Shamir's Secret Sharing (SSS)

This project is a web-based implementation of Shamir's Secret Sharing (SSS). It allows users to split sensitive text or images into multiple cryptographic shares. To recover the original secret, a minimum number of shares (the threshold) must be combined.

If fewer than the threshold number of shares are available, it is mathematically impossible to reconstruct the secret.

Features

• Text Encryption & Decryption
  Convert secret messages into cryptographic shares and reconstruct them later.

• Image Encryption & Decryption
  Image files are processed as byte data, split into chunks, and converted into shares stored inside a ZIP archive.

• Web Interface
  A brutalist-style frontend built using HTML, CSS, and JavaScript.

• FastAPI Backend
  A Python backend responsible for all cryptographic and mathematical operations.

• Terminal Version
  A standalone CLI script for command-line usage.

How the Math Works

The implementation is based on polynomial interpolation over a finite field, following the original scheme proposed by Adi Shamir.

1. The Polynomial

To hide a secret, we generate a random polynomial of degree k − 1, where k is the minimum number of shares required for recovery.

P(x) = a₀ + a₁x + a₂x² + … + aₖ₋₁xᵏ⁻¹

• a₀ (constant term)
  This is the secret. Recovering P(0) reveals the original secret.

• a₁ … aₖ₋₁
  Randomly generated coefficients, freshly chosen for each encryption.

2. Creating Shares

Each share corresponds to a point on the polynomial:

(xᵢ, yᵢ) where yᵢ = P(xᵢ)

• The x-value is a unique share index (e.g. 1, 2, 3, …).
• The y-value is the polynomial evaluated at that x.

Each share individually reveals no information about the secret.

3. Finite Field Arithmetic

All computations are performed inside a finite field to prevent integer overflow and ensure cryptographic security.

This project uses the Mersenne prime:

p = 2^521 − 1

Every operation is performed modulo ( p ):

• Addition
• Multiplication
• Exponentiation

result = result mod p

This guarantees that all values remain bounded and reversible.

4. Recovery (Lagrange Interpolation)

When at least ( k ) valid shares are available, the secret can be reconstructed using Lagrange interpolation.

Given ( k ) points, there exists exactly one polynomial of degree ( k - 1 ) that passes through all of them. The secret is recovered by computing:

P(0) = a₀

Lagrange interpolation directly evaluates this value without reconstructing the full polynomial explicitly.