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_mimc_proof(inp, logsteps, logprecision, output, proof):
p_root, d_root, branches, p_proof, d_proof = proof
start_time = time.time()
steps = 2**logsteps
precision = 2**logprecision
# Get (steps)th root of unity
root_of_unity = pow(7, (modulus - 1) // precision, modulus)
skips = precision // steps
# Verifies the low-degree proofs
assert verify_low_degree_proof(p_root, root_of_unity, p_proof, steps)
assert verify_low_degree_proof(d_root, root_of_unity, d_proof, steps * 2)
# Performs the spot checks
samples = spot_check_security_factor // (logprecision - logsteps)
positions = get_indices(blake(p_root + d_root), precision - skips, samples)
for i, pos in enumerate(positions):
# Check C(P(x)) = Z(x) * D(x)
x = pow(root_of_unity, pos, modulus)
p_of_x = verify_branch(p_root, pos, branches[i * 3])
p_of_rx = verify_branch(p_root, pos + skips, branches[i * 3 + 1])
d_of_x = verify_branch(d_root, pos, branches[i * 3 + 2])
zvalue = f.div(
pow(x, steps, modulus) - 1, x - pow(root_of_unity,
(steps - 1) * skips, modulus))
assert (p_of_rx - p_of_x**3 - x - zvalue * d_of_x) % modulus == 0
print('Verified %d consistency checks' % (spot_check_security_factor //
(logprecision - logsteps)))
print('Verified STARK in %.4f sec' % (time.time() - start_time))
return True
def get_indices(seed, modulus, count):
assert modulus < 2**24
data = seed
while len(data) < 4 * count:
data += blake(data[-32:])
return [
int.from_bytes(data[i:i + 4], 'big') % modulus
for i in range(0, count * 4, 4)
]
def get_pseudorandom_indices(seed, modulus, count, exclude_multiples_of=0):
assert modulus < 2**24
data = seed
while len(data) < 4 * count:
data += blake(data[-32:])
if exclude_multiples_of == 0:
return [
int.from_bytes(data[i:i + 4], 'big') % modulus
for i in range(0, count * 4, 4)
]
else:
real_modulus = modulus * (exclude_multiples_of -
1) // exclude_multiples_of
o = [
int.from_bytes(data[i:i + 4], 'big') % real_modulus
for i in range(0, count * 4, 4)
]
return [x + 1 + x // (exclude_multiples_of - 1) for x in o]
def verify_mimc_proof(inp, logsteps, logprecision, output, proof):
p_root, d_root, k_root, l_root, branches, fri_proof = proof
start_time = time.time()
steps = 2**logsteps
precision = 2**logprecision
# Get (steps)th root of unity
root_of_unity = pow(7, (modulus-1)//precision, modulus)
skips = precision // steps
# Verifies the low-degree proofs
assert verify_low_degree_proof(l_root, root_of_unity, fri_proof, steps * 2)
# Performs the spot checks
k = int.from_bytes(blake(p_root + d_root), 'big')
samples = spot_check_security_factor // (logprecision - logsteps)
positions = get_indices(l_root, precision - skips, samples)
for i, pos in enumerate(positions):
# Check C(P(x)) = Z(x) * D(x)
x = pow(root_of_unity, pos, modulus)
p_of_x = verify_branch(p_root, pos, branches[i*5])
p_of_rx = verify_branch(p_root, pos+skips, branches[i*5 + 1])
d_of_x = verify_branch(d_root, pos, branches[i*5 + 2])
k_of_x = verify_branch(k_root, pos, branches[i*5 + 3])
l_of_x = verify_branch(l_root, pos, branches[i*5 + 4])
zvalue = f.div(pow(x, steps, modulus) - 1,
x - pow(root_of_unity, (steps - 1) * skips, modulus))
assert (p_of_rx - p_of_x ** 3 - k_of_x - zvalue * d_of_x) % modulus == 0
assert (l_of_x - d_of_x - k * p_of_x * pow(x, steps, modulus)) % modulus == 0
print('Verified %d consistency checks' % (spot_check_security_factor // (logprecision - logsteps)))
print('Verified STARK in %.4f sec' % (time.time() - start_time))
print('Note: this does not include verifying the Merkle root of the constants tree')
print('This can be done by every client once as a precomputation')
return True
def mk_mimc_proof(inp, logsteps, logprecision):
start_time = time.time()
assert logsteps < logprecision <= 32
steps = 2**logsteps
precision = 2**logprecision
# Root of unity such that x^precision=1
root = pow(7, (modulus-1)//precision, modulus)
# Root of unity such that x^skips=1
skips = precision // steps
subroot = pow(root, skips)
# Powers of the root of unity, our computational trace will be
# along the sequence of roots of unity
xs = get_power_cycle(subroot)
# Generate the computational trace
constants = []
values = [inp]
k = 1
for i in range(steps-1):
values.append((values[-1]**3 + (k ^ 1)) % modulus)
constants.append(k ^ 1)
k = (k * 9) & ((1 << 256) - 1)
constants.append(0)
print('Done generating computational trace')
# Interpolate the computational trace into a polynomial
values_polynomial = fft(values, modulus, subroot, inv=True)
constants_polynomial = fft(constants, modulus, subroot, inv=True)
print('Converted computational steps and constants into a polynomial')
# Create the composed polynomial such that
# C(P(x), P(rx), K(x)) = P(rx) - P(x)**3 - K(x)
term1 = multiply_base(values_polynomial, subroot)
p_evaluations = fft(values_polynomial, modulus, root)
term2 = fft([pow(x, 3, modulus) for x in p_evaluations], modulus, root, inv=True)[:len(values_polynomial) * 3 - 2]
c_of_values = f.sub_polys(f.sub_polys(term1, term2), constants_polynomial)
print('Computed C(P, K) polynomial')
# Compute D(x) = C(P(x), P(rx), K(x)) / Z(x)
# Z(x) = (x^steps - 1) / (x - x_atlast_step)
d = divide_by_xnm1(f.mul_polys(c_of_values,
[modulus-xs[steps-1], 1]),
steps)
# assert f.mul_polys(d, z) == c_of_values
print('Computed D polynomial')
# Evaluate D and K across the entire subgroup
d_evaluations = fft(d, modulus, root)
k_evaluations = fft(constants_polynomial, modulus, root)
print('Evaluated P, D and K')
# Compute their Merkle roots
p_mtree = merkelize(p_evaluations)
d_mtree = merkelize(d_evaluations)
k_mtree = merkelize(k_evaluations)
print('Computed hash root')
# Based on the hashes of P and D, we select a random linear combination
# of P * x^steps and D, and prove the low-degreeness of that, instead of proving
# the low-degreeness of P and D separately
k = int.from_bytes(blake(p_mtree[1] + d_mtree[1]), 'big')
lincomb = f.add_polys(d, f.mul_by_const([0] * steps + values_polynomial, k))
l_evaluations = fft(lincomb, modulus, root)
l_mtree = merkelize(l_evaluations)
print('Computed random linear combination')
# Do some spot checks of the Merkle tree at pseudo-random coordinates
branches = []
samples = spot_check_security_factor // (logprecision - logsteps)
positions = get_indices(l_mtree[1], precision - skips, samples)
for pos in positions:
branches.append(mk_branch(p_mtree, pos))
branches.append(mk_branch(p_mtree, pos + skips))
branches.append(mk_branch(d_mtree, pos))
branches.append(mk_branch(k_mtree, pos))
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 = [p_mtree[1],
d_mtree[1],
k_mtree[1],
l_mtree[1],
branches,
prove_low_degree(lincomb, root, l_evaluations, steps * 2)]
print("STARK computed in %.4f sec" % (time.time() - start_time))
return o
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 roots
p_mtree = merkelize(p_evaluations)
d_mtree = merkelize(d_evaluations)
b_mtree = merkelize(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(p_mtree[1] + d_mtree[1] + b_mtree[1] + b'\x01'), 'big')
k2 = int.from_bytes(blake(p_mtree[1] + d_mtree[1] + b_mtree[1] + b'\x02'), 'big')
k3 = int.from_bytes(blake(p_mtree[1] + d_mtree[1] + b_mtree[1] + b'\x03'), 'big')
k4 = int.from_bytes(blake(p_mtree[1] + d_mtree[1] + b_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)
for pos in positions:
branches.append(mk_branch(p_mtree, pos))
branches.append(mk_branch(p_mtree, (pos + skips) % precision))
branches.append(mk_branch(d_mtree, pos))
branches.append(mk_branch(b_mtree, pos))
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 = [p_mtree[1],
d_mtree[1],
b_mtree[1],
l_mtree[1],
branches,
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
def mk_mimc_proof(inp, logsteps, logprecision):
start_time = time.time()
assert logsteps < logprecision <= 32
steps = 2**logsteps
precision = 2**logprecision
# Root of unity such that x^precision=1
root = pow(7, (modulus - 1) // precision, modulus)
# Root of unity such that x^skips=1
skips = precision // steps
subroot = pow(root, skips)
# Powers of the root of unity, our computational trace will be
# along the sequence of roots of unity
xs = get_power_cycle(subroot)
# Generate the computational trace
values = [inp]
for i in range(steps - 1):
values.append((values[-1]**3 + xs[i]) % modulus)
print('Done generating computational trace')
# Interpolate the computational trace into a polynomial
# values_polynomial = f.lagrange_interp(values, [pow(subroot, i, modulus) for i in range(steps)])
values_polynomial = fft(values, modulus, subroot, inv=True)
print('Computed polynomial')
#for x, v in zip(xs, values):
# assert f.eval_poly_at(values_polynomial, x) == v
# Create the composed polynomial such that
# C(P(x), P(rx)) = P(rx) - P(x)**3 - x
term1 = multiply_base(values_polynomial, subroot)
term2 = fft(
[pow(x, 3, modulus) for x in fft(values_polynomial, modulus, root)],
modulus,
root,
inv=True)[:len(values_polynomial) * 3 - 2]
c_of_values = f.sub_polys(f.sub_polys(term1, term2), [0, 1])
print('Computed C(P) polynomial')
# Compute D(x) = C(P(x)) / Z(x)
# Z(x) = (x^steps - 1) / (x - x_atlast_step)
d = divide_by_xnm1(f.mul_polys(c_of_values, [modulus - xs[steps - 1], 1]),
steps)
# assert f.mul_polys(d, z) == c_of_values
print('Computed D polynomial')
# Evaluate P and D across the entire subgroup
p_evaluations = fft(values_polynomial, modulus, root)
d_evaluations = fft(d, modulus, root)
print('Evaluated P and D')
# Compute their Merkle roots
p_mtree = merkelize(p_evaluations)
d_mtree = merkelize(d_evaluations)
print('Computed hash root')
# Do some spot checks of the Merkle tree at pseudo-random coordinates
branches = []
samples = spot_check_security_factor // (logprecision - logsteps)
positions = get_indices(blake(p_mtree[1] + d_mtree[1]), precision - skips,
samples)
for pos in positions:
branches.append(mk_branch(p_mtree, pos))
branches.append(mk_branch(p_mtree, pos + skips))
branches.append(mk_branch(d_mtree, pos))
print('Computed %d spot checks' % samples)
while len(d) < steps * 2:
d += [0]
# Return the Merkle roots of P and D, the spot check Merkle proofs,
# and low-degree proofs of P and D
o = [
p_mtree[1], d_mtree[1], branches,
prove_low_degree(values_polynomial, root, p_evaluations, steps),
prove_low_degree(d, root, d_evaluations, steps * 2)
]
print("STARK computed in %.4f sec" % (time.time() - start_time))
return o