Algorithm used

Motivation behind SSS

Shamir’s Secret Sharing uses the polynomial interpolation principle to perform threshold sharing in the following two phases:
Phase I: Generation of Shares
This phase involves the setup of the system as well as the generation of the shares.

Decide the values for the number of participants (n) and the threshold (t) to secure some secret value (s)
Construct a random polynomial, P(x), with degree t-1 by choosing random coefficients of the polynomial. Set the constant term in the polynomial (coefficient of zero degree term) to be equal to the secret value s
To generate the n shares, randomly pick n points lying on the polynomial P(x)
Distribute the picked coordinates in the previous step among the participants. These act as the shares in the system

Phase II: Reconstruction of Secret
For reconstruction of the secret, a minimum of t participants are required to pool their shares.

Collect t or more shares
Use an interpolation algorithm to reconstruct the polynomial, P'(x), from the shares. Lagrange’s Interpolation is an example of such an algorithm
Determine the value of the reconstructed polynomial for x = 0, i.e. calculate P'(0). This value reveals the constant term of the polynomial which happens to be the original secret. Thus, the secret is reconstructed

Below is the implementation.

Python3

import random
from math import ceil
from decimal import Decimal
 
FIELD_SIZE = 10**5
 
 
def reconstruct_secret(shares):
    """
    Combines individual shares (points on graph)
    using Lagranges interpolation.
 
    `shares` is a list of points (x, y) belonging to a
    polynomial with a constant of our key.
    """
    sums = 0
    prod_arr = []
 
    for j, share_j in enumerate(shares):
        xj, yj = share_j
        prod = Decimal(1)
 
        for i, share_i in enumerate(shares):
            xi, _ = share_i
            if i != j:
                prod *= Decimal(Decimal(xi)/(xi-xj))
 
        prod *= yj
        sums += Decimal(prod)
 
    return int(round(Decimal(sums), 0))
 
 
def polynom(x, coefficients):
    """
    This generates a single point on the graph of given polynomial
    in `x`. The polynomial is given by the list of `coefficients`.
    """
    point = 0
    # Loop through reversed list, so that indices from enumerate match the
    # actual coefficient indices
    for coefficient_index, coefficient_value in enumerate(coefficients[::-1]):
        point += x ** coefficient_index * coefficient_value
    return point
 
 
def coeff(t, secret):
    """
    Randomly generate a list of coefficients for a polynomial with
    degree of `t` - 1, whose constant is `secret`.
 
    For example with a 3rd degree coefficient like this:
        3x^3 + 4x^2 + 18x + 554
 
        554 is the secret, and the polynomial degree + 1 is 
        how many points are needed to recover this secret. 
        (in this case it's 4 points).
    """
    coeff = [random.randrange(0, FIELD_SIZE) for _ in range(t - 1)]
    coeff.append(secret)
    return coeff
 
 
def generate_shares(n, m, secret):
    """
    Split given `secret` into `n` shares with minimum threshold
    of `m` shares to recover this `secret`, using SSS algorithm.
    """
    coefficients = coeff(m, secret)
    shares = []
 
    for i in range(1, n+1):
        x = random.randrange(1, FIELD_SIZE)
        shares.append((x, polynom(x, coefficients)))
 
    return shares
 
 
# Driver code
if __name__ == '__main__':
 
    # (3,5) sharing scheme
    t, n = 3, 5
    secret = 1234
    print(f'Original Secret: {secret}')
 
    # Phase I: Generation of shares
    shares = generate_shares(n, t, secret)
    print(f'Shares: {", ".join(str(share) for share in shares)}')
 
    # Phase II: Secret Reconstruction
    # Picking t shares randomly for
    # reconstruction
    pool = random.sample(shares, t)
    print(f'Combining shares: {", ".join(str(share) for share in pool)}')
    print(f'Reconstructed secret: {reconstruct_secret(pool)}')

Output:

Original Secret: 1234
Shares: (79761, 4753361900938), (67842, 3439017561016), (42323, 1338629004828), (68237, 3479175081966), (32818, 804981007208)
Combining shares: (32818, 804981007208), (79761, 4753361900938), (68237, 3479175081966)
Reconstructed secret: 1234

Practical Applications

Secret sharing schemes are widely used in cryptosystems where trust is required to be distributed instead of centralized. Prominent examples of real world scenarios where secret sharing is used include:

Threshold Based Signatures for Bitcoin
Secure Multi-Party Computation
Private Machine Learning with Multi Party Computation
Management of Passwords

Implementing Shamir’s Secret Sharing Scheme in Python

Secret Sharing schemes are used in the distribution of a secret value among multiple participants (shareholders) by dividing the secret into fragments (shares). This is done in a manner that prevents a single shareholder from having any useful knowledge of the original secret at all. The only way to retrieve the original secret is to combine the shares, distributed among the participants. Hence, the control of the secret is distributed. These schemes are examples of Threshold Cryptosystems, which involve the division of secrets among multiple parties such that several parties (more than some threshold number) must cooperate to reconstruct the secret.

Fig 1: Depiction of Secret Sharing between n participants

In general, a secret may be split into n shares (for n shareholders), out of which, a minimum of t, (t < n) shares are required for successful reconstruction. Such a scheme is referred to as a (t, n) sharing-scheme. From the n participants, any subset of shareholders, of size greater or equal to t, can regenerate the secret. Importantly, even with any k (k < t) shares, no new information about the original secret is learned.

Tags:

#python-utility #Python #python

Motivation behind SSS

Algorithm used

Python3

Practical Applications

Implementing Shamir’s Secret Sharing Scheme in Python

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