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