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concrete/scripts.py
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import matplotlib.pyplot as plt
from sage.stats.distributions.discrete_gaussian_lattice import DiscreteGaussianDistributionIntegerSampler
from concrete_params import concrete_LWE_params, concrete_RLWE_params
import numpy as np
#from pytablewriter import MarkdownTableWriter
from hybrid_decoding import parameter_search
from random import uniform
# easier to just load the estimator
import estimator.estimator as est
# define the four cost models used for Concrete (2 classical, 2 quantum)
# note that classical and quantum are the two models used in the "HE Std"
def classical(beta, d, B):
return ZZ(2) ** RR(0.292 * beta + 16.4 + log(8 * d, 2))
def quantum(beta, d, B):
return ZZ(2) ** RR(0.265 * beta + 16.4 + log(8 * d, 2))
def classical_conservative(beta, d, B):
return ZZ(2) ** RR(0.292 * beta)
def quantum_conservative(beta, d, B):
return ZZ(2) ** RR(0.265 * beta)
# we add an enumeration model for completeness
cost_models = [classical, quantum, classical_conservative, quantum_conservative, est.BKZ.enum]
def estimate_lwe_nocrash(n, alpha, q, secret_distribution,
reduction_cost_model=est.BKZ.sieve, m=oo):
"""
A function to estimate the complexity of LWE, whilst skipping over any attacks which crash.s
"""
success = 0
try:
# we begin by trying all four attacks (usvp, dual, dec, mitm)
estimate = est.estimate_lwe(n, alpha, q, secret_distribution=secret_distribution,
reduction_cost_model=reduction_cost_model, m=oo,
skip={"bkw", "dec", "arora-gb"})
success = 1
except Exception as e:
print(e)
if success == 0:
try:
# dual crashes most often, so try skipping dual first
estimate = est.estimate_lwe(n, alpha, q, secret_distribution=secret_distribution,
reduction_cost_model=reduction_cost_model, m=oo,
skip={"bkw", "dec", "arora-gb", "dual"})
success = 1
except Exception as e:
print(e)
if success == 0:
try:
# next, skip mitm
estimate = est.estimate_lwe(n, alpha, q, secret_distribution=secret_distribution,
reduction_cost_model=reduction_cost_model, m=oo,
skip={"mitm", "bkw", "dec", "arora-gb", "dual"})
except Exception as e:
print(e)
security_level = get_security_level(estimate)
return security_level
def get_security_level(estimate, decimal_places = 2):
""" Function to get the security level from an LWE Estimator output,
i.e. returns only the bit-security level (without the attack params)
:param estimate: the input estimate
:param decimal_places: the number of decimal places
EXAMPLE:
sage: x = estimate_lwe(n = 256, q = 2**32, alpha = RR(8/2**32))
sage: get_security_level(x)
33.8016789754458
"""
levels = []
# use try/except to cover cases where we only consider one or two attacks
try:
levels.append(estimate["usvp"]["rop"])
except:
pass
try:
levels.append(estimate["dec"]["rop"])
except:
pass
try:
levels.append(estimate["dual"]["rop"])
except:
pass
try:
levels.append(estimate["mitm"]["rop"])
except:
pass
# take the minimum attack cost (in bits)
security_level = round(log(min(levels), 2), decimal_places)
return security_level
def get_all_security_levels(params):
""" A function which gets the security levels of a collection of TFHE parameters,
using the four cost models: classical, quantum, classical_conservative, and
quantum_conservative
:param params: a dictionary of LWE parameter sets (see concrete_params)
EXAMPLE:
sage: X = get_all_security_levels(concrete_LWE_params)
sage: X
[['LWE128_256',
126.692189756144,
117.566189756144,
98.6960000000000,
89.5700000000000], ...]
"""
RESULTS = []
for param in params:
results = [param]
x = params["{}".format(param)]
n = x["n"] * x["k"]
q = 2 ** 32
sd = 2 ** (x["sd"]) * q
alpha = sqrt(2 * pi) * sd / RR(q)
secret_distribution = (0, 1)
# assume access to an infinite number of samples
m = oo
for model in cost_models:
try:
model = model[0]
except:
model = model
estimate = parameter_search(mitm = True, reduction_cost_model = est.BKZ.sieve, n = n, q = q, alpha = alpha, m = m, secret_distribution = secret_distribution)
results.append(get_security_level(estimate))
RESULTS.append(results)
return RESULTS
def get_hybrid_security_levels(params):
""" A function which gets the security levels of a collection of TFHE parameters,
using the four cost models: classical, quantum, classical_conservative, and
quantum_conservative
:param params: a dictionary of LWE parameter sets (see concrete_params)
EXAMPLE:
sage: X = get_all_security_levels(concrete_LWE_params)
sage: X
[['LWE128_256',
126.692189756144,
117.566189756144,
98.6960000000000,
89.5700000000000], ...]
"""
RESULTS = []
for param in params:
results = [param]
x = params["{}".format(param)]
n = x["n"] * x["k"]
q = 2 ** 32
sd = 2 ** (x["sd"]) * q
alpha = sqrt(2 * pi) * sd / RR(q)
secret_distribution = (0, 1)
# assume access to an infinite number of papers
m = oo
model = est.BKZ.sieve
estimate = parameter_search(mitm = True, reduction_cost_model = est.BKZ.sieve, n = n, q = q, alpha = alpha, m = m, secret_distribution = secret_distribution)
results.append(get_security_level(estimate))
RESULTS.append(results)
return RESULTS
def latexit(results):
"""
A function which takes the output of get_all_security_levels() and
turns it into a latex table
:param results: the security levels
sage: X = get_all_security_levels(concrete_LWE_params)
sage: latextit(X)
\begin{tabular}{llllll}
LWE128_256 & $126.69$ & $117.57$ & $98.7$ & $89.57$ & $217.55$ \\
LWE128_512 & $135.77$ & $125.92$ & $106.58$ & $96.73$ & $218.53$ \\
LWE128_638 & $135.27$ & $125.49$ & $105.7$ & $95.93$ & $216.81$ \\
[...]
"""
return latex(table(results))
def markdownit(results, headings = ["Parameter Set", "Classical", "Quantum", "Classical (c)", "Quantum (c)", "Enum"]):
"""
A function which takes the output of get_all_security_levels() and
turns it into a markdown table
:param results: the security levels
sage: X = get_all_security_levels(concrete_LWE_params)
sage: markdownit(X)
# estimates
|Parameter Set|Classical|Quantum|Classical (c)|Quantum (c)| Enum |
|-------------|---------|-------|-------------|-----------|------|
|LWE128_256 |126.69 |117.57 |98.7 |89.57 |217.55|
|LWE128_512 |135.77 |125.92 |106.58 |96.73 |218.53|
|LWE128_638 |135.27 |125.49 |105.7 |95.93 |216.81|
[...]
"""
writer = MarkdownTableWriter(value_matrix = results, headers = headings, table_name = "estimates")
writer.write_table()
return writer
def inequality(x, y):
""" A function which compresses the conditions
x < y and x > y into a single condition via a
multiplier
"""
if x <= y:
return 1
if x > y:
return -1
def automated_param_select_n(sd, n=None, q=2 ** 32, reduction_cost_model=est.BKZ.sieve, secret_distribution=(0, 1),
target_security=128):
""" A function used to generate the smallest value of n which allows for
target_security bits of security, for the input values of (sd,q)
:param sd: the standard deviation of the error
:param n: an initial value of n to use in optimisation, guessed if None
:param q: the LWE modulus (q = 2**32, 2**64 in TFHE)
:param reduction_cost_model: the BKZ cost model considered, BKZ.sieve is default
:param secret_distribution: the LWE secret distribution
:param target_security: the target number of bits of security, 128 is default
EXAMPLE:
sage: X = automated_param_select_n(sd = -25, q = 2**32)
sage: X
1054
"""
if n is None:
# pick some random n which gets us close (based on concrete_LWE_params)
n = sd * (-25) * (target_security/80)
sd = 2 ** sd * q
alpha = sqrt(2 * pi) * sd / RR(q)
# initial estimate, to determine if we are above or below the target security level
security_level = estimate_lwe_nocrash(n, alpha, q, secret_distribution=secret_distribution,
reduction_cost_model=reduction_cost_model, m=oo)
z = inequality(security_level, target_security)
while z * security_level < z * target_security and n > 80:
n += z * 8
alpha = sqrt(2 * pi) * sd / RR(q)
security_level = estimate_lwe_nocrash(n, alpha, q, secret_distribution=secret_distribution,
reduction_cost_model=reduction_cost_model, m=oo)
if (-1 * sd > 0):
print("target security level is unatainable")
break
# final estimate (we went too far in the above loop)
if security_level < target_security:
n -= z * 8
alpha = sqrt(2 * pi) * sd / RR(q)
security_level = estimate_lwe_nocrash(n, alpha, q, secret_distribution=secret_distribution,
reduction_cost_model=reduction_cost_model, m=oo)
print("the finalised parameters are n = {}, log2(sd) = {}, log2(q) = {}, with a security level of {}-bits".format(n,
sd,
log(q,
2),
security_level))
# final sanity check so we don't return insecure (or inf) parameters
if security_level < target_security or security_level == oo:
n = None
return n
def automated_param_select_sd(n, sd=None, q=2**32, reduction_cost_model=est.BKZ.sieve, secret_distribution=(0, 1),
target_security=128):
""" A function used to generate the smallest value of sd which allows for
target_security bits of security, for the input values of (n,q)
:param n: the LWE dimension
:param sd: an initial value of sd to use in optimisation, guessed if None
:param q: the LWE modulus (q = 2**32, 2**64 in TFHE)
:param reduction_cost_model: the BKZ cost model considered, BKZ.sieve is default
:param secret_distribution: the LWE secret distribution
:param target_security: the target number of bits of security, 128 is default
EXAMPLE
sage: X = automated_param_select_sd(n = 1054, q = 2**32)
sage: X
-26
"""
if sd is None:
# pick some random sd which gets us close (based on concrete_LWE_params)
sd = round(n * 80 / (target_security * (-25)))
# make sure sd satisfies q * sd > 1
sd = max(sd, -(log(q,2) - 2))
sd_ = (2 ** sd) * q
alpha = sqrt(2 * pi) * sd_ / RR(q)
# initial estimate, to determine if we are above or below the target security level
security_level = estimate_lwe_nocrash(n, alpha, q, secret_distribution=secret_distribution,
reduction_cost_model=reduction_cost_model, m=oo)
z = inequality(security_level, target_security)
while z * security_level < z * target_security and sd > -log(q,2):
sd += z * (0.5)
sd_ = (2 ** sd) * q
alpha = sqrt(2 * pi) * sd_ / RR(q)
security_level = estimate_lwe_nocrash(n, alpha, q, secret_distribution=secret_distribution,
reduction_cost_model=reduction_cost_model, m=oo)
## THIS IS WHERE THE PROBLEM IS, CORRECT THIS CONDITION?
if (sd > log(q, 2)):
print("target security level is unatainable")
return None
# final estimate (we went too far in the above loop)
if security_level < target_security:
sd -= z * (0.5)
sd_ = (2 ** sd) * q
alpha = sqrt(2 * pi) * sd_ / RR(q)
security_level = estimate_lwe_nocrash(n, alpha, q, secret_distribution=secret_distribution,
reduction_cost_model=reduction_cost_model, m=oo)
print("the finalised parameters are n = {}, log2(sd) = {}, log2(q) = {}, with a security level of {}-bits".format(n,
sd,
log(q,
2),
security_level))
return sd
def generate_parameter_matrix(n_range, sd=None, q=2**32, reduction_cost_model=est.BKZ.sieve,
secret_distribution=(0, 1), target_security=128):
"""
:param n_range: a tuple (n_min, n_max) giving the values of n for which to generate parameters
:param sd: the standard deviation of the LWE error
:param q: the LWE modulus (q = 2**32, 2**64 in TFHE)
:param reduction_cost_model: the BKZ cost model considered, BKZ.sieve is default
:param secret_distribution: the LWE secret distribution
:param target_security: the target number of bits of security, 128 is default
TODO: we should probably parallelise this function for speed
TODO: code seems to fail when the initial estimate is < target_security bits
EXAMPLE:
sage: X = generate_parameter_matrix([788, 790])
sage: X
[(788, 4294967296, -20.0), (789, 4294967296, -20.0)]
"""
RESULTS = []
# grab min and max value/s of n, with a granularity (if given as input)
try:
(n_min, n_max, gran) = n_range
except:
(n_min, n_max) = n_range
gran = 1
sd_ = sd
for n in range(n_min, n_max + 1, gran):
sd = automated_param_select_sd(n, sd=sd_, q=q, reduction_cost_model=reduction_cost_model,
secret_distribution=secret_distribution, target_security=target_security)
sd_ = sd
RESULTS.append((n, q, sd))
return RESULTS
def generate_parameter_matrix_sd(sd_range, n=None, q=2**32, reduction_cost_model=est.BKZ.sieve,
secret_distribution=(0, 1), target_security=128):
"""
:param sd_range: a tuple (sd_min, sd_max) giving the values of sd for which to generate parameters
:param sd: the standard deviation of the LWE error
:param q: the LWE modulus (q = 2**32, 2**64 in TFHE)
:param reduction_cost_model: the BKZ cost model considered, BKZ.sieve is default
:param secret_distribution: the LWE secret distribution
:param target_security: the target number of bits of security, 128 is default
TODO: we should probably parallelise this function for speed
TODO: code seems to fail when the initial estimate is < target_security bits
EXAMPLE:
sage: X = generate_parameter_matrix([788, 790])
sage: X
[(788, 4294967296, -20.0), (789, 4294967296, -20.0)]
"""
RESULTS = []
# grab min and max value/s of n
(sd_min, sd_max) = sd_range
n = n
for sd in range(sd_min, sd_max + 1):
n = automated_param_select_n(sd, n=n, q=q, reduction_cost_model=reduction_cost_model,
secret_distribution=secret_distribution, target_security=target_security)
RESULTS.append((n, q, sd))
return RESULTS
def generate_parameter_step(results, label = None, torus_sd = True):
"""
Plot results
:param results: an output of generate_parameter_matrix
returns: a step plot of chosen parameters
EXAMPLE:
X = generate_parameter_matrix([700, 790])
generate_parameter_step(X)
plt.show()
"""
N = []
SD = []
for (n, q, sd) in results:
N.append(n)
if torus_sd:
SD.append(sd)
else:
SD.append(sd + log(q,2))
plt.plot(N, SD, label = label)
plt.legend(loc = "upper right")
return plt
# dual bug example
# n = 256; q = 2**32; sd = 2**(-4); reduction_cost_model = BKZ.sieve
# _ = estimate_lwe(n, alpha, q, reduction_cost_model)
def test_params(n, q, sd, secret_distribution):
sd = sd * q
alpha = RR(sqrt(2*pi) * sd / q)
est = est.estimate_lwe(n, alpha, q, secret_distribution = secret_distribution, reduction_cost_model = est.BKZ.sieve, skip = ("arora-gb", "bkw", "dec"))
return est
def generate_iso_lines(N = [256, 2048], SD = [0, 32], q = 2**32):
RESULTS = []
for n in range(N[0], N[1] + 1, 1):
for sd in range(SD[0], SD[1] + 1, 1):
sd = 2**sd
alpha = sqrt(2*pi) * sd / q
try:
est = est.estimate_lwe(n, alpha, q, secret_distribution = (0,1), reduction_cost_model = est.BKZ.sieve, skip = ("bkw", "arora-gb", "dec"))
est = get_security_level(est, 2)
except:
est = est.estimate_lwe(n, alpha, q, secret_distribution = (0,1), reduction_cost_model = est.BKZ.sieve, skip = ("bkw", "arora-gb", "dual", "dec"))
est = get_security_level(est, 2)
RESULTS.append((n, sd, est))
return RESULTS
def plot_iso_lines(results):
x1 = []
x2 = []
for z in results:
x1.append(z[0])
# use log(q)
# use -ve values to match Pascal's diagram
x2.append(z[2])
plt.plot(x1, x2)
return plt
def test_multiple_sd(n, q, secret_distribution, reduction_cost_model, split = 33, m = oo):
est = []
Y = []
for sd_ in np.linspace(0,32,split):
Y.append(sd_)
sd = (2** (-1 * sd_))* q
alpha = sqrt(2*pi) * sd / q
try:
es = est.estimate_lwe(n=512, alpha=alpha, q=q, secret_distribution=(0, 1), reduction_cost_model = reduction_cost_model,
skip=("bkw", "dec", "arora-gb"), m = m)
except:
es = est.estimate_lwe(n=512, alpha=alpha, q=q, secret_distribution=(0, 1), reduction_cost_model = reduction_cost_model,
skip=("bkw", "dec", "arora-gb", "dual"), m = m)
est.append(get_security_level(es,2))
return est, Y
## parameter curves
def get_parameter_curves_data_sd(sec_levels, sd_range, q):
Results = []
for sec in sec_levels:
try:
result_sec = generate_parameter_matrix_sd(n = None, sd_range=sd_range, q=q, reduction_cost_model=est.BKZ.sieve,
secret_distribution=(0,1), target_security=sec)
Results.append(result_sec)
except:
pass
return Results
def get_parameter_curves_data_n(sec_levels, n_range, q):
Results = []
for sec in sec_levels:
try:
result_sec = generate_parameter_matrix(n_range, sd = None, q=q, reduction_cost_model=est.BKZ.sieve,
secret_distribution=(0,1), target_security=sec)
Results.append(result_sec)
except:
pass
return Results
def interpolate_result(result, log_q):
import numpy as np
# linear function interpolation
x = []
y = []
# 1. filter out any points which reccomend sd = -log(q) + 2
new_result= []
for res in result:
if res[2] >= - log_q + 2:
new_result.append(res)
result = new_result
for res in result:
x.append(res[0])
y.append(res[2])
(a,b) = np.polyfit(x, y, 1)
return (a,b)
def plot_interpolants(interpolants, n_range, log_q, degree = 1):
for x in interpolants:
if degree == 1:
vals = [x[0] * n + x[1] for n in range(n_range[0],n_range[1])]
elif degree == 2:
vals = [x[0] * n**2 + x[1]*n + x[2] for n in range(n_range[0],n_range[1])]
# any values which fall outside of the range and edited to give at least two bits of noise.
vvals = []
for v in vals:
if v < -log_q + 2:
vvals.append(-log_q + 2)
else:
vvals.append(v)
plt.plot(range(n_range[0], n_range[0] + len(vvals)), vvals)
return 0
## currently running
# sage: n_range = (256, 2048, 16)
# sage: sec_levels = [80 + 16*k for k in range(0,12)]
# sage: results = get_parameter_curves_data_n(sec_levels, n_range, q = 2**64)
def verify_results(results, security_level, secret_distribution = (0,1), reduction_cost_model = est.BKZ.sieve):
""" A function which verifies that a set of results match a given security level
:param results : a set of tuples of the form (n, q, sd)
:param security_level: the target security level for these params
"""
estimates = []
# 1. Grab the parameters
for (n, q, sd) in results:
if sd is not None:
sd = 2**sd
alpha = sqrt(2*pi) * sd
# 2. Test that these parameters satisfy the given security level
try:
estimate = est.estimate_lwe(n, alpha, q, secret_distribution=secret_distribution,
reduction_cost_model=reduction_cost_model, m=oo, skip = {"bkw","dec","arora-gb"})
except:
estimate = est.estimate_lwe(n, alpha, q, secret_distribution=secret_distribution,
reduction_cost_model=reduction_cost_model, m=oo,
skip={"bkw", "dec", "arora-gb", "dual"})
estimates.append(estimate)
return estimates
def verify_interpolants(interpolant, n_range, log_q, secret_distribution = (0,1), reduction_cost_model = est.BKZ.sieve):
estimates = []
q = 2**log_q
(a, b) = interpolant
for n in range(n_range[0], n_range[1]):
print(n)
# we take the max here to ensure that the "cut-off" for the minimal error is met.
sd = max(a * n + b, -log_q + 2)
sd = 2 ** sd
alpha = sqrt(2*pi) * sd
security_level = estimate_lwe_nocrash(n, alpha, q, secret_distribution=secret_distribution,
reduction_cost_model=reduction_cost_model, m=oo)
print(security_level)
if security_level == oo:
security_level = 0
estimates.append(security_level)
return estimates
def get_zama_curves():
# hardcode the parameters for now
n_range = [128, 2048, 32]
sec_levels = [80 + 16*k for k in range(0,12)]
results = get_parameter_curves_data_n(sec_levels, n_range, q = 2**64)
return results
def test_curves():
# a small hardcoded example for testing purposes
n_range = [256, 1024, 128]
sec_levels = [80, 128, 256]
results = get_parameter_curves_data_n(sec_levels, n_range, q = 2**64)
return results
def find_nalpha(l, sec_lvl):
for j in range(len(l)):
if l[j] != oo and l[j] > sec_lvl:
return j
## we start with 80/128/192/256-bits of security
## lambda = 80
## z = verify_interpolants(interps[0], (128,2048), 64)
## i = 0
## min(z[i:]) = 80.36
## so the model is sd(n) = max(-0.04047677865612648 * n + 1.1433465085639063, log_q - 2), n >= 128
## lambda = 128
## z = verify_interpolants(interps[3], (128,2048), 64)
## i = 83
## min(z[i:]) = 128.02
## so the model is sd(n) = max(-0.026374888765705498 * n + 2.012143923330495, log_q - 2), n >= 211 ( = 128 + 83)
## lambda = 192
## z = verify_interpolants(interps[7], (128,2048), 64)
## i = 304
## min(z[i:]) = 192.19
## so the model is sd(n) = max(-0.018504919354426233 * n + 2.6634073426215843, log_q - 2), n >= 432 ( = 128 + 212)
## lambda = 256
## z = verify_interpolants(interps[-1], (128,2048), 64)
## i = 653
## min(z[i:]) = 256.25
## so the model is sd(n) = max(-0.014327640360322604 * n + 2.899270827311091), log_q - 2), n >= 781 ( = 128 + 653)
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