"""
Minimal pure-Python implementation of
`Shamir's Secret Sharing scheme <https://en.wikipedia.org/wiki/Shamir%27s_Secret_Sharing>`__.
"""
from __future__ import annotations
import doctest
import warnings
from typing import Union, Optional, Sequence
from collections.abc import Iterable
import base64
import secrets
import lagrange
MODULUS_DEFAULT = (2 ** 127) - 1
"""
Default prime modulus (equivalent to ``(2 ** 127) - 1``) that is used for
creating secret shares if a prime modulus is not specified explicitly.
"""
def _randint(bound: int) -> int:
"""
Generate a random integer according to an approximately uniform distribution
via rejection sampling.
"""
length = 1 + (bound.bit_length() // 8)
value = int.from_bytes(secrets.token_bytes(length), 'little')
while value >= bound:
value = int.from_bytes(secrets.token_bytes(length), 'little')
return value
[docs]class share:
"""
Data structure for representing an individual secret share. Normally, the
:obj:`shares` function should be used to construct a sequence of :obj:`share`
objects.
>>> isinstance(shares(1, 3, modulus=31)[0], share)
True
>>> len(shares(1, 3, modulus=31))
3
>>> interpolate(shares(123, 12, modulus=15485867))
123
>>> interpolate(shares(2**100, 100)) == 2**100
True
It must be possible to represent the index integer using 32 bits. There is no
bound on the size of the value.
>>> share(4294967296, 123)
Traceback (most recent call last):
...
ValueError: index must be an integer that can be represented using at most 32 bits
"""
def __init__(self: share, index: int, value: int, modulus: Optional[int] = MODULUS_DEFAULT):
"""
Create a share instance according to the supplied parameters.
"""
if index > 4294967295:
raise ValueError(
'index must be an integer that can be represented using at most 32 bits'
)
self.index = index
self.value = value
self.modulus = modulus
[docs] @staticmethod
def from_bytes(bs: Union[bytes, bytearray]) -> share:
"""
Convert a secret share represented as a bytes-like object
into a :obj:`share` object.
>>> s = share.from_bytes(bytes.fromhex('7b00000002000000c801fd03'))
>>> (s.index, s.value, s.modulus)
(123, 456, 1021)
>>> s = share.from_bytes(share(1, 2**100).to_bytes())
>>> (s.index, s.value) == (1, 2**100)
True
"""
length = int.from_bytes(bs[4: 8], 'little')
return share(
int.from_bytes(bs[:4], 'little'),
int.from_bytes(bs[8: (8 + length)], 'little'),
int.from_bytes(bs[(8 + length):], 'little')
)
[docs] @staticmethod
def from_base64(s: str) -> share:
"""
Convert a secret share represented as a Base64 encoding of
a bytes-like object into a :obj:`share` object.
>>> s = share.from_base64('ewAAAAIAAADIAf0D')
>>> (s.index, s.value, s.modulus)
(123, 456, 1021)
>>> s = share.from_base64(share(3, 2**100).to_base64())
>>> (s.index, s.value) == (3, 2**100)
True
"""
return share.from_bytes(base64.standard_b64decode(s))
def __add__(self: share, other: Union[share, int]) -> share:
"""
Add two secret shares (represented as :obj:`share` objects).
>>> (r, s, t) = shares(123, 3)
>>> (u, v, w) = shares(456, 3)
>>> interpolate([r + u, s + v, t + w])
579
The integer constant ``0`` is supported as an input to accommodate the
base case required by the built-in :obj:`sum` function.
>>> share(123, 456, 1021) + 0
share(123, 456, 1021)
>>> ts = [shares(n, quantity=3) for n in [123, 456, 789]]
>>> interpolate([sum(ss) for ss in zip(*ts)])
1368
When secret shares are added, it is not possible to determine
whether the sum of the values they represent exceeds the maximum
value that can be represented. If the sum does exceed the value,
then the value reconstructed from the shares will wrap around.
This corresponds to the usual behavior of field elements under
addition within a field.
>>> (a, b) = shares(1020, quantity=2, modulus=1021) # One-byte integer.
>>> (c, d) = shares(2, quantity=2, modulus=1021) # One-byte integer.
>>> interpolate([a + c, b + d]) == (1020 + 2) % 1021 == 1
True
Any attempt to add secret shares that are represented using
different finite fields -- or that have different indices --
raises an exception.
>>> (r, s, t) = shares(2, quantity=3, modulus=5)
>>> (u, v, w) = shares(3, quantity=3, modulus=7)
>>> r + u
Traceback (most recent call last):
...
ValueError: shares being added must have the same index and modulus
>>> (r, s, t) = shares(2, quantity=3, modulus=5)
>>> (u, v, w) = shares(3, quantity=3, modulus=5)
>>> r + v
Traceback (most recent call last):
...
ValueError: shares being added must have the same index and modulus
The examples below test this addition method for a range of share
quantities and addition operation counts.
>>> for quantity in range(2, 20):
... for operations in range(2, 20):
... vs = [
... int.from_bytes(secrets.token_bytes(2), 'little')
... for _ in range(operations)
... ]
... sss = [shares(v, quantity) for v in vs]
... assert(interpolate([sum(ss) for ss in zip(*sss)]) == sum(vs))
"""
if isinstance(other, int) and other == 0:
return self
if self.index == other.index and self.modulus == other.modulus:
return share(
self.index,
(self.value + other.value) % self.modulus,
self.modulus
)
raise ValueError(
'shares being added must have the same index and modulus'
)
def __radd__(self: share, other: Union[share, int]) -> share:
"""
Add two secret shares (represented as :obj:`share` objects).
>>> (r, s, t) = shares(123, 3)
>>> (u, v, w) = shares(456, 3)
>>> interpolate([r + u, s + v, t + w])
579
The integer constant ``0`` is supported as an input to accommodate the
base case required by the built-in :obj:`sum` function.
>>> 0 + share(123, 456, 1021)
share(123, 456, 1021)
>>> ts = [shares(n, quantity=3) for n in [123, 456, 789]]
>>> interpolate([sum(ss) for ss in zip(*ts)])
1368
"""
if isinstance(other, int) and other == 0:
return self
return other + self # pragma: no cover
def __mul__(self: share, scalar: int) -> share:
"""
Multiply this secret share by an integer scalar. Note that all
secret shares must be multiplied by the same integer scalar in
order for the reconstructed value to reflect the correct result.
>>> (r, s, t) = shares(123, 3)
>>> r = r * 2
>>> s = s * 2
>>> t = t * 2
>>> interpolate([r, s, t])
246
When secret shares are multiplied by a scalar, it is not possible to
determine whether the result exceeds the range of values that can be
represented. If the result does fall outside the range, then the value
reconstructed from the shares will wrap around. This corresponds to
the usual behavior of field elements under scalar multiplication
within a field.
>>> (s, t) = shares(512, quantity=2, modulus=1021)
>>> s = s * 2
>>> t = t * 2
>>> interpolate([s, t]) == (512 * 2) % 1021 == 3
True
The scalar argument must be a nonnegative integer.
>>> (r, s, t) = shares(123, 3)
>>> s = s * 2.0
Traceback (most recent call last):
...
TypeError: scalar must be an integer
>>> (r, s, t) = shares(123, 3)
>>> s = s * -2
Traceback (most recent call last):
...
ValueError: scalar must be a nonnegative integer
The examples below test this scalar multiplication method for a range
of share quantities and a number of random scalar values.
>>> for quantity in range(2, 20):
... for _ in range(100):
... v = int.from_bytes(secrets.token_bytes(2), 'little')
... c = int.from_bytes(secrets.token_bytes(1), 'little')
... ss = shares(v, quantity)
... assert(interpolate([c * s for s in ss]) == c * v)
"""
if not isinstance(scalar, int):
raise TypeError('scalar must be an integer')
if scalar < 0:
raise ValueError('scalar must be a nonnegative integer')
return share(
self.index,
(self.value * scalar) % self.modulus,
self.modulus
)
def __rmul__(self: share, scalar: int) -> share:
"""
Multiply this secret share by an integer scalar. Note that all
secret shares must be multiplied by the same integer scalar in
order for the reconstructed value to reflect the correct result.
>>> (r, s, t) = shares(123, 3)
>>> r = r * 2
>>> s = s * 2
>>> t = t * 2
>>> interpolate([r, s, t])
246
"""
return self * scalar
def __int__(self: share) -> int:
"""
Return the least nonnegative residue of the field element corresponding
to this instance.
>>> int(share(123, 456, 1021))
456
"""
return self.value % self.modulus
def __len__(self: share) -> int:
"""
Return the modulus corresponding to the field (within which this instance
represents an element).
>>> len(share(123, 456, 1021))
1021
"""
return self.modulus
[docs] def to_bytes(self: share) -> bytes:
"""
Return a bytes-like object that encodes this :obj:`share` object.
>>> share(123, 456, 1021).to_bytes().hex()
'7b00000002000000c801fd03'
>>> s = share.from_bytes(share(3, 2**100).to_bytes())
>>> (s.index, s.value) == (3, 2**100)
True
"""
length = (self.modulus.bit_length() + 7) // 8
return (
int(self.index).to_bytes(4, 'little') + \
int(length).to_bytes(4, 'little') + \
int(self.value).to_bytes(length, 'little') + \
int(self.modulus).to_bytes(length, 'little')
)
[docs] def to_base64(self: share) -> str:
"""
Return a Base64 string representation of this :obj:`share` object.
>>> share(123, 456, 1021).to_base64()
'ewAAAAIAAADIAf0D'
>>> s = share.from_base64(share(3, 2**100).to_base64())
>>> (s.index, s.value) == (3, 2**100)
True
"""
return base64.standard_b64encode(self.to_bytes()).decode('utf-8')
def __str__(self: share) -> str:
"""
Return the string representation of this :obj:`share` object.
>>> str(share(123, 456, 1021))
'share(123, 456, 1021)'
"""
return 'share(' + ', '.join([
str(self.index),
str(self.value),
str(self.modulus)
]) + ')'
def __repr__(self: share) -> str:
"""
Return the string representation of this :obj:`share` object.
>>> share(123, 456, 1021)
share(123, 456, 1021)
"""
return str(self)
[docs]def shares(
value: int,
quantity: int,
modulus: Optional[int] = MODULUS_DEFAULT,
threshold: Optional[int] = None
) -> Sequence[share]:
"""
Transforms an integer value into the specified number of secret shares, with
recovery of the original value possible using the returned sequence of secret
shares (via the :obj:`interpolate` function).
:param value: Integer value to be split into secret shares.
:param quantity: Number of secret shares (at least two) to construct
and return.
:param modulus: Prime modulus corresponding to the finite field used for
creating secret shares.
:param threshold: Minimum number of shares that will be required to
reconstruct a value.
>>> len(shares(1, 3, modulus=31))
3
>>> len(shares(17, 10, modulus=41))
10
>>> len(shares(123, 100))
100
Attempts to transform a value that is greater than the supplied prime
modulus raise an exception.
>>> shares(256, 3, modulus=31)
Traceback (most recent call last):
...
ValueError: value cannot be greater than the prime modulus
Other invocations with invalid parameter values also raise exceptions.
>>> shares('abc', 3, 17)
Traceback (most recent call last):
...
TypeError: value must be an integer
>>> shares(1, 'abc', 17)
Traceback (most recent call last):
...
TypeError: quantity of shares must be an integer
>>> shares(1, 3, 'abc')
Traceback (most recent call last):
...
TypeError: prime modulus must be an integer
>>> shares(-2, 3, 17)
Traceback (most recent call last):
...
ValueError: value must be a nonnegative integer
>>> shares(1, 1, 17)
Traceback (most recent call last):
...
ValueError: quantity of shares must be at least 2
>>> shares(1, 2**32, 17)
Traceback (most recent call last):
...
ValueError: quantity of shares must be an integer that can be represented using at most 32 bits
>>> shares(1, 3, 1)
Traceback (most recent call last):
...
ValueError: prime modulus must be at least 2
Requesting fewer shares than needed to reconstruct is permitted (but a
warning is issued).
>>> len(shares(1, quantity=3, modulus=11, threshold=7))
3
Requesting a larger set of shares than is necessary to reconstruct the
original value is permitted.
>>> len(shares(1, quantity=7, modulus=11, threshold=3))
7
"""
if not isinstance(value, int):
raise TypeError('value must be an integer')
if value < 0:
raise ValueError('value must be a nonnegative integer')
if not isinstance(quantity, int):
raise TypeError('quantity of shares must be an integer')
if quantity < 2:
raise ValueError('quantity of shares must be at least 2')
if quantity >= 2 ** 32:
raise ValueError(
'quantity of shares must be an integer that can be represented using at most 32 bits'
)
if not isinstance(modulus, int):
raise TypeError('prime modulus must be an integer')
if modulus < 2:
raise ValueError('prime modulus must be at least 2')
if value >= modulus:
raise ValueError('value cannot be greater than the prime modulus')
# Use the maximum threshold if one is not specified.
threshold = threshold or quantity
if threshold > quantity:
warnings.warn(
'quantity of shares should be at least the threshold to be reconstructable'
)
# Add the base coefficient.
coefficients = [value] + [_randint(modulus - 1) for _ in range(1, threshold - 1)]
# Compute each share value such that ``shares[i] = f(i)`` if the polynomial
# is ``f``.
shares_ = [
sum(
c_j * i ** j % modulus
for j, c_j in enumerate(coefficients)
) % modulus
for i in range(1, quantity + 1)
]
# Embed each shares index (x-coordinate) by shifting right and using the new lowest 32-bits.
shares_ = [share(index + 1, value, modulus) for (index, value) in enumerate(shares_)]
return shares_
[docs]def interpolate(
shares: Iterable[share], # pylint: disable=redefined-outer-name
threshold: int = None
) -> int:
"""
Reassemble an integer value from a sequence of secret shares using
Lagrange interpolation (via the :obj:`~lagrange.lagrange.interpolate` function
exported by the `lagrange <https://pypi.org/project/lagrange>`__ library).
:param shares: Iterable of shares from which to reconstruct a value.
:param threshold: Minimum number of shares that will be required to
reconstruct a value.
>>> interpolate(shares(5, 3, modulus=31))
5
>>> interpolate(shares(123, 12))
123
The appropriate order for the secret shares is already encoded in the
individual :obj:`share` instances (assuming they were created using
the :obj:`shares` function). Thus, they can be supplied in any order.
>>> interpolate(reversed(shares(123, 12)))
123
If the threshold is known to be different than the number of shares,
it should be specified as such. In the example below, the value 123
was shared with twenty parties such that at least twelve of them
must collaborate to reconstruct the value.
>>> interpolate(shares(123, 20, 1223, 12)[:12], 12) # Use first twelve shares.
123
>>> interpolate(shares(123, 20, 1223, 12)[20-12:], 12) # Use last twelve shares.
123
>>> interpolate(shares(123, 20, 1223, 12)[:15], 12) # Use first fifteen shares.
123
>>> interpolate(shares(123, 20, 1223, 12)[:11], 12) # Try using only eleven shares.
Traceback (most recent call last):
...
ValueError: not enough points for a unique interpolation
Invocations with invalid parameter values raise exceptions.
>>> interpolate([1, 2, 3])
Traceback (most recent call last):
...
TypeError: input must contain share objects
>>> interpolate(shares(123, 3, 1223) + shares(123, 3, 1021))
Traceback (most recent call last):
...
ValueError: all shares must have the same modulus
"""
shares = list(shares) # Store shares for reuse, even if an iterable is supplied.
if not all (isinstance(s, share) for s in shares):
raise TypeError('input must contain share objects')
moduli = [s.modulus for s in shares]
if len(set(moduli)) > 1:
raise ValueError('all shares must have the same modulus')
return lagrange.interpolate(
[(s.index, s.value) for s in shares],
moduli[0],
(threshold or len(shares)) - 1
)
if __name__ == '__main__': # pragma: no cover
doctest.testmod()