darkfi/scripts/halo/halo2.sage

import numpy as np

q = 0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001
K = GF(q)
P.<X> = K[]

# GENERATOR^{2^s} where t * 2^s + 1 = q with t odd.
# In other words, this is a t root of unity.
generator = K(5)
# There is a large 2^32 order subgroup in this curve because it is 2-adic
#t = (K(q) - 1) / 2^32
delta = generator^(2^32)

def get_omega():
    generator = K(5)
    assert (q - 1) % 2^32 == 0
    # Root of unity
    t = (q - 1) / 2^32
    omega = generator**t

    assert omega != 1
    assert omega^(2^16) != 1
    assert omega^(2^31) != 1
    assert omega^(2^32) == 1

    return omega

# Order of this element is 2^32
omega = get_omega()
k = 4
n = 2^k
omega = omega^(2^32 / n)
assert omega^n == 1

# Arithmetization for:
# sxy + (s - 1)(x + y) - z = 0
# s(s - 1) = 0

# F1(A1 - 1) + F2 (A1 - I) + F3((1 - A1)(A2 + A3) - A4) + F4(A1 A2 A3 - A4) = 0
A = []
F = []

var_one = K(1)
var_zero = K(0)
var_x = K(4)
var_y = K(6)
var_s = K(1)
var_sxy = var_s * var_x * var_y
var_1s_xy = (1 - var_s) * (var_x + var_y)
# Public
var_z = var_sxy + var_1s_xy

# 4 advice columns
# 4 fixed columns
# 1 instance column

# Row 1
A_1_1, A_2_1, A_3_1, A_4_1 = var_one, 0, 0, 0
F_1_1, F_2_1, F_3_1, F_4_1 = 1, 0, 0, 0
I_1 = 0

# Row 2
A_1_2, A_2_2, A_3_2, A_4_2 = var_zero, 0, 0, 0
F_1_2, F_2_2, F_3_2, F_4_2 = 0, 1, 0, 0
I_2 = 0

# Row 3
A_1_3, A_2_3, A_3_3, A_4_3 = var_s, var_s, var_zero, var_zero
F_1_3, F_2_3, F_3_3, F_4_3 = 0, 0, 1, 0
I_3 = 0

# Row 4
A_1_4, A_2_4, A_3_4, A_4_4 = var_s, var_x, var_y, var_sxy
F_1_4, F_2_4, F_3_4, F_4_4 = 0, 0, 0, 1
I_4 = 0

# Row 5
A_1_5, A_2_5, A_3_5, A_4_5 = var_s, var_x, var_y, var_1s_xy
F_1_5, F_2_5, F_3_5, F_4_5 = 0, 0, 1, 0
I_5 = 0

# Row 6
A_1_6, A_2_6, A_3_6, A_4_6 = var_zero, var_sxy, var_1s_xy, var_z
F_1_6, F_2_6, F_3_6, F_4_6 = 0, 0, 1, 0
I_6 = 0

# Row 7
# z = public z
A_1_7, A_2_7, A_3_7, A_4_7 = var_z, 0, 0, 0
F_1_7, F_2_7, F_3_7, F_4_7 = 0, 1, 0, 0
I_7 = var_z

A1 = [A_1_1, A_1_2, A_1_3, A_1_4, A_1_5, A_1_6, A_1_7]
A2 = [A_2_1, A_2_2, A_2_3, A_2_4, A_2_5, A_2_6, A_2_7]
A3 = [A_3_1, A_3_2, A_3_3, A_3_4, A_3_5, A_3_6, A_3_7]
A4 = [A_4_1, A_4_2, A_4_3, A_4_4, A_4_5, A_4_6, A_4_7]
F1 = [F_1_1, F_1_2, F_1_3, F_1_4, F_1_5, F_1_6, F_1_7]
F2 = [F_2_1, F_2_2, F_2_3, F_2_4, F_2_5, F_2_6, F_2_7]
F3 = [F_3_1, F_3_2, F_3_3, F_3_4, F_3_5, F_3_6, F_3_7]
F4 = [F_4_1, F_4_2, F_4_3, F_4_4, F_4_5, F_4_6, F_4_7]
I  = [I_1,   I_2,   I_3,   I_4,   I_5,   I_6,   I_7]

# There should be 5 unused blinding rows.
# see src/plonk/circuit.rs: fn blinding_factors(&self) -> usize;
# We have 9 so we are perfectly fine.

# Add 9 empty rows
assert n - len(A1) == 9
for i in range(9):
    A1.append(K.random_element())
    A2.append(K.random_element())
    A3.append(K.random_element())
    A4.append(K.random_element())
    F1.append(0)
    F2.append(0)
    F3.append(0)
    F4.append(0)
    I.append(K.random_element())

assert (len(A1) == len(A2) == len(A3) == len(A4) == len(F1) == len(F2)
        == len(F3) == len(F4) == len(I) == n)

for A_1_i, A_2_i, A_3_i, A_4_i, F_1_i, F_2_i, F_3_i, F_4_i, I_i in zip(
    A1, A2, A3, A4, F1, F2, F3, F4, I):
    assert (F_1_i * (A_1_i - 1)
            + F_2_i * (A_1_i - I_i)
            + F_3_i * ((1 - A_1_i) * (A_2_i + A_3_i) - A_4_i)
            + F_4_i * (A_1_i * A_2_i * A_3_i - A_4_i)) == 0

a_1_X = P.lagrange_polynomial((omega^i, A_1_i) for i, A_1_i in enumerate(A1))
a_2_X = P.lagrange_polynomial((omega^i, A_2_i) for i, A_2_i in enumerate(A2))
a_3_X = P.lagrange_polynomial((omega^i, A_3_i) for i, A_3_i in enumerate(A3))
a_4_X = P.lagrange_polynomial((omega^i, A_4_i) for i, A_4_i in enumerate(A4))
f_1_X = P.lagrange_polynomial((omega^i, F_1_i) for i, F_1_i in enumerate(F1))
f_2_X = P.lagrange_polynomial((omega^i, F_2_i) for i, F_2_i in enumerate(F2))
f_3_X = P.lagrange_polynomial((omega^i, F_3_i) for i, F_3_i in enumerate(F3))
f_4_X = P.lagrange_polynomial((omega^i, F_4_i) for i, F_4_i in enumerate(F4))
# Treat the instance wire as a 5th advice wire
a_5_X = P.lagrange_polynomial((omega^i, A_5_i) for i, A_5_i in enumerate(I))

for i, (A_1_i, A_2_i, A_3_i, A_4_i, F_1_i, F_2_i, F_3_i, F_4_i, I_i) in \
    enumerate(zip(A1, A2, A3, A4, F1, F2, F3, F4, I)):
    assert a_1_X(omega^i) == A_1_i
    assert a_2_X(omega^i) == A_2_i
    assert a_3_X(omega^i) == A_3_i
    assert a_4_X(omega^i) == A_4_i
    assert a_5_X(omega^i) == I_i
    assert f_1_X(omega^i) == F_1_i
    assert f_2_X(omega^i) == F_2_i
    assert f_3_X(omega^i) == F_3_i
    assert f_4_X(omega^i) == F_4_i

# beta, gamma

#       0   1   2    3             4           5    6       ...     15
# A1:   1,  0,  s,   s,            s,          0,   z
#      16  17  18   19            20          21   22       ...     31
# A2:   -,  -,  s,   x,            x,        sxy,   -
#      32  33  34   35            36          37   38       ...     47
# A3:   -,  -,  0,   y,            y, (1-s)(x+y),   -
#      48  49  50   51            52          53   54       ...     63
# A4:   -,  -,  0, sxy, (1-s)(x + y),          z,   -
#      64  65  66   67            68          69   70       ...     79
# A5:   -,  -,  -,   -,            -,          -,   z
permuted_indices_A1 = []

y = K.random_element()

gate_0 = f_1_X * (a_1_X - 1)
gate_1 = f_2_X * (a_1_X - a_5_X)
gate_2 = f_3_X * ((1 - a_1_X) * (a_2_X + a_3_X) - a_4_X)
gate_3 = f_4_X * (a_1_X * a_2_X * a_3_X - a_4_X)

h = gate_0 + y * gate_1 + y^2 * gate_2 + y^3 * gate_3
t = X^n - 1
for i in range(n):
    assert h(omega^i) == 0