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#
# SecretSharing.py : distribute a secret amongst a group of participants
#
# ===================================================================
#
# Copyright (c) 2014, Legrandin <helderijs@gmail.com>
# All rights reserved.
#
# Redistribution and use in source and binary forms, with or without
# modification, are permitted provided that the following conditions
# are met:
#
# 1. Redistributions of source code must retain the above copyright
# notice, this list of conditions and the following disclaimer.
# 2. Redistributions in binary form must reproduce the above copyright
# notice, this list of conditions and the following disclaimer in
# the documentation and/or other materials provided with the
# distribution.
#
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
# "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
# LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
# FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
# COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
# INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
# BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
# LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
# CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
# LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
# ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
# POSSIBILITY OF SUCH DAMAGE.
# ===================================================================
from Crypto.Util.py3compat import is_native_int
from Crypto.Util import number
from Crypto.Util.number import long_to_bytes, bytes_to_long
from Crypto.Random import get_random_bytes as rng
def _mult_gf2(f1, f2):
"""Multiply two polynomials in GF(2)"""
# Ensure f2 is the smallest
if f2 > f1:
f1, f2 = f2, f1
z = 0
while f2:
if f2 & 1:
z ^= f1
f1 <<= 1
f2 >>= 1
return z
def _div_gf2(a, b):
"""
Compute division of polynomials over GF(2).
Given a and b, it finds two polynomials q and r such that:
a = b*q + r with deg(r)<deg(b)
"""
if (a < b):
return 0, a
deg = number.size
q = 0
r = a
d = deg(b)
while deg(r) >= d:
s = 1 << (deg(r) - d)
q ^= s
r ^= _mult_gf2(b, s)
return (q, r)
class _Element(object):
"""Element of GF(2^128) field"""
# The irreducible polynomial defining this field is 1+x+x^2+x^7+x^128
irr_poly = 1 + 2 + 4 + 128 + 2 ** 128
def __init__(self, encoded_value):
"""Initialize the element to a certain value.
The value passed as parameter is internally encoded as
a 128-bit integer, where each bit represents a polynomial
coefficient. The LSB is the constant coefficient.
"""
if is_native_int(encoded_value):
self._value = encoded_value
elif len(encoded_value) == 16:
self._value = bytes_to_long(encoded_value)
else:
raise ValueError("The encoded value must be an integer or a 16 byte string")
def __eq__(self, other):
return self._value == other._value
def __int__(self):
"""Return the field element, encoded as a 128-bit integer."""
return self._value
def encode(self):
"""Return the field element, encoded as a 16 byte string."""
return long_to_bytes(self._value, 16)
def __mul__(self, factor):
f1 = self._value
f2 = factor._value
# Make sure that f2 is the smallest, to speed up the loop
if f2 > f1:
f1, f2 = f2, f1
if self.irr_poly in (f1, f2):
return _Element(0)
mask1 = 2 ** 128
v, z = f1, 0
while f2:
# if f2 ^ 1: z ^= v
mask2 = int(bin(f2 & 1)[2:] * 128, base=2)
z = (mask2 & (z ^ v)) | ((mask1 - mask2 - 1) & z)
v <<= 1
# if v & mask1: v ^= self.irr_poly
mask3 = int(bin((v >> 128) & 1)[2:] * 128, base=2)
v = (mask3 & (v ^ self.irr_poly)) | ((mask1 - mask3 - 1) & v)
f2 >>= 1
return _Element(z)
def __add__(self, term):
return _Element(self._value ^ term._value)
def inverse(self):
"""Return the inverse of this element in GF(2^128)."""
# We use the Extended GCD algorithm
# http://en.wikipedia.org/wiki/Polynomial_greatest_common_divisor
if self._value == 0:
raise ValueError("Inversion of zero")
r0, r1 = self._value, self.irr_poly
s0, s1 = 1, 0
while r1 > 0:
q = _div_gf2(r0, r1)[0]
r0, r1 = r1, r0 ^ _mult_gf2(q, r1)
s0, s1 = s1, s0 ^ _mult_gf2(q, s1)
return _Element(s0)
def __pow__(self, exponent):
result = _Element(self._value)
for _ in range(exponent - 1):
result = result * self
return result
class Shamir(object):
"""Shamir's secret sharing scheme.
A secret is split into ``n`` shares, and it is sufficient to collect
``k`` of them to reconstruct the secret.
"""
@staticmethod
def split(k, n, secret, ssss=False):
"""Split a secret into ``n`` shares.
The secret can be reconstructed later using just ``k`` shares
out of the original ``n``.
Each share must be kept confidential to the person it was
assigned to.
Each share is associated to an index (starting from 1).
Args:
k (integer):
The sufficient number of shares to reconstruct the secret (``k < n``).
n (integer):
The number of shares that this method will create.
secret (byte string):
A byte string of 16 bytes (e.g. the AES 128 key).
ssss (bool):
If ``True``, the shares can be used with the ``ssss`` utility.
Default: ``False``.
Return (tuples):
``n`` tuples. A tuple is meant for each participant and it contains two items:
1. the unique index (an integer)
2. the share (a byte string, 16 bytes)
"""
#
# We create a polynomial with random coefficients in GF(2^128):
#
# p(x) = \sum_{i=0}^{k-1} c_i * x^i
#
# c_0 is the encoded secret
#
coeffs = [_Element(rng(16)) for i in range(k - 1)]
coeffs.append(_Element(secret))
# Each share is y_i = p(x_i) where x_i is the public index
# associated to each of the n users.
def make_share(user, coeffs, ssss):
idx = _Element(user)
share = _Element(0)
for coeff in coeffs:
share = idx * share + coeff
if ssss:
share += _Element(user) ** len(coeffs)
return share.encode()
return [(i, make_share(i, coeffs, ssss)) for i in range(1, n + 1)]
@staticmethod
def combine(shares, ssss=False):
"""Recombine a secret, if enough shares are presented.
Args:
shares (tuples):
The *k* tuples, each containin the index (an integer) and
the share (a byte string, 16 bytes long) that were assigned to
a participant.
ssss (bool):
If ``True``, the shares were produced by the ``ssss`` utility.
Default: ``False``.
Return:
The original secret, as a byte string (16 bytes long).
"""
#
# Given k points (x,y), the interpolation polynomial of degree k-1 is:
#
# L(x) = \sum_{j=0}^{k-1} y_i * l_j(x)
#
# where:
#
# l_j(x) = \prod_{ \overset{0 \le m \le k-1}{m \ne j} }
# \frac{x - x_m}{x_j - x_m}
#
# However, in this case we are purely interested in the constant
# coefficient of L(x).
#
k = len(shares)
gf_shares = []
for x in shares:
idx = _Element(x[0])
value = _Element(x[1])
if any(y[0] == idx for y in gf_shares):
raise ValueError("Duplicate share")
if ssss:
value += idx ** k
gf_shares.append((idx, value))
result = _Element(0)
for j in range(k):
x_j, y_j = gf_shares[j]
numerator = _Element(1)
denominator = _Element(1)
for m in range(k):
x_m = gf_shares[m][0]
if m != j:
numerator *= x_m
denominator *= x_j + x_m
result += y_j * numerator * denominator.inverse()
return result.encode()
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