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falcon.py
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falcon.py
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"""
Python implementation of Falcon:
https://falcon-sign.info/.
"""
from common import q
from numpy import set_printoptions
from math import sqrt
from fft import fft, ifft, sub, neg, add_fft, mul_fft
from ntt import sub_zq, mul_zq, div_zq
from ffsampling import gram, ffldl_fft, ffsampling_fft
from ntrugen import ntru_gen
from encoding import compress, decompress
# https://pycryptodome.readthedocs.io/en/latest/src/hash/shake256.html
from Crypto.Hash import SHAKE256
# Randomness
from os import urandom
from rng import ChaCha20
# For debugging purposes
import sys
if sys.version_info >= (3, 4):
from importlib import reload # Python 3.4+ only.
set_printoptions(linewidth=200, precision=5, suppress=True)
logn = {
2: 1,
4: 2,
8: 3,
16: 4,
32: 5,
64: 6,
128: 7,
256: 8,
512: 9,
1024: 10
}
# Bytelength of the signing salt and header
HEAD_LEN = 1
SALT_LEN = 40
SEED_LEN = 56
# Parameter sets for Falcon:
# - n is the dimension/degree of the cyclotomic ring
# - sigma is the std. dev. of signatures (Gaussians over a lattice)
# - sigmin is a lower bounds on the std. dev. of each Gaussian over Z
# - sigbound is the upper bound on ||s0||^2 + ||s1||^2
# - sig_bytelen is the bytelength of signatures
Params = {
# FalconParam(2, 2)
2: {
"n": 2,
"sigma": 144.81253976308423,
"sigmin": 1.1165085072329104,
"sig_bound": 101498,
"sig_bytelen": 44,
},
# FalconParam(4, 2)
4: {
"n": 4,
"sigma": 146.83798833523608,
"sigmin": 1.1321247692325274,
"sig_bound": 208714,
"sig_bytelen": 47,
},
# FalconParam(8, 2)
8: {
"n": 8,
"sigma": 148.83587593064718,
"sigmin": 1.147528535373367,
"sig_bound": 428865,
"sig_bytelen": 52,
},
# FalconParam(16, 4)
16: {
"n": 16,
"sigma": 151.78340713845503,
"sigmin": 1.170254078853483,
"sig_bound": 892039,
"sig_bytelen": 63,
},
# FalconParam(32, 8)
32: {
"n": 32,
"sigma": 154.6747794602761,
"sigmin": 1.1925466358390344,
"sig_bound": 1852696,
"sig_bytelen": 82,
},
# FalconParam(64, 16)
64: {
"n": 64,
"sigma": 157.51308555044122,
"sigmin": 1.2144300507766141,
"sig_bound": 3842630,
"sig_bytelen": 122,
},
# FalconParam(128, 32)
128: {
"n": 128,
"sigma": 160.30114421975344,
"sigmin": 1.235926056771981,
"sig_bound": 7959734,
"sig_bytelen": 200,
},
# FalconParam(256, 64)
256: {
"n": 256,
"sigma": 163.04153322607107,
"sigmin": 1.2570545284063217,
"sig_bound": 16468416,
"sig_bytelen": 356,
},
# FalconParam(512, 128)
512: {
"n": 512,
"sigma": 165.7366171829776,
"sigmin": 1.2778336969128337,
"sig_bound": 34034726,
"sig_bytelen": 666,
},
# FalconParam(1024, 256)
1024: {
"n": 1024,
"sigma": 168.38857144654395,
"sigmin": 1.298280334344292,
"sig_bound": 70265242,
"sig_bytelen": 1280,
},
}
def print_tree(tree, pref=""):
"""
Display a LDL tree in a readable form.
Args:
T: a LDL tree
Format: coefficient or fft
"""
leaf = "|_____> "
top = "|_______"
son1 = "| "
son2 = " "
width = len(top)
a = ""
if len(tree) == 3:
if (pref == ""):
a += pref + str(tree[0]) + "\n"
else:
a += pref[:-width] + top + str(tree[0]) + "\n"
a += print_tree(tree[1], pref + son1)
a += print_tree(tree[2], pref + son2)
return a
else:
return (pref[:-width] + leaf + str(tree) + "\n")
def normalize_tree(tree, sigma):
"""
Normalize leaves of a LDL tree (from values ||b_i||**2 to sigma/||b_i||).
Args:
T: a LDL tree
sigma: a standard deviation
Format: coefficient or fft
"""
if len(tree) == 3:
normalize_tree(tree[1], sigma)
normalize_tree(tree[2], sigma)
else:
tree[0] = sigma / sqrt(tree[0].real)
tree[1] = 0
class PublicKey:
"""
This class contains methods for performing public key operations in Falcon.
"""
def __init__(self, sk):
"""Initialize a public key."""
self.n = sk.n
self.h = sk.h
self.hash_to_point = sk.hash_to_point
self.signature_bound = sk.signature_bound
self.verify = sk.verify
def __repr__(self):
"""Print the object in readable form."""
rep = "Public for n = {n}:\n\n".format(n=self.n)
rep += "h = {h}\n".format(h=self.h)
return rep
class SecretKey:
"""
This class contains methods for performing
secret key operations (and also public key operations) in Falcon.
One can:
- initialize a secret key for:
- n = 128, 256, 512, 1024,
- phi = x ** n + 1,
- q = 12 * 1024 + 1
- find a preimage t of a point c (both in ( Z[x] mod (Phi,q) )**2 ) such that t*B0 = c
- hash a message to a point of Z[x] mod (Phi,q)
- sign a message
- verify the signature of a message
"""
def __init__(self, n, polys=None):
"""Initialize a secret key."""
# Public parameters
self.n = n
self.sigma = Params[n]["sigma"]
self.sigmin = Params[n]["sigmin"]
self.signature_bound = Params[n]["sig_bound"]
self.sig_bytelen = Params[n]["sig_bytelen"]
# Compute NTRU polynomials f, g, F, G verifying fG - gF = q mod Phi
if polys is None:
self.f, self.g, self.F, self.G = ntru_gen(n)
else:
[f, g, F, G] = polys
assert all((len(poly) == n) for poly in [f, g, F, G])
self.f = f[:]
self.g = g[:]
self.F = F[:]
self.G = G[:]
# From f, g, F, G, compute the basis B0 of a NTRU lattice
# as well as its Gram matrix and their fft's.
B0 = [[self.g, neg(self.f)], [self.G, neg(self.F)]]
G0 = gram(B0)
self.B0_fft = [[fft(elt) for elt in row] for row in B0]
G0_fft = [[fft(elt) for elt in row] for row in G0]
self.T_fft = ffldl_fft(G0_fft)
# Normalize Falcon tree
normalize_tree(self.T_fft, self.sigma)
# The public key is a polynomial such that h*f = g mod (Phi,q)
self.h = div_zq(self.g, self.f)
def __repr__(self, verbose=False):
"""Print the object in readable form."""
rep = "Private key for n = {n}:\n\n".format(n=self.n)
rep += "f = {f}\n".format(f=self.f)
rep += "g = {g}\n".format(g=self.g)
rep += "F = {F}\n".format(F=self.F)
rep += "G = {G}\n".format(G=self.G)
if verbose:
rep += "\nFFT tree\n"
rep += print_tree(self.T_fft, pref="")
return rep
def hash_to_point(self, message, salt):
"""
Hash a message to a point in Z[x] mod(Phi, q).
Inspired by the Parse function from NewHope.
"""
n = self.n
if q > (1 << 16):
raise ValueError("The modulus is too large")
k = (1 << 16) // q
# Create a SHAKE object and hash the salt and message.
shake = SHAKE256.new()
shake.update(salt)
shake.update(message)
# Output pseudorandom bytes and map them to coefficients.
hashed = [0 for i in range(n)]
i = 0
j = 0
while i < n:
# Takes 2 bytes, transform them in a 16 bits integer
twobytes = shake.read(2)
elt = (twobytes[0] << 8) + twobytes[1] # This breaks in Python 2.x
# Implicit rejection sampling
if elt < k * q:
hashed[i] = elt % q
i += 1
j += 1
return hashed
def sample_preimage(self, point, seed=None):
"""
Sample a short vector s such that s[0] + s[1] * h = point.
"""
[[a, b], [c, d]] = self.B0_fft
# We compute a vector t_fft such that:
# (fft(point), fft(0)) * B0_fft = t_fft
# Because fft(0) = 0 and the inverse of B has a very specific form,
# we can do several optimizations.
point_fft = fft(point)
t0_fft = [(point_fft[i] * d[i]) / q for i in range(self.n)]
t1_fft = [(-point_fft[i] * b[i]) / q for i in range(self.n)]
t_fft = [t0_fft, t1_fft]
# We now compute v such that:
# v = z * B0 for an integral vector z
# v is close to (point, 0)
if seed is None:
# If no seed is defined, use urandom as the pseudo-random source.
z_fft = ffsampling_fft(t_fft, self.T_fft, self.sigmin, urandom)
else:
# If a seed is defined, initialize a ChaCha20 PRG
# that is used to generate pseudo-randomness.
chacha_prng = ChaCha20(seed)
z_fft = ffsampling_fft(t_fft, self.T_fft, self.sigmin,
chacha_prng.randombytes)
v0_fft = add_fft(mul_fft(z_fft[0], a), mul_fft(z_fft[1], c))
v1_fft = add_fft(mul_fft(z_fft[0], b), mul_fft(z_fft[1], d))
v0 = [int(round(elt)) for elt in ifft(v0_fft)]
v1 = [int(round(elt)) for elt in ifft(v1_fft)]
# The difference s = (point, 0) - v is such that:
# s is short
# s[0] + s[1] * h = point
s = [sub(point, v0), neg(v1)]
return s
def sign(self, message, randombytes=urandom):
"""
Sign a message. The message MUST be a byte string or byte array.
Optionally, one can select the source of (pseudo-)randomness used
(default: urandom).
"""
int_header = 0x30 + logn[self.n]
header = int_header.to_bytes(1, "little")
salt = randombytes(SALT_LEN)
hashed = self.hash_to_point(message, salt)
# We repeat the signing procedure until we find a signature that is
# short enough (both the Euclidean norm and the bytelength)
while(1):
if (randombytes == urandom):
s = self.sample_preimage(hashed)
else:
seed = randombytes(SEED_LEN)
s = self.sample_preimage(hashed, seed=seed)
norm_sign = sum(coef ** 2 for coef in s[0])
norm_sign += sum(coef ** 2 for coef in s[1])
# Check the Euclidean norm
if norm_sign <= self.signature_bound:
enc_s = compress(s[1], self.sig_bytelen - HEAD_LEN - SALT_LEN)
# Check that the encoding is valid (sometimes it fails)
if (enc_s is not False):
return header + salt + enc_s
def verify(self, message, signature):
"""
Verify a signature.
"""
# Unpack the salt and the short polynomial s1
salt = signature[HEAD_LEN:HEAD_LEN + SALT_LEN]
enc_s = signature[HEAD_LEN + SALT_LEN:]
s1 = decompress(enc_s, self.sig_bytelen - HEAD_LEN - SALT_LEN, self.n)
# Check that the encoding is valid
if (s1 is False):
print("Invalid encoding")
return False
# Compute s0 and normalize its coefficients in (-q/2, q/2]
hashed = self.hash_to_point(message, salt)
s0 = sub_zq(hashed, mul_zq(s1, self.h))
s0 = [(coef + (q >> 1)) % q - (q >> 1) for coef in s0]
# Check that the (s0, s1) is short
norm_sign = sum(coef ** 2 for coef in s0)
norm_sign += sum(coef ** 2 for coef in s1)
if norm_sign > self.signature_bound:
print("Squared norm of signature is too large:", norm_sign)
return False
# If all checks are passed, accept
return True