Add Inner Product Argument (IPA) implementations

Add Inner Product Argument (IPA) implementations:
- Bulletproofs version
- Halo paper version (modified IPA)

ipa.sage

@@ -0,0 +1,291 @@
+# 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(object):
+    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(object):
+    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], b[0], G[0], l, r, L, R
+
+    def verify(self, P, a, v, x_powers, r, u, U, lj, rj, L, R, b_ipa, G_ipa):
+        print("methid verify()")
+        #  b = x_powers
+        #  G = self.gs
+        b = b_ipa # TODO b_0 & G_0 will be computed by the client
+        G = G_ipa
+
+        #  k = int(math.log(self.d, 2))
+        #  s = build_s_from_us(u, k)
+
+        # 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
+
+
+#  def build_s_from_us(u, k):
+#      s = None*k
+#      for i in range(k):
+#          e = 1
+#          for j in range(k):
+#              e = e*u[j]
+#          #  s[i] = 
+#      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
+
+

ipa_test.sage

@@ -0,0 +1,161 @@
+import unittest, operator
+load("ipa.sage")
+
+# Halo paper: https://eprint.iacr.org/2019/1021.pdf
+# Bulletproofs paper: https://eprint.iacr.org/2017/1066.pdf
+
+# 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(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
+            #  r = int(ipa.F.random_element())
+
+            # 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(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
+            
+            # verifier
+            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
+            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()
+
+            P = P + int(v) * U
+
+            # prover
+            a_ipa, b_ipa, G_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, b_ipa, G_ipa)
+            assert verif == True
+
+
+if __name__ == '__main__':
+        unittest.main()

ring-signatures.sage

@@ -80,7 +80,7 @@ def verify(g, R, m, key_image, sig):
     assert len(R) == len(r)
 
     # check that key_image \in G (EC), by l * key_image == 0
-    assert q * key_image == 0 # in sage 0 EC point is interpreted as (0:1:0)
+    assert q * key_image == 0 # in sage 0 EC point is represented as (0:1:0)
 
 
     # for i \in 1,2,...,n