Threshold cryptography allows secure splitting of a secret into multiple pieces, called “shares.”
Using a technique like Shamir’s Secret Sharing, we can split a secret so that only a minimum number of shares (a threshold) are needed to reconstruct it. This method is useful for secure data sharing across multiple parties without relying on a single, central key.
In this article, we’ll build a simple threshold cryptography library in Rust that includes modular arithmetic for secure calculations.
We’ll use Shamir’s Secret Sharing to split a secret into shares and a prime modulus for modular arithmetic. This setup makes our calculations secure within a finite field, preventing overflow and adding security.
Dependencies
First, set up a new Rust project and add dependencies to handle large integers and randomness. In your Cargo.toml file, add:
[dependencies]
num-bigint = "0.4"
rand = "0.8"
- num-bigint: Handles big integers needed for secure cryptography.
- rand: Helps generate random numbers securely.
Define the Large Prime Modulus
In cryptography, calculations often use a prime modulus to stay within a fixed-size number range. We’ll use a large 256-bit prime, similar to those in elliptic curve cryptography. Here’s the constant definition in the code:
use num_bigint::{BigInt, RandBigInt, ModInverse};
use rand::rngs::OsRng;
// Define a large prime modulus
const PRIME_MODULUS: &str = "115792089237316195423570985008687907853269984665640564039457584007913129639319";
lazy_static::lazy_static! {
static ref PRIME: BigInt = PRIME_MODULUS.parse().unwrap();
}
This constant, PRIME, will be used in all calculations to ensure they stay within a safe range.
Setting Up the Structs for Threshold Secret Sharing
We’ll create a ThresholdSecretSharing struct to manage secret splitting and reconstruction. Each share will have two parts: x and y. The secret is a point at x = 0 on a polynomial we’ll generate randomly.
Here’s the struct setup:
pub struct Share {
pub x: BigInt,
pub y: BigInt,
}
pub struct ThresholdSecretSharing {
threshold: usize,
total_shares: usize,
}
impl ThresholdSecretSharing {
pub fn new(threshold: usize, total_shares: usize) -> Self {
assert!(threshold <= total_shares, "Threshold must be <= total shares.");
ThresholdSecretSharing {
threshold,
total_shares,
}
}
}
This setup includes a Share struct to hold each piece of the secret and the ThresholdSecretSharing struct to manage operations like splitting and reconstructing the secret.
Split the Secret
In Shamir’s Secret Sharing, a secret is split by generating a random polynomial. Each share is a point on this polynomial, and the polynomial’s constant term is the secret.
impl ThresholdSecretSharing {
pub fn split_secret(&self, secret: &BigInt) -> Vec<Share> {
let mut rng = OsRng;
let mut coefficients = vec![secret.clone() % &*PRIME]; // Secret as the constant term
for _ in 1..self.threshold {
coefficients.push(rng.gen_bigint(256) % &*PRIME); // Random coefficients
}
(1..=self.total_shares)
.map(|x| {
let x_val = BigInt::from(x);
let y_val = evaluate_polynomial(&coefficients, &x_val);
Share { x: x_val, y: y_val }
})
.collect()
}
}
// Helper function to evaluate polynomial at x with modular arithmetic
fn evaluate_polynomial(coefficients: &[BigInt], x: &BigInt) -> BigInt {
coefficients.iter().rev().fold(BigInt::from(0), |acc, coeff| {
(acc * x + coeff) % &*PRIME
})
}
The split_secret function generates random coefficients and evaluates the polynomial at different x-values, giving us shares. Each y value in the shares is calculated using modular arithmetic with our PRIME.
Reconstruct the Secret
We can reconstruct the secret using Lagrange interpolation. Given t shares, we calculate the polynomial’s constant term (the secret) using modular arithmetic to stay within the finite field.
impl ThresholdSecretSharing {
pub fn reconstruct_secret(&self, shares: &[Share]) -> BigInt {
assert!(shares.len() >= self.threshold, "Insufficient shares to reconstruct the secret.");
let mut secret = BigInt::from(0);
for i in 0..shares.len() {
let mut num = BigInt::from(1);
let mut denom = BigInt::from(1);
for j in 0..shares.len() {
if i != j {
num = (num * &shares[j].x) % &*PRIME;
denom = (denom * (&shares[j].x - &shares[i].x)) % &*PRIME;
}
}
let denom_inv = denom.mod_inverse(&*PRIME).unwrap(); // Modular inverse
let lagrange_coeff = (num * denom_inv) % &*PRIME;
secret = (secret + (&shares[i].y * lagrange_coeff)) % &*PRIME;
}
secret
}
}
The reconstruct_secret function uses modular arithmetic and Lagrange interpolation. The mod_inverse function calculates the inverse modulo PRIME, ensuring we stay within the finite field.
Example Usage
Now we can split and reconstruct a secret using the following code:
fn main() {
let secret = BigInt::from(1234567890);
let threshold = 3;
let total_shares = 5;
let tss = ThresholdSecretSharing::new(threshold, total_shares);
let shares = tss.split_secret(&secret);
// Select 3 shares to reconstruct the secret
let selected_shares = vec![shares[0].clone(), shares[1].clone(), shares[2].clone()];
let reconstructed_secret = tss.reconstruct_secret(&selected_shares);
println!("Original Secret: {}", secret);
println!("Reconstructed Secret: {}", reconstructed_secret);
}
Adding a few more useful features
Let’s now enhance the foundational implementation by adding:
Hash-Based Commitments for Share Verification
Adding a hash-based commitment allows each party to verify that a share is valid without reconstructing the secret. By storing the hash of each share, any recipient can check if their share has been tampered with. This feature enhances security, especially when shares are distributed over untrusted channels.
A Share Expiration Mechanism
To increase security, we could add an expiration mechanism to the shares. For instance, a timestamp can be embedded in each share, making them invalid after a certain period. This helps prevent stale shares from being used in sensitive applications where data access should expire.
Encrypting Shares for Secure Distribution
By encrypting each share before distribution, you can ensure that only authorized recipients can use their respective shares. This feature is especially useful when distributing shares over untrusted networks. We’ll use symmetric encryption to protect each share.
Enhanced Implementation
1. Add Hash-Based Commitments
First, let’s modify the Share struct to include a hash commitment for each share. We’ll use SHA-256 as the hash function for simplicity.
