Changeset - 0957647049ef
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Laman - 4 years ago 2020-12-18 12:13:06

merged FFT
6 files changed with 197 insertions and 8 deletions:
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src/shamira/core.py
Show inline comments
 
@@ -6,6 +6,7 @@ import base64
 
import binascii
 

	
 
from . import gf256
 
from . import fft
 

	
 

	
 
class SException(Exception): pass
 
@@ -15,13 +16,17 @@ class DecodingException(SException): pas
 
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):
 
@@ -31,8 +36,9 @@ def generate_raw(secret, k, n):
 
	: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):
src/shamira/fft.py
Show inline comments
 
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
Show inline comments
 
@@ -29,7 +29,6 @@ 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
 
@@ -39,6 +38,19 @@ def gfmul(a, b):
 
	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.
 

	
src/shamira/tests/test_fft.py
Show inline comments
 
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
Show inline comments
 
@@ -22,6 +22,26 @@ class TestGF256(TestCase):
 
			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)
src/shamira/tests/test_shamira.py
Show inline comments
 
@@ -5,7 +5,7 @@ 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):
 
@@ -29,7 +29,7 @@ class TestShamira(TestCase):
 
			_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"))
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