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Shamir's Secret Sharing

Shamir's Secret Sharing is an algorithm in cryptography created by Adi Shamir. It is a form of secret sharing, where a secret is divided into parts, giving each participant its own unique part.

To reconstruct the original secret, a minimum number of parts is required. In the threshold scheme this number is less than the total number of parts. Otherwise all participants are needed to reconstruct the original secret.

High-level explanationEdit

Shamir's Secret Sharing is used to secure a secret in a distributed way, most often to secure other encryption keys. The secret is split into multiple parts, called shares. These shares are used to reconstruct the original secret.

To unlock the secret via Shamir's secret sharing, you need a minimum number of shares. This is called the threshold, and is used to denote the minimum number of shares needed to unlock the secret. Let us walk through an example:

Problem: Company XYZ needs to secure their vault's passcode. They could use something standard, such as AES, but what if the holder of the key is unavailable or dies? What if the key is compromised via a malicious hacker or the holder of the key turns rogue, and uses their power over the vault to their benefit?

This is where SSS comes in. It can be used to encrypt the vault's passcode and generate a certain number of shares, where a certain number of shares can be allocated to each executive within Company XYZ. Now, only if they pool their shares can they unlock the vault. The threshold can be appropriately set for the number of executives, so the vault is always able to be accessed by the authorized individuals. Should a share or two fall into the wrong hands, they couldn't open the passcode unless the other executives cooperated.

Mathematical definitionEdit

The aim is to divide secret   (for example, the combination to a safe) into   pieces of data   in such a way that:

  1. Knowledge of any   or more   pieces makes   easy to compute. That is, the complete secret   can be reconstructed from any combination of   pieces of data.
  2. Knowledge of any   or fewer   pieces leaves   completely undetermined, in the sense that the possible values for   seem as likely as with knowledge of   pieces. Said another way, the secret   cannot be reconstructed with fewer than   pieces.

This scheme is called   threshold scheme. If   then every piece of the original secret   is required to reconstruct the secret.

Shamir's secret sharing schemeEdit

 
One can draw an infinite number of polynomials of degree 2 through 2 points. 3 points are required to define a unique polynomial of degree 2. This image is for illustration purposes only — Shamir's scheme uses polynomials over a finite field, not representable on a 2-dimensional plane.

The essential idea of Adi Shamir's threshold scheme is that 2 points are sufficient to define a line, 3 points are sufficient to define a parabola, 4 points to define a cubic curve and so forth. That is, it takes   points to define a polynomial of degree  .

Suppose we want to use a   threshold scheme to share our secret  , without loss of generality assumed to be an element in a finite field   of size   where   and   is a prime number.

Choose at random   positive integers   with  , and let  . Build the polynomial  . Let us construct any   points out of it, for instance set   to retrieve  . Every participant is given a point (a non-zero integer input to the polynomial, and the corresponding integer output) along with the prime which defines the finite field to use. Given any subset of   of these pairs, we can find the coefficients of the polynomial using interpolation. The secret is the constant term  .

UsageEdit

ExampleEdit

The following example illustrates the basic idea. Note, however, that calculations in the example are done using integer arithmetic rather than using finite field arithmetic. Therefore the example below does not provide perfect secrecy and is not a true example of Shamir's scheme. So we'll explain this problem and show the right way to implement it (using finite field arithmetic).

PreparationEdit

Suppose that our secret is 1234  .

We wish to divide the secret into 6 parts  , where any subset of 3 parts   is sufficient to reconstruct the secret. At random we obtain   numbers: 166 and 94.

  where   is secret

Our polynomial to produce secret shares (points) is therefore:

 

We construct six points   from the polynomial:

 

We give each participant a different single point (both   and  ). Because we use   instead of   the points start from   and not  . This is necessary because   is the secret.

ReconstructionEdit

In order to reconstruct the secret any 3 points will be enough.

Consider  .

We will compute Lagrange basis polynomials:

 
 
 

Therefore

 

Recall that the secret is the free coefficient, which means that  , and we are done.

Computationally efficient approachEdit

Considering that the goal of using polynomial interpolation is to find a constant in a source polynomial   using Lagrange polynomials "as it is" is not efficient, since unused constants are calculated.

An optimized approach to use Lagrange polynomials to find   is defined as follows:

 

ProblemEdit

Although the simplified version of the method demonstrated above, which uses integer arithmetic rather than finite field arithmetic, works fine, there is a security problem: Eve gains a lot of information about   with every   that she finds.

Suppose that she finds the 2 points   and  , she still doesn't have   points so in theory she shouldn't have gained any more info about  . But she combines the info from the 2 points with the public info:   and she :

  1. fills the  -formula with   and the value of  
  2. fills (i) with the values of  's   and  
  3. fills (i) with the values of  's   and  
  4. does (iii)-(ii):   and rewrites this as  
  5. knows that   so she starts replacing   in (iv) with 0, 1, 2, 3, ... to find all possible values for  :
    •  
    •  
    •  
    •  
    •  
    •  
    After   she stops because she reasons that if she continues she would get negative values for   (which is impossible because  ), she can now conclude  
  6. replaces   by (iv) in (ii):  
  7. replaces in (vi)   by the values found in (v) so she gets   which leads her to the information:
 
She now only has 150 numbers to guess from instead of an infinite number of natural numbers.

SolutionEdit

 
This is a polynomial curve over a finite field—now the order of the polynomial has seemingly little to do with the shape of the graph.

Geometrically this attack exploits the fact that we know the order of the polynomial and so gain insight into the paths it may take between known points. This reduces possible values of unknown points since it must lie on a smooth curve.

This problem can be fixed by using finite field arithmetic. A field of size   is used. The graph shows a polynomial curve over a finite field, in contrast to the usual smooth curve it appears very disorganised and disjointed.

In practice this is only a small change, it just means that we should choose a prime   that is bigger than the number of participants and every   (including  ) and we have to calculate the points as   instead of  .

Since everyone who receives a point also has to know the value of  , it may be considered to be publicly known. Therefore, one should select a value for   that is not too low.

Low values of   are risky because Eve knows  , so the lower one sets  , the fewer possible values Eve has to guess from to get  .

For this example we choose  , so our polynomial becomes   which gives the points:  

This time Eve doesn't win any info when she finds a   (until she has   points).

Suppose again that Eve finds   and  , this time the public info is:   so she:

  1. fills the  -formula with   and the value of   and  :  
  2. fills (i) with the values of  's   and  
  3. fills (i) with the values of  's   and  
  4. does (iii)-(ii):   and rewrites this as  
  5. knows that   so she starts replacing   in (iv) with 0, 1, 2, 3, ... to find all possible values for  :
    •  
    •  
    •  
    •  

This time she can't stop because   could be any integer (even negative if  ) so there are an infinite amount of possible values for  . She knows that   always decreases by 3 so if   was divisible by   she could conclude   but because it's prime she can't even conclude that and so she didn't win any information.

Python exampleEdit

'''
The following Python implementation of Shamir's Secret Sharing is
released into the Public Domain under the terms of CC0 and OWFa:
https://creativecommons.org/publicdomain/zero/1.0/
http://www.openwebfoundation.org/legal/the-owf-1-0-agreements/owfa-1-0

See the bottom few lines for usage. Tested on Python 2 and 3.
'''

from __future__ import division
from __future__ import print_function

import random
import functools

# 12th Mersenne Prime
# (for this application we want a known prime number as close as
# possible to our security level; e.g.  desired security level of 128
# bits -- too large and all the ciphertext is large; too small and
# security is compromised)
_PRIME = 2**127 - 1
# 13th Mersenne Prime is 2**521 - 1

_RINT = functools.partial(random.SystemRandom().randint, 0)

def _eval_at(poly, x, prime):
    '''evaluates polynomial (coefficient tuple) at x, used to generate a
    shamir pool in make_random_shares below.
    '''
    accum = 0
    for coeff in reversed(poly):
        accum *= x
        accum += coeff
        accum %= prime
    return accum

def make_random_shares(minimum, shares, prime=_PRIME):
    '''
    Generates a random shamir pool, returns the secret and the share
    points.
    '''
    if minimum > shares:
        raise ValueError("pool secret would be irrecoverable")
    poly = [_RINT(prime) for i in range(minimum)]
    points = [(i, _eval_at(poly, i, prime))
              for i in range(1, shares + 1)]
    return poly[0], points

def _extended_gcd(a, b):
    '''
    division in integers modulus p means finding the inverse of the
    denominator modulo p and then multiplying the numerator by this
    inverse (Note: inverse of A is B such that A*B % p == 1) this can
    be computed via extended Euclidean algorithm
    http://en.wikipedia.org/wiki/Modular_multiplicative_inverse#Computation
    '''
    x = 0
    last_x = 1
    y = 1
    last_y = 0
    while b != 0:
        quot = a // b
        a, b = b, a%b
        x, last_x = last_x - quot * x, x
        y, last_y = last_y - quot * y, y
    return last_x, last_y

def _divmod(num, den, p):
    '''compute num / den modulo prime p

    To explain what this means, the return value will be such that
    the following is true: den * _divmod(num, den, p) % p == num
    '''
    inv, _ = _extended_gcd(den, p)
    return num * inv

def _lagrange_interpolate(x, x_s, y_s, p):
    '''
    Find the y-value for the given x, given n (x, y) points;
    k points will define a polynomial of up to kth order
    '''
    k = len(x_s)
    assert k == len(set(x_s)), "points must be distinct"
    def PI(vals):  # upper-case PI -- product of inputs
        accum = 1
        for v in vals:
            accum *= v
        return accum
    nums = []  # avoid inexact division
    dens = []
    for i in range(k):
        others = list(x_s)
        cur = others.pop(i)
        nums.append(PI(x - o for o in others))
        dens.append(PI(cur - o for o in others))
    den = PI(dens)
    num = sum([_divmod(nums[i] * den * y_s[i] % p, dens[i], p)
               for i in range(k)])
    return (_divmod(num, den, p) + p) % p

def recover_secret(shares, prime=_PRIME):
    '''
    Recover the secret from share points
    (x,y points on the polynomial)
    '''
    if len(shares) < 2:
        raise ValueError("need at least two shares")
    x_s, y_s = zip(*shares)
    return _lagrange_interpolate(0, x_s, y_s, prime)

def main():
    '''main function'''
    secret, shares = make_random_shares(minimum=3, shares=6)

    print('secret:                                                     ',
          secret)
    print('shares:')
    if shares:
        for share in shares:
            print('  ', share)

    print('secret recovered from minimum subset of shares:             ',
          recover_secret(shares[:3]))
    print('secret recovered from a different minimum subset of shares: ',
          recover_secret(shares[-3:]))

if __name__ == '__main__':
    main()

PropertiesEdit

Some of the useful properties of Shamir's   threshold scheme are:

  1. Secure: Information theoretic security.
  2. Minimal: The size of each piece does not exceed the size of the original data.
  3. Extensible: When   is kept fixed,   pieces can be dynamically added or deleted without affecting the other pieces.
  4. Dynamic: Security can be easily enhanced without changing the secret, but by changing the polynomial occasionally (keeping the same free term) and constructing new shares to the participants.
  5. Flexible: In organizations where hierarchy is important, we can supply each participant different number of pieces according to their importance inside the organization. For instance, the president can unlock the safe alone, whereas 3 secretaries are required together to unlock it.

A known issue in Shamir's Secret Sharing scheme is the verification of correctness of the retrieved shares during the reconstruction process, which is known as verifiable secret sharing. Verifiable secret sharing aims at verifying that shareholders are honest and not submitting fake shares.

See alsoEdit

ReferencesEdit

  • Shamir, Adi (1979), "How to share a secret", Communications of the ACM, 22 (11): 612–613, doi:10.1145/359168.359176.
  • Benzekki, K. (2017), "A Verifiable Secret Sharing Approach for Secure MultiCloud Storage", In Ubiquitous Networking, Casablanca: Springer, 10542.

External linksEdit