shamir

Shamir's Secret Sharing

A implementation of Shamir's Secret Sharing algorithm over GF(256) in both Java and JavaScript.

Add to your Java project

<dependency>
  <groupId>com.codahale</groupId>
  <artifactId>shamir</artifactId>
  <version>0.7.0</version>
</dependency>

Note: module name for Java 9+ is com.codahale.shamir.

Use the thing in Java

import com.codahale.shamir.Scheme;
import java.nio.charset.StandardCharsets;
import java.security.SecureRandom;
import java.util.Map;
 
class Example {
  void doIt() {
    final Scheme scheme = new Scheme(new SecureRandom(), 5, 3);
    final byte[] secret = "hello there".getBytes(StandardCharsets.UTF_8);
    final Map<Integer, byte[]> parts = scheme.split(secret);
    final byte[] recovered = scheme.join(parts);
    System.out.println(new String(recovered, StandardCharsets.UTF_8));
  } 
}

Use the thing in JavaScript

const { split, join } = require('shamir');
const { randomBytes } = require('crypto');
 
const PARTS = 5;
const QUORUM = 3;
 
function doIt() {
    const secret = 'hello there';
    // you can use any polyfill to covert string to Uint8Array
    const utf8Encoder = new TextEncoder();
    const utf8Decoder = new TextDecoder();
    const secretBytes = utf8Encoder.encode('hello there');
    // parts is a object whos keys are the part number and values are an Uint8Array
    const parts = split(randomBytes, PARTS, QUORUM, secretBytes);
    // we only need QUORUM of the parts to recover the secret
    delete parts[2];
    delete parts[3];
    // recovered is an Unit8Array
    const recovered = join(parts);
    // prints 'hello there'
    console.log(utf8Decoder.decode(recovered));
}

How it works

Shamir's Secret Sharing algorithm is a way to split an arbitrary secret S into N parts, of which at least K are required to reconstruct S. For example, a root password can be split among five people, and if three or more of them combine their parts, they can recover the root password.

Splitting secrets

Splitting a secret works by encoding the secret as the constant in a random polynomial of K degree. For example, if we're splitting the secret number 42 among five people with a threshold of three (N=5,K=3), we might end up with the polynomial:

f(x) = 71x^3 - 87x^2 + 18x + 42

To generate parts, we evaluate this polynomial for values of x greater than zero:

f(1) =   44
f(2) =  298
f(3) = 1230
f(4) = 3266
f(5) = 6822

These (x,y) pairs are then handed out to the five people.

Joining parts

When three or more of them decide to recover the original secret, they pool their parts together:

f(1) =   44
f(3) = 1230
f(4) = 3266

Using these points, they construct a Lagrange polynomial, g, and calculate g(0). If the number of parts is equal to or greater than the degree of the original polynomial (i.e. K), then f and g will be exactly the same, and f(0) = g(0) = 42, the encoded secret. If the number of parts is less than the threshold K, the polynomial will be different and g(0) will not be 42.

Implementation details

Shamir's Secret Sharing algorithm only works for finite fields, and this library performs all operations in GF(256). Each byte of a secret is encoded as a separate GF(256) polynomial, and the resulting parts are the aggregated values of those polynomials.

Using GF(256) allows for secrets of arbitrary length and does not require additional parameters, unlike GF(Q), which requires a safe modulus. It's also much faster than GF(Q): splitting and combining a 1KiB secret into 8 parts with a threshold of 3 takes single-digit milliseconds, whereas performing the same operation over GF(Q) takes several seconds, even using per-byte polynomials. Treating the secret as a single y coordinate over GF(Q) is even slower, and requires a modulus larger than the secret.

Java Performance

It's fast. Plenty fast.

For a 1KiB secret split with a n=4,k=3 scheme:

Benchmark         (n)  (secretSize)  Mode  Cnt     Score    Error  Units
Benchmarks.join     4          1024  avgt  200   196.787 ±  0.974  us/op
Benchmarks.split    4          1024  avgt  200   396.708 ±  1.520  us/op

N.B.: split is quadratic with respect to the number of shares being combined.

JavaScript Performance

For a 1KiB secret split with a n=4,k=3 scheme running on NodeJS v10.16.0:

Benchmark         (n)  (secretSize)  Cnt   Score   Units
Benchmarks.join     4          1024  200   2.08    ms/op
Benchmarks.split    4          1024  200   2.78    ms/op

Split is dominated by the calls to get random polynomials per byte of the secet. Using a more realistic 128 bit secret with n=4,k=3 scheme running on NodeJS v10.16.0:

Benchmark         (n)  (secretSize)  Cnt   Score   Units
Benchmarks.join     5          1024  200   0.083    ms/op
Benchmarks.split    5          1024  200   0.081    ms/op

Tiered sharing

Some usages of secret sharing involve levels of access: e.g. recovering a secret requires two admin shares and three user shares. As @ba1ciu discovered, these can be implemented by building a tree of shares:

class BuildTree {
  public static void shareTree(String... args) {
    final byte[] secret = "this is a secret".getBytes(StandardCharsets.UTF_8);
    
    // tier 1 of the tree 
    final Scheme adminScheme = new Scheme(new SecureRandom(), 5, 2);
    final Map<Integer, byte[]> admins = adminScheme.split(secret);
 
    // tier 2 of the tree 
    final Scheme userScheme = Scheme.of(4, 3);
    final Map<Integer, Map<Integer, byte[]>> admins =
        users.entrySet()
            .stream()
            .collect(Collectors.toMap(Map.Entry::getKey, e -> userScheme.split(e.getValue())));
    
    System.out.println("Admin shares:");
    System.out.printf("%d = %s\n", 1, Arrays.toString(admins.get(1)));
    System.out.printf("%d = %s\n", 2, Arrays.toString(admins.get(2)));
 
    System.out.println("User shares:");
    System.out.printf("%d = %s\n", 1, Arrays.toString(users.get(3).get(1)));
    System.out.printf("%d = %s\n", 2, Arrays.toString(users.get(3).get(2)));
    System.out.printf("%d = %s\n", 3, Arrays.toString(users.get(3).get(3)));
    System.out.printf("%d = %s\n", 4, Arrays.toString(users.get(3).get(4)));
  }
}

By discarding the third admin share and the first two sets of user shares, we have a set of shares which can be used to recover the original secret as long as either two admins or one admin and three users agree.

License

Copyright © 2017 Coda Hale

Copyright © 2019 Simon Massey

Distributed under the Apache License 2.0.