def prove_low_degree(values,
root_of_unity,
maxdeg_plus_1,
modulus,
exclude_multiples_of=0):
f = PrimeField(modulus)
print('Proving %d values are degree <= %d' % (len(values), maxdeg_plus_1))
# If the degree we are checking for is less than or equal to 32,
# use the polynomial directly as a proof
if maxdeg_plus_1 <= 16:
print('Produced FRI proof')
return [[x.to_bytes(32, 'big') for x in values]]
# Calculate the set of x coordinates
xs = get_power_cycle(root_of_unity, modulus)
assert len(values) == len(xs)
# Put the values into a Merkle tree. This is the root that the
# proof will be checked against
m = merkelize(values)
# Select a pseudo-random x coordinate
special_x = int.from_bytes(m[1], 'big') % modulus
# Calculate the "column" at that x coordinate
# (see https://vitalik.ca/general/2017/11/22/starks_part_2.html)
# We calculate the column by Lagrange-interpolating each row, and not
# directly from the polynomial, as this is more efficient
quarter_len = len(xs) // 4
x_polys = f.multi_interp_4(
[[xs[i + quarter_len * j] for j in range(4)]
for i in range(quarter_len)],
[[values[i + quarter_len * j] for j in range(4)]
for i in range(quarter_len)])
column = [f.eval_quartic(p, special_x) for p in x_polys]
m2 = merkelize(column)
# Pseudo-randomly select y indices to sample
ys = get_pseudorandom_indices(m2[1],
len(column),
40,
exclude_multiples_of=exclude_multiples_of)
# Compute the Merkle branches for the values in the polynomial and the column
branches = []
for y in ys:
branches.append(
[mk_branch(m2, y)] +
[mk_branch(m, y + (len(xs) // 4) * j) for j in range(4)])
# This component of the proof
o = [m2[1], branches]
# Recurse...
return [o] + prove_low_degree(column,
f.exp(root_of_unity, 4),
maxdeg_plus_1 // 4,
modulus,
exclude_multiples_of=exclude_multiples_of)
def verify_mimc_proof(inp, steps, round_constants, output, proof):
p_root, d_root, b_root, l_root, branches, fri_proof = proof
start_time = time.time()
assert steps <= 2**32 // extension_factor
assert is_a_power_of_2(steps) and is_a_power_of_2(len(round_constants))
assert len(round_constants) < steps
precision = steps * extension_factor
# Get (steps)th root of unity
G2 = f.exp(7, (modulus-1)//precision)
skips = precision // steps
# Gets the polynomial representing the round constants
skips2 = steps // len(round_constants)
constants_mini_polynomial = fft(round_constants, modulus, f.exp(G2, extension_factor * skips2), inv=True)
# Verifies the low-degree proofs
assert verify_low_degree_proof(l_root, G2, fri_proof, steps * 2, modulus, exclude_multiples_of=extension_factor)
# Performs the spot checks
k1 = int.from_bytes(blake(p_root + d_root + b_root + b'\x01'), 'big')
k2 = int.from_bytes(blake(p_root + d_root + b_root + b'\x02'), 'big')
k3 = int.from_bytes(blake(p_root + d_root + b_root + b'\x03'), 'big')
k4 = int.from_bytes(blake(p_root + d_root + b_root + b'\x04'), 'big')
samples = spot_check_security_factor
positions = get_pseudorandom_indices(l_root, precision, samples,
exclude_multiples_of=extension_factor)
last_step_position = f.exp(G2, (steps - 1) * skips)
for i, pos in enumerate(positions):
x = f.exp(G2, pos)
x_to_the_steps = f.exp(x, steps)
p_of_x = verify_branch(p_root, pos, branches[i*5])
p_of_g1x = verify_branch(p_root, (pos+skips)%precision, branches[i*5 + 1])
d_of_x = verify_branch(d_root, pos, branches[i*5 + 2])
b_of_x = verify_branch(b_root, pos, branches[i*5 + 3])
l_of_x = verify_branch(l_root, pos, branches[i*5 + 4])
zvalue = f.div(f.exp(x, steps) - 1,
x - last_step_position)
k_of_x = f.eval_poly_at(constants_mini_polynomial, f.exp(x, skips2))
# Check transition constraints C(P(x)) = Z(x) * D(x)
assert (p_of_g1x - p_of_x ** 3 - k_of_x - zvalue * d_of_x) % modulus == 0
# Check boundary constraints B(x) * Q(x) + I(x) = P(x)
interpolant = f.lagrange_interp_2([1, last_step_position], [inp, output])
zeropoly2 = f.mul_polys([-1, 1], [-last_step_position, 1])
assert (p_of_x - b_of_x * f.eval_poly_at(zeropoly2, x) -
f.eval_poly_at(interpolant, x)) % modulus == 0
# Check correctness of the linear combination
assert (l_of_x - d_of_x -
k1 * p_of_x - k2 * p_of_x * x_to_the_steps -
k3 * b_of_x - k4 * b_of_x * x_to_the_steps) % modulus == 0
print('Verified %d consistency checks' % spot_check_security_factor)
print('Verified STARK in %.4f sec' % (time.time() - start_time))
return True
def verify_low_degree_proof(merkle_root, root_of_unity, proof, maxdeg_plus_1, modulus, exclude_multiples_of=0):
f = PrimeField(modulus)
# Calculate which root of unity we're working with
testval = root_of_unity
roudeg = 1
while testval != 1:
roudeg *= 2
testval = (testval * testval) % modulus
# Powers of the given root of unity 1, p, p**2, p**3 such that p**4 = 1
quartic_roots_of_unity = [1,
f.exp(root_of_unity, roudeg // 4),
f.exp(root_of_unity, roudeg // 2),
f.exp(root_of_unity, roudeg * 3 // 4)]
# Verify the recursive components of the proof
for prf in proof[:-1]:
root2, branches = prf
print('Verifying degree <= %d' % maxdeg_plus_1)
# Calculate the pseudo-random x coordinate
special_x = int.from_bytes(merkle_root, 'big') % modulus
# Calculate the pseudo-randomly sampled y indices
ys = get_pseudorandom_indices(root2, roudeg // 4, 40,
exclude_multiples_of=exclude_multiples_of)
# For each y coordinate, get the x coordinates on the row, the values on
# the row, and the value at that y from the column
xcoords = []
rows = []
columnvals = []
for i, y in enumerate(ys):
# The x coordinates from the polynomial
x1 = f.exp(root_of_unity, y)
xcoords.append([(quartic_roots_of_unity[j] * x1) % modulus for j in range(4)])
# The values from the original polynomial
row = [verify_branch(merkle_root, y + (roudeg // 4) * j, prf)
for j, prf in zip(range(4), branches[i][1:])]
rows.append(row)
columnvals.append(verify_branch(root2, y, branches[i][0]))
# Verify for each selected y coordinate that the four points from the
# polynomial and the one point from the column that are on that y
# coordinate are on the same deg < 4 polynomial
polys = f.multi_interp_4(xcoords, rows)
for p, c in zip(polys, columnvals):
assert f.eval_quartic(p, special_x) == c
# Update constants to check the next proof
merkle_root = root2
root_of_unity = f.exp(root_of_unity, 4)
maxdeg_plus_1 //= 4
roudeg //= 4
# Verify the direct components of the proof
data = [int.from_bytes(x, 'big') for x in proof[-1]]
print('Verifying degree <= %d' % maxdeg_plus_1)
assert maxdeg_plus_1 <= 16
# Check the Merkle root matches up
mtree = merkelize(data)
assert mtree[1] == merkle_root
# Check the degree of the data
powers = get_power_cycle(root_of_unity, modulus)
if exclude_multiples_of:
pts = [x for x in range(len(data)) if x % exclude_multiples_of]
else:
pts = range(len(data))
poly = f.lagrange_interp([powers[x] for x in pts[:maxdeg_plus_1]],
[data[x] for x in pts[:maxdeg_plus_1]])
for x in pts[maxdeg_plus_1:]:
assert f.eval_poly_at(poly, powers[x]) == data[x]
print('FRI proof verified')
return True
def mk_mimc_proof(inp, steps, round_constants):
start_time = time.time()
# Some constraints to make our job easier
assert steps <= 2**32 // extension_factor
assert is_a_power_of_2(steps) and is_a_power_of_2(len(round_constants))
assert len(round_constants) < steps
precision = steps * extension_factor
# Root of unity such that x^precision=1
G2 = f.exp(7, (modulus-1)//precision)
# Root of unity such that x^steps=1
skips = precision // steps
G1 = f.exp(G2, skips)
# Powers of the higher-order root of unity
xs = get_power_cycle(G2, modulus)
last_step_position = xs[(steps-1)*extension_factor]
# Generate the computational trace
computational_trace = [inp]
for i in range(steps-1):
computational_trace.append(
(computational_trace[-1]**3 + round_constants[i % len(round_constants)]) % modulus
)
output = computational_trace[-1]
print('Done generating computational trace')
# Interpolate the computational trace into a polynomial P, with each step
# along a successive power of G1
computational_trace_polynomial = fft(computational_trace, modulus, G1, inv=True)
p_evaluations = fft(computational_trace_polynomial, modulus, G2)
print('Converted computational steps into a polynomial and low-degree extended it')
skips2 = steps // len(round_constants)
constants_mini_polynomial = fft(round_constants, modulus, f.exp(G1, skips2), inv=True)
constants_polynomial = [0 if i % skips2 else constants_mini_polynomial[i//skips2] for i in range(steps)]
constants_mini_extension = fft(constants_mini_polynomial, modulus, f.exp(G2, skips2))
print('Converted round constants into a polynomial and low-degree extended it')
# Create the composed polynomial such that
# C(P(x), P(g1*x), K(x)) = P(g1*x) - P(x)**3 - K(x)
c_of_p_evaluations = [(p_evaluations[(i+extension_factor)%precision] -
f.exp(p_evaluations[i], 3) -
constants_mini_extension[i % len(constants_mini_extension)])
% modulus for i in range(precision)]
print('Computed C(P, K) polynomial')
# Compute D(x) = C(P(x), P(g1*x), K(x)) / Z(x)
# Z(x) = (x^steps - 1) / (x - x_atlast_step)
z_num_evaluations = [xs[(i * steps) % precision] - 1 for i in range(precision)]
z_num_inv = f.multi_inv(z_num_evaluations)
z_den_evaluations = [xs[i] - last_step_position for i in range(precision)]
d_evaluations = [cp * zd * zni % modulus for cp, zd, zni in zip(c_of_p_evaluations, z_den_evaluations, z_num_inv)]
print('Computed D polynomial')
# Compute interpolant of ((1, input), (x_atlast_step, output))
interpolant = f.lagrange_interp_2([1, last_step_position], [inp, output])
i_evaluations = [f.eval_poly_at(interpolant, x) for x in xs]
zeropoly2 = f.mul_polys([-1, 1], [-last_step_position, 1])
inv_z2_evaluations = f.multi_inv([f.eval_poly_at(zeropoly2, x) for x in xs])
b_evaluations = [((p - i) * invq) % modulus for p, i, invq in zip(p_evaluations, i_evaluations, inv_z2_evaluations)]
print('Computed B polynomial')
# Compute their Merkle root
mtree = merkelize([pval.to_bytes(32, 'big') +
dval.to_bytes(32, 'big') +
bval.to_bytes(32, 'big') for
pval, dval, bval in zip(p_evaluations, d_evaluations, b_evaluations)])
print('Computed hash root')
# Based on the hashes of P, D and B, we select a random linear combination
# of P * x^steps, P, B * x^steps, B and D, and prove the low-degreeness of that,
# instead of proving the low-degreeness of P, B and D separately
k1 = int.from_bytes(blake(mtree[1] + b'\x01'), 'big')
k2 = int.from_bytes(blake(mtree[1] + b'\x02'), 'big')
k3 = int.from_bytes(blake(mtree[1] + b'\x03'), 'big')
k4 = int.from_bytes(blake(mtree[1] + b'\x04'), 'big')
# Compute the linear combination. We don't even both calculating it in
# coefficient form; we just compute the evaluations
G2_to_the_steps = f.exp(G2, steps)
powers = [1]
for i in range(1, precision):
powers.append(powers[-1] * G2_to_the_steps % modulus)
l_evaluations = [(d_evaluations[i] +
p_evaluations[i] * k1 + p_evaluations[i] * k2 * powers[i] +
b_evaluations[i] * k3 + b_evaluations[i] * powers[i] * k4) % modulus
for i in range(precision)]
l_mtree = merkelize(l_evaluations)
print('Computed random linear combination')
# Do some spot checks of the Merkle tree at pseudo-random coordinates, excluding
# multiples of `extension_factor`
branches = []
samples = spot_check_security_factor
positions = get_pseudorandom_indices(l_mtree[1], precision, samples,
exclude_multiples_of=extension_factor)
augmented_positions = sum([[x, (x + skips) % precision] for x in positions], [])
#for pos in positions:
# branches.append(mk_branch(mtree, pos))
# branches.append(mk_branch(mtree, (pos + skips) % precision))
# branches.append(mk_branch(l_mtree, pos))
print('Computed %d spot checks' % samples)
# Return the Merkle roots of P and D, the spot check Merkle proofs,
# and low-degree proofs of P and D
o = [mtree[1],
l_mtree[1],
mk_multi_branch(mtree, augmented_positions),
mk_multi_branch(l_mtree, positions),
prove_low_degree(l_evaluations, G2, steps * 2, modulus, exclude_multiples_of=extension_factor)]
print("STARK computed in %.4f sec" % (time.time() - start_time))
return o