⁻¹
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print(" u_j", u_)
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print(" u_j^-1", u_inv)
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a = vec_add(vec_scalar_mul_field(a_lo, u_), vec_scalar_mul_field(a_hi, u_inv))
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b = vec_add(vec_scalar_mul_field(b_lo, u_inv), vec_scalar_mul_field(b_hi, u_))
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G = vec_add(vec_scalar_mul_point(G_lo, u_inv), vec_scalar_mul_point(G_hi, u_))
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print(" new a, b, G")
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print(" a =", a)
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print(" b =", b)
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print(" G =", G)
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assert len(a)==1
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assert len(b)==1
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assert len(G)==1
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# a, b, G have length=1
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# l, r are random blinding factors
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# L, R are the "cross-terms" of the inner product
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return a[0], l, r, L, R
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def verify(self, P, a, v, x_powers, r, u, U, lj, rj, L, R):
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print("method verify()")
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# compute P' = P + [v] U
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P = P + int(v) * U
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s = build_s_from_us(u, self.d)
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b = inner_product_field(s, x_powers)
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G = inner_product_point(s, self.gs)
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# synthetic blinding factor
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# r' = r + ∑ ( lⱼ uⱼ² + rⱼ uⱼ⁻²)
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print(" synthetic blinding factor r' = r + ∑ ( lⱼ uⱼ² + rⱼ uⱼ⁻²)")
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r_ = r
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print(" r_ =", r_)
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# Q_0 = P' ⋅ ∑ ( [uⱼ²] Lⱼ + [uⱼ⁻²] Rⱼ)
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print(" Q_0 = P' ⋅ ∑ ( [uⱼ²] Lⱼ + [uⱼ⁻²] Rⱼ)")
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Q_0 = P
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print(" Q_0 =", Q_0)
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for j in range(len(u)):
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print(" j =", j)
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u_ = u[j] # uⱼ
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u_inv = u[j]^(-1) # uⱼ⁻²
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# ∑ ( [uⱼ²] Lⱼ + [uⱼ⁻²] Rⱼ)
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Q_0 = Q_0 + int(u[j]^2) * L[j] + int(u_inv^2) * R[j]
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print(" Q_0 =", Q_0)
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r_ = r_ + lj[j] * (u_^2) + rj[j] * (u_inv^2)
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print(" r_ =", r_)
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Q_1 = int(a) * G + int(r_) * self.h + int(a * b)*U
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print(" Q_1", Q_1)
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# Q_1_ = int(a) * (G + int(b)*U) + int(r_) * self.h
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return Q_0 == Q_1
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# s = (
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# u₁⁻¹ u₂⁻¹ … uₖ⁻¹,
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# u₁ u₂⁻¹ … uₖ⁻¹,
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# u₁⁻¹ u₂ … uₖ⁻¹,
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# u₁ u₂ … uₖ⁻¹,
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# ⋮ ⋮ ⋮
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# u₁ u₂ … uₖ
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# )
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def build_s_from_us(u, d):
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k = int(math.log(d, 2))
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s = [1]*d
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t = d
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for j in reversed(range(k)):
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t = t/2
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c = 0
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for i in range(d):
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if c<t:
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s[i] = s[i] * u[j]^(-1)
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else:
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s[i] = s[i] * u[j]
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c = c+1
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if c>=t*2:
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c=0
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return s
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def powers_of(g, d):
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r = [None] * d
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for i in range(d):
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r[i] = g^i
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return r
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def multiples_of(g, d):
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r = [None] * d
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for i in range(d):
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r[i] = g*i
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return r
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def random_values(G, d):
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r = [None] * d
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for i in range(d):
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r[i] = G.random_element()
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return r
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def inner_product_field(a, b):
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assert len(a) == len(b)
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c = 0
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for i in range(len(a)):
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c = c + a[i] * b[i]
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return c
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def inner_product_point(a, b):
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assert len(a) == len(b)
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c = 0
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for i in range(len(a)):
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c = c + int(a[i]) * b[i]
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return c
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def vec_add(a, b):
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assert len(a) == len(b)
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return [x + y for x, y in zip(a, b)]
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def vec_mul(a, b):
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assert len(a) == len(b)
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return [x * y for x, y in zip(a, b)]
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def vec_scalar_mul_field(a, n):
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r = [None]*len(a)
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for i in range(len(a)):
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r[i] = a[i]*n
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return r
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def vec_scalar_mul_point(a, n):
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r = [None]*len(a)
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for i in range(len(a)):
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r[i] = a[i]*int(n)
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return r
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# Tests
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import unittest, operator
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# Ethereum elliptic curve
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p = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F
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a = 0
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b = 7
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Fp = GF(p)
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E = EllipticCurve(Fp, [a,b])
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GX = 0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798
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GY = 0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8
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g = E(GX,GY)
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n = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141
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h = 1
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q = g.order()
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Fq = GF(q)
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# simpler curve values
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# p = 19
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# Fp = GF(p)
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# E = EllipticCurve(Fp,[0,3])
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# g = E(1, 2)
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# q = g.order()
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# Fq = GF(q)
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print(E)
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print(Fq)
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assert is_prime(p)
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assert is_prime(q)
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assert g * q == 0
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class TestUtils(unittest.TestCase):
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def test_vecs(self):
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a = [1, 2, 3, 4, 5]
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b = [1, 2, 3, 4, 5]
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c = vec_scalar_mul_field(a, 10)
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assert c == [10, 20, 30, 40, 50]
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c = inner_product_field(a, b)
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assert c == 55
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# check that <a, b> with b = (1, x, x^2, ..., x^{d-1}) is the same
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# than evaluating p(x) with coefficients a_i, at x
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a = [Fq(1), Fq(2), Fq(3), Fq(4), Fq(5), Fq(6), Fq(7), Fq(8)]
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z = Fq(3)
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b = powers_of(z, 8)
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c = inner_product_field(a, b)
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x = PolynomialRing(Fq, 'x').gen()
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px = 1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6 + 8*x^7
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assert c == px(x=z)
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class TestIPA_bulletproofs(unittest.TestCase):
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def test_inner_product_argument(self):
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d = 8
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ipa = IPA_bulletproofs(Fq, E, g, d)
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# prover
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# p(x) = 1 + 2x + 3x² + 4x³ + 5x⁴ + 6x⁵ + 7x⁶ + 8x⁷
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a = [ipa.F(1), ipa.F(2), ipa.F(3), ipa.F(4), ipa.F(5), ipa.F(6), ipa.F(7), ipa.F(8)]
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x = ipa.F(3)
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b = powers_of(x, ipa.d) # = b
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# prover
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P = ipa.commit(a, b)
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print("commit", P)
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v = ipa.evaluate(a, b)
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print("v", v)
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# verifier generate random challenges {uᵢ} ∈ 𝕀 and U ∈ 𝔾
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U = ipa.E.random_element()
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k = int(math.log(d, 2))
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u = [None] * k
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for j in reversed(range(0, k)):
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u[j] = ipa.F.random_element()
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while (u[j] == 0): # prevent u[j] from being 0
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u[j] = ipa.F.random_element()
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P = P + int(inner_product_field(a, b)) * U
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# prover
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a_ipa, b_ipa, G_ipa, H_ipa, L, R = ipa.ipa(a, b, u, U)
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# verifier
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print("P", P)
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print("a_ipa", a_ipa)
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verif = ipa.verify(P, a_ipa, v, b, u, U, L, R, b_ipa, G_ipa, H_ipa)
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print("Verification:", verif)
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assert verif == True
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class TestIPA_halo(unittest.TestCase):
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def test_homomorphic_property(self):
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ipa = IPA_halo(Fq, E, g, 5)
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a = [1, 2, 3, 4, 5]
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b = [1, 2, 3, 4, 5]
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c = vec_add(a, b)
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assert c == [2,4,6,8,10]
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r = int(ipa.F.random_element())
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s = int(ipa.F.random_element())
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vc_a = ipa.commit(a, r)
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vc_b = ipa.commit(b, s)
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# com(a, r) + com(b, s) == com(a+b, r+s)
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expected_vc_c = ipa.commit(vec_add(a, b), r+s)
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vc_c = vc_a + vc_b
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assert vc_c == expected_vc_c
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def test_inner_product_argument(self):
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d = 8
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ipa = IPA_halo(Fq, E, g, d)
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# prover
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# p(x) = 1 + 2x + 3x² + 4x³ + 5x⁴ + 6x⁵ + 7x⁶ + 8x⁷
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a = [ipa.F(1), ipa.F(2), ipa.F(3), ipa.F(4), ipa.F(5), ipa.F(6), ipa.F(7), ipa.F(8)]
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x = ipa.F(3)
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x_powers = powers_of(x, ipa.d) # = b
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# blinding factor
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r = int(ipa.F.random_element())
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# prover
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P = ipa.commit(a, r)
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print("commit", P)
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v = ipa.evaluate(a, x_powers)
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print("v", v)
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# verifier generate random challenges {uᵢ} ∈ 𝕀 and U ∈ 𝔾
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# This might be obtained from the hash of the transcript
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# (Fiat-Shamir heuristic for non-interactive version)
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U = ipa.E.random_element()
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k = int(math.log(ipa.d, 2))
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u = [None] * k
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for j in reversed(range(0, k)):
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u[j] = ipa.F.random_element()
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while (u[j] == 0): # prevent u[j] from being 0
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u[j] = ipa.F.random_element()
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# prover
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a_ipa, lj, rj, L, R = ipa.ipa(a, x_powers, u, U)
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# verifier
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print("P", P)
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print("a_ipa", a_ipa)
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print("\n Verify:")
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verif = ipa.verify(P, a_ipa, v, x_powers, r, u, U, lj, rj, L, R)
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assert verif == True
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if __name__ == '__main__':
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unittest.main()
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