Shamira Changeset - 0957647049ef

Changeset - 0957647049ef

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Laman - 5 years ago 2020-12-18 12:13:06

merged FFT

6 files changed with 197 insertions and 8 deletions:

src/shamira/core.py

src/shamira/fft.py

src/shamira/gf256.py

src/shamira/tests/test_fft.py

src/shamira/tests/test_gf256.py

src/shamira/tests/test_shamira.py

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src/shamira/core.py

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 # GNU GPLv3, see LICENSE
 import os
 import re
 import base64
 import binascii
 from . import gf256
 from . import fft
 class SException(Exception): pass
 class InvalidParams(SException): pass
 class DetectionException(SException): pass
 class DecodingException(SException): pass
 class MalformedShare(SException): pass
 def compute_x(n):
 	return fft.precompute_x(fft.ceil_size(n))[:n]
 def _share_byte(secret_b, k, n):
 	if not k<=n<255:
 		raise InvalidParams("Failed k<=n<255, k={0}, n={1}".format(k, n))
 	# we might be concerned with zero coefficients degenerating our polynomial, but there's no reason - we still need k shares to determine it is the case
 	coefs = [int(b) for b in os.urandom(k-1)]+[int(secret_b)]
 	points = [gf256.evaluate(coefs, i) for i in range(1, n+1)]
 	return points
 	# we might be concerned with zero coefficients degenerating our polynomial,
 	# but there's no reason - we still need k shares to determine it is the case
 	coefs = [int(secret_b)]+[int(b) for b in os.urandom(k-1)]
 	return fft.evaluate(coefs, n)
 def generate_raw(secret, k, n):
 	"""Splits secret into shares.
 	:param secret: (bytes)
 	:param k: number of shares necessary for secret recovery. 1 <= k <= n
 	:param n: (int) number of shares generated. 1 <= n < 255
 	:return: [(i, (bytes) share), ...]"""
 	xs = compute_x(n)
 	shares = [_share_byte(b, k, n) for b in secret]
-	return [(i+1, bytes([s[i] for s in shares])) for i in range(n)]
+	return [(xi, bytes([s[i] for s in shares])) for (i, xi) in enumerate(xs)]
 def reconstruct_raw(*shares):
 	"""Tries to recover the secret from its shares.
 	:param shares: (((int) i, (bytes) share), ...)
 	:return: (bytes) reconstructed secret. Too few shares return garbage."""
 	if len({x for (x, _) in shares}) < len(shares):
 		raise MalformedShare("Found a non-unique share. Please check your inputs.")
 	(xs, payloads) = zip(*shares)
 	secret_len = len(payloads[0])
 	res = [None]*secret_len
 	weights = gf256.compute_weights(xs)
 	for i in range(secret_len):
 		ys = [s[i] for s in payloads]
 		res[i] = (gf256.get_constant_coef(weights, ys))
 	return bytes(res)
 def generate(secret, k, n, encoding="b32", label="", omit_k_n=False):
 	"""Wraps generate_raw().
 	:param secret: (str or bytes)
 	:param k: number of shares necessary for secret recovery
 	:param n: number of shares generated
 	:param encoding: {hex, b32, b64} desired output encoding. Hexadecimal, Base32 or Base64.
 	:param label: (str) any label to prefix the shares with
 	:param omit_k_n: (boolean) suppress the default shares prefix
 	:return: [(str) share, ...]"""
 	if isinstance(secret,str):
 		secret = secret.encode("utf-8")
 	shares = generate_raw(secret, k, n)
 	prefix = ""
 	if label:
 		prefix = label + "."
 	if not omit_k_n:
 		prefix += "{0}.{1}.".format(k, n)
 	return [prefix + encode(s, encoding) for s in shares]
 def reconstruct(*shares, encoding="", raw=False):
 	"""Wraps reconstruct_raw.
 	:param shares: ((str) share, ...)
 	:param encoding: {hex, b32, b64, ""} encoding of share strings. If not provided or empty, the function tries to guess it.

src/shamira/fft.py

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@@ new file 100644 @@
 import math
 import cmath
 import itertools
 from .gf256 import gfmul, gfpow
 # divisors of 255 and their factors in natural numbers
 DIVISORS = [3, 5, 15, 17, 51, 85, 255]
 FACTORS = {3: [3], 5: [5], 15: [3, 5], 17: [17], 51: [3, 17], 85: [5, 17], 255: [3, 5, 17]}
 # values of n-th square roots in GF256
 SQUARE_ROOTS = {3: 189, 5: 12, 15: 225, 17: 53, 51: 51, 85: 15, 255: 3}
 def ceil_size(n):
 	assert n <= DIVISORS[-1]
 	for (i, ni) in enumerate(DIVISORS):
 		if ni >= n:
 			break
 	return ni
 def precompute_x(n):
 	"""Return a geometric sequence [1, w, w**2, ..., w**(n-1)], where w**n==1.
 	This can be done only for certain values of n."""
 	assert n in SQUARE_ROOTS, n
 	w = SQUARE_ROOTS[n]  # primitive N-th square root of 1
 	return list(itertools.accumulate([1]+[w]*(n-1), gfmul))
 def complex_dft(p):
 	"""Quadratic formula from the definition. The basic case in complex numbers."""
 	N = len(p)
 	w = cmath.exp(-2*math.pi*1j/N)  # primitive N-th square root of 1
 	y = [0]*N
 	for k in range(N):
 		xk = w**k
 		for n in range(N):
 			y[k] += p[n] * xk**n
 	return y
 def dft(p):
 	"""Quadratic formula from the definition. In GF256."""
 	N = len(p)
 	x = precompute_x(N)
 	y = [0]*N
 	for k in range(N):
 		for n in range(N):
 			y[k] ^= gfmul(p[n], gfpow(x[k], n))
 	return y
 def compute_inverse(N1, N2):
 	for i in range(N2):
 		if N1*i % N2 == 1:
 			return i
 	raise ValueError("Failed to find an inverse to {0} mod {1}.".format(N1, N2))
 def prime_fft(p, divisors, basic_dft=dft):
 	"""https://en.wikipedia.org/wiki/Prime-factor_FFT_algorithm"""
 	if len(divisors) == 1:
 		return basic_dft(p)
 	N = len(p)
 	N1 = divisors[0]
 	N2 = N//N1
 	N1_inv = compute_inverse(N1, N2)
 	N2_inv = compute_inverse(N2, N1)
 	ys = []
 	for n1 in range(N1):  # compute rows
 		p_ = [p[(n2*N1+n1*N2) % N] for n2 in range(N2)]
 		ys.append(prime_fft(p_, divisors[1:], basic_dft))
 	for k2 in range(N2):  # compute cols
 		p_ = [row[k2] for row in ys]
 		y_ = basic_dft(p_)
 		for (yi, row) in zip(y_, ys):  # update col
 			row[k2] = yi
 	# remap and output
 	res = [0]*N
 	for k1 in range(N1):
 		for k2 in range(N2):
 			res[(k1*N2*N2_inv+k2*N1*N1_inv) % N] = ys[k1][k2]
 	return res
 def evaluate(coefs, n):
 	ni = ceil_size(n)
 	extended_coefs = coefs + [0]*(ni-len(coefs))
 	ys = prime_fft(extended_coefs, FACTORS[ni])
 	return ys[:n]

src/shamira/gf256.py

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 # GNU GPLv3, see LICENSE
 """Arithmetic operations on Galois Field 2**8. See https://en.wikipedia.org/wiki/Finite_field_arithmetic"""
 from functools import reduce, cache
 import operator
 def _gfmul(a, b):
 	"""Basic multiplication. Russian peasant algorithm."""
 	res = 0
 	while a and b:
 		if b&1: res ^= a
 		if a&0x80: a = 0xff&(a<<1)^0x1b
 		else: a <<= 1
 		b >>= 1
 	return res
 g = 3  # generator
 E = [None]*256  # exponentials
 L = [None]*256  # logarithms
 acc = 1
 for i in range(256):
 	E[i] = acc
 	L[acc] = i
 	acc = _gfmul(acc, g)
 L[1] = 0
 INV = [E[255-L[i]] if i!=0 else None for i in range(256)]  # multiplicative inverse
 @cache
 def gfmul(a, b):
 	"""Fast multiplication. Basic multiplication is expensive. a*b==g**(log(a)+log(b))"""
 	assert 0<=a<=255, 0<=b<=255
 	if a==0 or b==0: return 0
 	t = L[a]+L[b]
 	if t>255: t -= 255
 	return E[t]
 def gfpow(x, k):
 	"""Compute x**k."""
 	i = 1
 	res = 1
 	while i <= k:
 		if k&i:
 			res = gfmul(res, x)
 		x = gfmul(x, x)
 		i <<= 1
 	return res
 def evaluate(coefs, x):
 	"""Evaluate polynomial's value at x.
 	:param coefs: [an, ..., a1, a0]."""
 	res = 0
 	for a in coefs:  # Horner's rule
 		res = gfmul(res, x)
 		res ^= a
 	return res
 def get_constant_coef(weights, y_coords):
 	"""Compute constant polynomial coefficient given the points.
 	See https://en.wikipedia.org/wiki/Shamir's_Secret_Sharing#Computationally_Efficient_Approach"""
 	return reduce(
 		operator.xor,
 		map(lambda ab: gfmul(*ab), zip(weights, y_coords))
+	)
 def compute_weights(x_coords):
 	assert x_coords
 	res = [
 			reduce(
 				gfmul,
 				(gfmul(xj, INV[xj^xi]) for xj in x_coords if xi!=xj),
 			) for xi in x_coords
+	]
 	return res

src/shamira/tests/test_fft.py

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@@ new file 100644 @@
 # GNU GPLv3, see LICENSE
 import random
 import functools
 import operator
 from unittest import TestCase
 from .. import gf256
 from ..fft import *
 def batch_evaluate(coefs, xs):
 	return [gf256.evaluate(coefs, x) for x in xs]
 class TestFFT(TestCase):
 	def test_complex_dft(self):
 		self.assertEqual(complex_dft([0]), [0+0j])
 		self.assertEqual(complex_dft([1]), [1+0j])
 		self.assertEqual(complex_dft([2]), [2+0j])
 		all(self.assertAlmostEqual(a, b) for (a, b) in zip(complex_dft([3, 1]), [4+0j, 2+0j]))
 		all(self.assertAlmostEqual(a, b) for (a, b) in zip(complex_dft([3, 1, 4]), [8+0j, 0.5+2.59807621j, 0.5-2.59807621j]))
 		all(self.assertAlmostEqual(a, b) for (a, b) in zip(complex_dft([3, 1, 4, 1]), [9+0j, -1+0j, 5+0j, -1+0j]))
 		all(self.assertAlmostEqual(a, b) for (a, b) in zip(
 			complex_dft([3, 1, 4, 1, 5]),
 			[14+0j, 0.80901699+2.04087031j, -0.30901699+5.20431056j, -0.30901699-5.20431056j, 0.80901699-2.04087031j]
 		))
 	def test_complex_prime_fft(self):
 		random.seed(1918)
 		for divisors in [[3], [2, 3], [3, 5], [3, 5, 17], [2, 3, 5, 7, 11]]:
 			n = functools.reduce(operator.mul, divisors)
 			coefficients = [random.randint(-128, 127) for i in range(n)]
 			a = prime_fft(coefficients, divisors, complex_dft)
 			b = complex_dft(coefficients)
 			all(self.assertAlmostEqual(ai, bi) for (ai, bi) in zip(a, b))
 	def test_finite_dft(self):
 		random.seed(1918)
 		x = {i: precompute_x(i) for i in [3, 5, 15, 17]}  # all sets of xs
 		for n in [3, 5, 15, 17]:
 			coefficients = [random.randint(0, 255) for i in range(n)]
 			self.assertEqual(
 				dft(coefficients),
 				batch_evaluate(coefficients[::-1], x[n])
+			)
 	def test_finite_prime_fft(self):
 		random.seed(1918)
 		for divisors in [[3], [3, 5], [3, 17], [5, 17], [3, 5, 17]]:
 			n = functools.reduce(operator.mul, divisors)
 			coefficients = [random.randint(0, 255) for i in range(n)]
 			a = prime_fft(coefficients, divisors)
 			b = dft(coefficients)
 			all(self.assertAlmostEqual(ai, bi) for (ai, bi) in zip(a, b))

src/shamira/tests/test_gf256.py

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 # GNU GPLv3, see LICENSE
 import random
 import unittest
 from unittest import TestCase
 from ..gf256 import _gfmul
 from ..gf256 import *
 class TestGF256(TestCase):
 	def test__gfmul(self):
 		self.assertEqual(_gfmul(0, 0), 0)
 		self.assertEqual(_gfmul(1, 1), 1)
 		self.assertEqual(_gfmul(2, 2), 4)
 		self.assertEqual(_gfmul(0, 21), 0)
 		self.assertEqual(_gfmul(0x53, 0xca), 0x01)
 		self.assertEqual(_gfmul(0xff, 0xff), 0x13)
 	def test_gfmul(self):
 		for a in range(256):
 			for b in range(256):
 				self.assertEqual(_gfmul(a, b), gfmul(a, b))
 	def test_gfpow(self):
 		self.assertEqual(gfpow(0, 0), 1)
 		for i in range(1, 256):
 			self.assertEqual(gfpow(i, 0), 1)
 			self.assertEqual(gfpow(i, 1), i)
 			self.assertEqual(gfpow(0, i), 0)
 			self.assertEqual(gfpow(1, i), 1)
 			self.assertEqual(gfpow(i, 256), i)
 			self.assertEqual(gfpow(i, 2), gfmul(i, i))
 		random.seed(1918)
 		for i in range(256):
 			j = random.randint(2, 255)
 			k = random.randint(3, 255)
 			y = 1
 			for m in range(k):
 				y = gfmul(y, j)
 			self.assertEqual(gfpow(j, k), y)
 	def test_evaluate(self):
 		for x in range(256):
 			(a0, a1, a2, a3) = (x, x>>1, x>>2, x>>3)
 			self.assertEqual(evaluate([17], x), 17)  # constant polynomial
 			self.assertEqual(evaluate([a3, a2, a1, a0], 0), x)  # any polynomial at 0
 			self.assertEqual(evaluate([a3, a2, a1, a0], 1), a0^a1^a2^a3)  # polynomial at 1 == sum of coefficients
 	def test_get_constant_coef(self):
 		weights = compute_weights((1, 2, 3))
 		ys = (1, 2, 3)
 		self.assertEqual(get_constant_coef(weights, ys), 0)
 		random.seed(17)
 		random_matches = 0
 		for i in range(10):
 			k = random.randint(2, 255)
 			# exact
 			res = self.check_coefs_match(k, k)
 			self.assertEqual(res[0], res[1])
 			# overdetermined
 			res = self.check_coefs_match(k, 256)
 			self.assertEqual(res[0], res[1])
 			# underdetermined => random
 			res = self.check_coefs_match(k, k-1)
 			if res[0]==res[1]:
 				random_matches += 1
 		self.assertLess(random_matches, 2)  # with a chance (255/256)**10=0.96 there should be no match
 	def check_coefs_match(self, k, m):
 		coefs = [random.randint(0, 255) for i in range(k)]
 		points = [(j, evaluate(coefs, j)) for j in range(1, 256)]
 		random.shuffle(points)
 		(xs, ys) = zip(*points[:m])
 		weights = compute_weights(xs)
 		return (get_constant_coef(weights, ys), coefs[-1])
 if __name__=='__main__':
 	unittest.main()

src/shamira/tests/test_shamira.py

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 # GNU GPLv3, see LICENSE
 import os
 import random
 from unittest import TestCase
 from .. import *
 from .. import gf256
 from ..core import encode, decode,detect_encoding, _share_byte
+from ..core import encode, decode, detect_encoding, _share_byte, compute_x
 class TestShamira(TestCase):
 	_urandom = os.urandom
 	@classmethod
 	def setUpClass(cls):
 		random.seed(17)
 		os.urandom = lambda n: bytes(random.randint(0, 255) for i in range(n))
 	@classmethod
 	def tearDownClass(cls):
 		os.urandom = cls._urandom
 	def test_share_byte(self):
 		with self.assertRaises(InvalidParams):  # too few shares
 			_share_byte(b"a", 5, 4)
 		with self.assertRaises(InvalidParams):  # too many shares
 			_share_byte(b"a", 5, 255)
 		with self.assertRaises(ValueError):  # not castable to int
 			_share_byte("x", 2, 3)
 		ys = _share_byte(ord(b"a"), 2, 3)
-		xs = list(range(1, 4))
+		xs = compute_x(3)
 		weights = gf256.compute_weights(xs)
 		self.assertEqual(gf256.get_constant_coef(weights, ys), ord(b"a"))
 		weights = gf256.compute_weights(xs[:2])
 		self.assertEqual(gf256.get_constant_coef(weights, ys[:2]), ord(b"a"))
 		weights = gf256.compute_weights(xs[:1])
 		self.assertNotEqual(gf256.get_constant_coef(weights, ys[:1]), ord(b"a"))  # underdetermined => random
 	def test_generate_reconstruct_raw(self):
 		for (k, n) in [(2, 3), (254, 254)]:
 			shares = generate_raw(b"abcd", k, n)
 			random.shuffle(shares)
 			self.assertEqual(reconstruct_raw(*shares[:k]), b"abcd")
 			self.assertNotEqual(reconstruct_raw(*shares[:k-1]), b"abcd")
 	def test_generate_reconstruct(self):
 		for encoding in ["hex", "b32", "b64"]:
 			for secret in [b"abcd", "abcde", "ěščřžý"]:
 				for (k, n) in [(2, 3), (254, 254)]:
 					raw = isinstance(secret, bytes)
 					with self.subTest(enc=encoding, r=raw, sec=secret, k=k, n=n):
 						shares = generate(secret, k, n, encoding)
 						random.shuffle(shares)
 						self.assertEqual(reconstruct(*shares[:k], encoding=encoding, raw=raw), secret)
 						self.assertEqual(reconstruct(*shares[:k], raw=raw), secret)
 						s = secret if raw else secret.encode("utf-8")
 						self.assertNotEqual(reconstruct(*shares[:k-1], encoding=encoding, raw=True), s)
 		shares = generate(b"\xfeaa", 2, 3)
 		with self.assertRaises(DecodingException):
 			reconstruct(*shares)
 	def test_encode(self):
 		share = (2, b"\x00\x01\x02")
 		for (encoding, encoded_str) in [("hex", '000102'), ("b32", 'AAAQE==='), ("b64", 'AAEC')]:
 			with self.subTest(enc=encoding):
 				self.assertEqual(encode(share, encoding), "2."+encoded_str)
 	def test_decode(self):
 		with self.assertRaises(MalformedShare):
 			decode("AAA")
 			decode("1.")
 			decode(".AAA")
 			decode("1AAA")
 			decode("1.0001020f", "hex")
 			decode("1.000102030", "hex")
 			decode("1.AAAQEAY")

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