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

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

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

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));
  } 
}

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 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.

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.

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.

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.

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.

Copyright © 2017 Coda Hale

Distributed under the Apache License 2.0.