A Developer's Guide to Shamir's Secret Sharing over GF(256)

In B2B data exchange systems, storing raw cryptographic keys in a single file storage center exposes the system to compromise. Shamir's Secret Sharing (SSS) solves this by splitting a secret key \(S\) into \(N\) parts. Any subset of size \(M\) can reconstruct \(S\), while fewer than \(M\) shares reveal zero information about the secret.

To implement SSS correctly in code, we must perform arithmetic operations inside a Galois Field, specifically \(GF(2^8)\). This guarantees that our coefficients and calculated shares remain exactly 8-bit bytes (0-255), preventing byte size inflation.

Polynomial Splitting Mechanics

We generate a random polynomial of degree \(M-1\), where the constant term is our secret byte \(S\):

We evaluate this polynomial at \(x = 1..N\) to obtain our distributed shares. Since this is Galois Field arithmetic, addition and subtraction are equivalent to the bitwise XOR operator (\(\oplus\)), and multiplication is computed using the Russian Peasant algorithm.

GF(256) Russian Peasant Multiplication Code

Here is the direct JavaScript implementation of Russian peasant multiplication over the generator polynomial \(x^8 + x^4 + x^3 + x + 1\) (0x11B):

function gfMultiply(a, b) {
  let p = 0;
  for (let i = 0; i < 8; i++) {
    if (b & 1) p ^= a;
    const hiBit = a & 0x80;
    a = (a << 1) & 0xFF;
    if (hiBit) a ^= 0x1B; // Reduce mod 0x11B
    b >>= 1;
  }
  return p;
}

Lagrange Interpolation Reconstruction

To reconstruct the master secret from \(M\) shares, we compute the Lagrange basis polynomials at \(x = 0\):

The sum of the products of each share value and its Lagrange basis polynomial gives the original secret.