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 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 the row, and not
# directly from the polynomial, as this is more efficient
column = []
for i in range(len(xs) // 4):
x_poly = f.lagrange_interp_4(
[xs[i + len(xs) * j // 4] for j in range(4)],
[values[i + len(values) * j // 4] for j in range(4)],
)
column.append(f.eval_poly_at(x_poly, special_x))
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]
# Interpolate the polynomial for the column
# sub_xs = [xs[i] for i in range(0, len(xs), 4)]
# ypoly = fft(column[:len(sub_xs)], modulus,
# f.exp(root_of_unity, 4), inv=True)
# Recurse...
return [o] + prove_low_degree(column,
f.exp(root_of_unity, 4),
maxdeg_plus_1 // 4,
modulus,
exclude_multiples_of=exclude_multiples_of)