144 lines
5.5 KiB
Python
144 lines
5.5 KiB
Python
from merkle_tree import merkelize, mk_branch, verify_branch
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from utils import get_power_cycle, get_pseudorandom_indices
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from poly_utils import PrimeField
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# Generate an FRI proof that the polynomial that has the specified
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# values at successive powers of the specified root of unity has a
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# degree lower than maxdeg_plus_1
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#
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# We use maxdeg+1 instead of maxdeg because it's more mathematically
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# convenient in this case.
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def prove_low_degree(values, root_of_unity, maxdeg_plus_1, modulus, exclude_multiples_of=0):
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f = PrimeField(modulus)
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print('Proving %d values are degree <= %d' % (len(values), maxdeg_plus_1))
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# If the degree we are checking for is less than or equal to 32,
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# use the polynomial directly as a proof
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if maxdeg_plus_1 <= 16:
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print('Produced FRI proof')
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return [[x.to_bytes(32, 'big') for x in values]]
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# Calculate the set of x coordinates
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xs = get_power_cycle(root_of_unity, modulus)
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assert len(values) == len(xs)
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# Put the values into a Merkle tree. This is the root that the
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# proof will be checked against
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m = merkelize(values)
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# Select a pseudo-random x coordinate
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special_x = int.from_bytes(m[1], 'big') % modulus
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# Calculate the "column" at that x coordinate
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# (see https://vitalik.ca/general/2017/11/22/starks_part_2.html)
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# We calculate the column by Lagrange-interpolating each row, and not
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# directly from the polynomial, as this is more efficient
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quarter_len = len(xs)//4
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x_polys = f.multi_interp_4(
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[[xs[i+quarter_len*j] for j in range(4)] for i in range(quarter_len)],
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[[values[i+quarter_len*j] for j in range(4)] for i in range(quarter_len)]
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)
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column = [f.eval_quartic(p, special_x) for p in x_polys]
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m2 = merkelize(column)
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# Pseudo-randomly select y indices to sample
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ys = get_pseudorandom_indices(m2[1], len(column), 40, exclude_multiples_of=exclude_multiples_of)
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# Compute the Merkle branches for the values in the polynomial and the column
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branches = []
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for y in ys:
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branches.append([mk_branch(m2, y)] +
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[mk_branch(m, y + (len(xs) // 4) * j) for j in range(4)])
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# This component of the proof
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o = [m2[1], branches]
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# Recurse...
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return [o] + prove_low_degree(column, f.exp(root_of_unity, 4),
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maxdeg_plus_1 // 4, modulus, exclude_multiples_of=exclude_multiples_of)
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# Verify an FRI proof
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def verify_low_degree_proof(merkle_root, root_of_unity, proof, maxdeg_plus_1, modulus, exclude_multiples_of=0):
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f = PrimeField(modulus)
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# Calculate which root of unity we're working with
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testval = root_of_unity
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roudeg = 1
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while testval != 1:
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roudeg *= 2
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testval = (testval * testval) % modulus
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# Powers of the given root of unity 1, p, p**2, p**3 such that p**4 = 1
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quartic_roots_of_unity = [1,
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f.exp(root_of_unity, roudeg // 4),
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f.exp(root_of_unity, roudeg // 2),
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f.exp(root_of_unity, roudeg * 3 // 4)]
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# Verify the recursive components of the proof
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for prf in proof[:-1]:
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root2, branches = prf
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print('Verifying degree <= %d' % maxdeg_plus_1)
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# Calculate the pseudo-random x coordinate
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special_x = int.from_bytes(merkle_root, 'big') % modulus
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# Calculate the pseudo-randomly sampled y indices
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ys = get_pseudorandom_indices(root2, roudeg // 4, 40,
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exclude_multiples_of=exclude_multiples_of)
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# For each y coordinate, get the x coordinates on the row, the values on
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# the row, and the value at that y from the column
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xcoords = []
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rows = []
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columnvals = []
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for i, y in enumerate(ys):
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# The x coordinates from the polynomial
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x1 = f.exp(root_of_unity, y)
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xcoords.append([(quartic_roots_of_unity[j] * x1) % modulus for j in range(4)])
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# The values from the original polynomial
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row = [verify_branch(merkle_root, y + (roudeg // 4) * j, prf, output_as_int=True)
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for j, prf in zip(range(4), branches[i][1:])]
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rows.append(row)
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columnvals.append(verify_branch(root2, y, branches[i][0], output_as_int=True))
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# Verify for each selected y coordinate that the four points from the
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# polynomial and the one point from the column that are on that y
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# coordinate are on the same deg < 4 polynomial
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polys = f.multi_interp_4(xcoords, rows)
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for p, c in zip(polys, columnvals):
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assert f.eval_quartic(p, special_x) == c
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# Update constants to check the next proof
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merkle_root = root2
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root_of_unity = f.exp(root_of_unity, 4)
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maxdeg_plus_1 //= 4
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roudeg //= 4
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# Verify the direct components of the proof
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data = [int.from_bytes(x, 'big') for x in proof[-1]]
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print('Verifying degree <= %d' % maxdeg_plus_1)
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assert maxdeg_plus_1 <= 16
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# Check the Merkle root matches up
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mtree = merkelize(data)
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assert mtree[1] == merkle_root
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# Check the degree of the data
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powers = get_power_cycle(root_of_unity, modulus)
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if exclude_multiples_of:
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pts = [x for x in range(len(data)) if x % exclude_multiples_of]
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else:
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pts = range(len(data))
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poly = f.lagrange_interp([powers[x] for x in pts[:maxdeg_plus_1]],
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[data[x] for x in pts[:maxdeg_plus_1]])
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for x in pts[maxdeg_plus_1:]:
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assert f.eval_poly_at(poly, powers[x]) == data[x]
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print('FRI proof verified')
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return True
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