Shamir's Secret Sharing

Web Crypto Version of S4

If the share is a k-degree polynomial then it needs k+1 points to generate the original polynomial with the secret.
If you needs 5 partial keys then you just have to generate 5 points that are on the polynomial.

Math¶

\[ Ax^4 + Bx^3 + Cx^2 + Dx + E \\ A, B, C, D, E\in\mathbb{Z} \]

\[ \mathcal{M} := \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 & Kx_1^4 & Kx_2^4 & Kx_3^4\cr 0 & 1 & 0 & 0 & 0 & 0 & Kx_1^3 & Kx_2^3 & Kx_3^3\cr 0 & 0 & 1 & 0 & 0 & 0 & Kx_1^2 & Kx_2^2 & Kx_3^2\cr 0 & 0 & 0 & 1 & 0 & 0 & Kx_1 & Kx_2 & Kx_3\cr 0 & 0 & 0 & 0 & 1 & 0 & K & K & K\cr 0 & 0 & 0 & 0 & 0 & 1 & -Kf(x_1) & -Kf(x_2) & -Kf(x_3)\cr \end{pmatrix} \]

Usage¶

from cryptopals_lib import *


from binascii import hexlify
from Crypto.Cipher import AES
from Crypto.Random import get_random_bytes
from Crypto.Protocol.SecretSharing import Shamir

#Generate Key
key = get_random_bytes(16)
print(key)

#Generate Shares
shares = Shamir.split(2, 5, key, ssss=True)

#Print Shares
for idx, share in shares:
    print("Index #{}: {}".format(idx, hexlify(share)))


#Use Two shares to get secret
key = Shamir.combine(shares[:2])
print(key)

Implementation¶

Security¶

When choosing random values for the shares make sure that the number cant be 0 this removes a term and than makes it easier for the secret to be generated

Each share must not repeat

Shamir's Secret Sharing