4.4 - proof of operation

scripts/zk/4.4-proof-of-operation.py

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+from bls_py import bls12381
+from bls_py import pairing
+from bls_py import ec
+from bls_py.fields import Fq, Fq2, Fq6, Fq12, bls12381_q as Q
+import random
+import numpy as np
+
+# Section 3.6 from "Why and How zk-SNARK Works"
+
+def rand_scalar():
+    return random.randrange(1, bls12381.q)
+
+#x = rand_scalar()
+#y = ec.y_for_x(x)
+
+g1 = ec.generator_Fq(bls12381)
+g2 = ec.generator_Fq2(bls12381)
+
+null = ec.AffinePoint(Fq(Q, 0), Fq(Q, 1), True, bls12381)
+assert g1 + null == g1
+null2 = ec.AffinePoint(Fq2.zero(Q), Fq2.zero(Q), True, bls12381)
+assert null2 + g2 == g2
+
+#################################
+# Verifier (trusted setup)
+#################################
+
+# samples a random value (a secret)
+s = rand_scalar()
+
+# calculate the shift
+a = rand_scalar()
+
+# calculates encryptions of s for all powers i in 0 to d
+# E(s^i) = g^s^i
+d = 10
+encrypted_powers = [
+    g1 * (s**i) for i in range(d)
+]
+encrypted_powers_g2 = [
+    g2 * (s**i) for i in range(d)
+]
+encrypted_shifted_powers = [
+    g1 * (a * s**i) for i in range(d)
+]
+encrypted_shifted_powers_g2 = [
+    g2 * (a * s**i) for i in range(d)
+]
+
+# evaluates unencrypted target polynomial with s: t(s)
+target = (s - 1) * (s - 2)
+# CRS = common reference string = trusted setup parameters
+target_crs = g1 * target
+alpha_crs = g2 * a
+alpha_crs_g1 = g1 * a
+
+# Proving key = (encrypted_powers, encrypted_shifted_powers)
+# Verify key = (target_crs, alpha_crs)
+
+# encrypted values of s provided to the prover
+# Actual values of s are toxic waste and discarded
+
+#################################
+# Prover
+#################################
+
+# delta shift
+# Removed for now
+#delta = rand_scalar()
+
+# E(p(s)) = p(s)G
+#         = c_d s^d G + ... + c_1 s^1 G + c_0 s^0 G
+#         = s^3 G - 3 s^2 G + 2 s G
+# E(h(s)) = sG
+# t(s) = s^2 - 3s + 2
+# E(h(s)) t(s) = s^3 G - 3 s^2 G + 2 s G
+
+# Lets test these manually:
+
+e_s = encrypted_powers
+e_p_s = e_s[3] - 3 * e_s[2] + 2 * e_s[1]
+e_h_s = e_s[1]
+t_s = s**2 - 3*s + 2
+assert t_s == target
+assert e_p_s == e_h_s * t_s
+
+e_as = encrypted_shifted_powers
+e_p_as = e_as[3] - 3 * e_as[2] + 2 * e_as[1]
+assert e_p_s * a == e_p_as
+
+#############################
+
+left_poly = np.poly1d([3])
+right_poly = np.poly1d([2])
+out_poly = np.poly1d([6])
+
+# x^3 - 3x^2 + 2x
+main_poly = left_poly * right_poly - out_poly
+# (x - 1)
+target_poly = np.poly1d([1, -1])
+
+# Calculates polynomial h(x) = p(x) / t(x)
+cofactor, remainder = main_poly / target_poly
+assert remainder == np.poly1d([0])
+
+# Using encrypted powers and coefficients, evaluates
+# E(p(s)) and E(h(s))
+def evaluate(poly, encrypted_powers, identity):
+    coeffs = list(poly.coef)[::-1]
+    result = identity
+    for power, coeff in zip(encrypted_powers, coeffs):
+        #print(coeff, power)
+        coeff = int(coeff)
+        # I have to do this for some strange reason
+        # Because if coeff is negative and I do += power * coeff
+        # then it gives me a different result than what I expect
+        if coeff < 0:
+            result -= power * (-coeff)
+        else:
+            result += power * coeff
+    # Add delta to the result
+    # Free extra obfuscation to the polynomial
+    return result
+
+assert left_poly * right_poly == out_poly
+
+encrypted_left_poly = evaluate(left_poly, encrypted_powers, null)
+encrypted_right_poly = evaluate(right_poly, encrypted_powers_g2, null2)
+encrypted_out_poly = evaluate(out_poly, encrypted_powers, null)
+
+#assert encrypted_poly == e_p_s
+encrypted_cofactor = evaluate(cofactor, encrypted_powers_g2, null2)
+
+# Alpha shifted powers
+encrypted_shift_left_poly = evaluate(left_poly, encrypted_shifted_powers, null)
+encrypted_shift_right_poly = evaluate(right_poly, encrypted_shifted_powers_g2, null2)
+encrypted_shift_out_poly = evaluate(out_poly, encrypted_shifted_powers, null)
+
+# resulting g^p and g^h are provided to the verifier
+
+# proof = (encrypted_poly, encrypted_cofactor, encrypted_shift_poly)
+
+#################################
+# Verifier
+#################################
+
+# Last check that p = t(s) h
+
+assert pairing.ate_pairing(2 * g1, g2) == pairing.ate_pairing(g1, g2) * pairing.ate_pairing(g1, g2)
+
+# Verify (g^p)^a == g^p'
+# Check polynomial restriction:
+
+def check_polynomial_restriction(encrypted_shift_poly, encrypted_poly):
+    res1 = pairing.ate_pairing(encrypted_shift_poly, g2)
+    res2 = pairing.ate_pairing(encrypted_poly, alpha_crs)
+    assert res1 == res2
+
+def check_polynomial_restriction_swapped(encrypted_shift_poly, encrypted_poly):
+    res1 = pairing.ate_pairing(g1, encrypted_shift_poly)
+    res2 = pairing.ate_pairing(alpha_crs_g1, encrypted_poly)
+    assert res1 == res2
+
+check_polynomial_restriction(encrypted_shift_left_poly, encrypted_left_poly)
+check_polynomial_restriction_swapped(encrypted_shift_right_poly, encrypted_right_poly)
+check_polynomial_restriction(encrypted_shift_out_poly, encrypted_out_poly)
+
+# Valid operation check
+# e(g^l, g^r) == e(g^t, g^h) * e(g^o, g)
+res1 = pairing.ate_pairing(encrypted_left_poly, encrypted_right_poly)
+res2 = pairing.ate_pairing(target_crs, encrypted_cofactor) * \
+       pairing.ate_pairing(encrypted_out_poly, g2)
+assert res1 == res2
+