Edit File: SecretSharing.py
# # 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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