问题
I tryed to implement Shamir's Secret Sharing in Java but I have some problem.
When I put K>10 the secret is no more reconstructed. Who can help me? This is what i've done. What's the problem?
Initially I choose N and K, next I have the generation of coefficients, the creation of shares and finally the reconstruction.
import java.math.BigInteger;
import java.util.Random;
public class Main {
public static void main(String[] args){
//INIT
int N = 55;
int K = 11;
BigInteger secret = new BigInteger("123");
modLength = secret.bitLength() + 1;
BigInteger primeNum = genPrime();
BigInteger[] coeff = new BigInteger[K-1];
BigInteger[] partecipants = new BigInteger[K];
for (int i=0;i<K;i++)
partecipants[i] = new BigInteger(Integer.toString(i+1));
System.out.println("Prime Number: "+primeNum);
for (int i=0;i<K-1;i++){
coeff[i] = randomZp(primeNum);
System.out.println("a"+(i+1)+": "+coeff[i]);
}
//SHARES
BigInteger[] shares = new BigInteger[N];
for(int i=0;i<N;i++){
BigInteger toAdd= secret;
for(int j=0;j<K-1;j++){
BigInteger power = new BigInteger(Integer.toString((int)(Math.pow((i+1),(j+1)))));
toAdd=toAdd.add(coeff[j].multiply(power));
}
shares[i] = toAdd.mod(primeNum);
System.out.println("Share n."+(i+1)+": "+shares[i]);
}
//INTERPOLAZIONE
BigInteger secret2= new BigInteger("0");
double term;
for (int i=0;i<K;i++){
term = 1;
BigInteger sharePartecipanteNow = shares[(partecipants[i].intValue())-1];
for (int j=0;j<K;j++){
if (partecipants[i].intValue()!=partecipants[j].intValue()){
BigInteger num = new BigInteger(Integer.toString(partecipants[j].intValue()*(-1)));
BigInteger den = new BigInteger(Integer.toString(partecipants[i].intValue()-partecipants[j].intValue()));
term = term*(num.doubleValue())/(den.doubleValue());
}
}
term = term*sharePartecipanteNow.intValue();
secret2 = secret2.add(new BigInteger(Integer.toString((int)term)));
}
while(secret2.intValue()<0)
secret2 = secret2.add(primeNum);
System.out.println("The secret is: "+secret2.mod(primeNum));
}
private static BigInteger genPrime() {
BigInteger p=null;
boolean ok=false;
do{
p=BigInteger.probablePrime(modLength, new Random());
if(p.isProbablePrime(CERTAINTY))
ok=true;
}while(ok==false);
return p;
}
private static BigInteger randomZp(BigInteger p) {
BigInteger r;
do{
r = new BigInteger(modLength, new Random());
} while (r.compareTo(BigInteger.ZERO) < 0 || r.compareTo(p) >= 0);
return r;
}
private static final int CERTAINTY = 50;
private static int modLength;
}
回答1:
Looks like your implementation suffers from numerical errors in secret reconstruction part (using double causes rounding errors).
However there is also a problem in shares generation part, however I'm not able to point it out.
I have rewritten your code using Java example from Shamir's Secret Sharing. This is quick and dirty but correct (I hope).
import java.math.BigInteger;
import java.util.Random;
public final class Shamir {
public final class SecretShare {
public SecretShare(final int num, final BigInteger share) {
this.num = num;
this.share = share;
}
public int getNum() {
return num;
}
public BigInteger getShare() {
return share;
}
@Override
public String toString() {
return "SecretShare [num=" + num + ", share=" + share + "]";
}
private final int num;
private final BigInteger share;
}
public Shamir(final int k, final int n) {
this.k = k;
this.n = n;
random = new Random();
}
public SecretShare[] split(final BigInteger secret) {
final int modLength = secret.bitLength() + 1;
prime = new BigInteger(modLength, CERTAINTY, random);
final BigInteger[] coeff = new BigInteger[k - 1];
System.out.println("Prime Number: " + prime);
for (int i = 0; i < k - 1; i++) {
coeff[i] = randomZp(prime);
System.out.println("a" + (i + 1) + ": " + coeff[i]);
}
final SecretShare[] shares = new SecretShare[n];
for (int i = 1; i <= n; i++) {
BigInteger accum = secret;
for (int j = 1; j < k; j++) {
final BigInteger t1 = BigInteger.valueOf(i).modPow(BigInteger.valueOf(j), prime);
final BigInteger t2 = coeff[j - 1].multiply(t1).mod(prime);
accum = accum.add(t2).mod(prime);
}
shares[i - 1] = new SecretShare(i - 1, accum);
System.out.println("Share " + shares[i - 1]);
}
return shares;
}
public BigInteger getPrime() {
return prime;
}
public BigInteger combine(final SecretShare[] shares, final BigInteger primeNum) {
BigInteger accum = BigInteger.ZERO;
for (int i = 0; i < k; i++) {
BigInteger num = BigInteger.ONE;
BigInteger den = BigInteger.ONE;
for (int j = 0; j < k; j++) {
if (i != j) {
num = num.multiply(BigInteger.valueOf(-j - 1)).mod(primeNum);
den = den.multiply(BigInteger.valueOf(i - j)).mod(primeNum);
}
}
System.out.println("den: " + den + ", num: " + den + ", inv: " + den.modInverse(primeNum));
final BigInteger value = shares[i].getShare();
final BigInteger tmp = value.multiply(num).multiply(den.modInverse(primeNum)).mod(primeNum);
accum = accum.add(primeNum).add(tmp).mod(primeNum);
System.out.println("value: " + value + ", tmp: " + tmp + ", accum: " + accum);
}
System.out.println("The secret is: " + accum);
return accum;
}
private BigInteger randomZp(final BigInteger p) {
while (true) {
final BigInteger r = new BigInteger(p.bitLength(), random);
if (r.compareTo(BigInteger.ZERO) > 0 && r.compareTo(p) < 0) {
return r;
}
}
}
private BigInteger prime;
private final int k;
private final int n;
private final Random random;
private static final int CERTAINTY = 50;
public static void main(final String[] args) {
final Shamir shamir = new Shamir(11, 20);
final BigInteger secret = new BigInteger("1234567890123456789012345678901234567890");
final SecretShare[] shares = shamir.split(secret);
final BigInteger prime = shamir.getPrime();
final Shamir shamir2 = new Shamir(11, 20);
final BigInteger result = shamir2.combine(shares, prime);
}
}
回答2:
karbi79's Shamir's Secret Sharing implementation is not valid. It could look like fine answer [basic test works fine], but it's not!
Proper implementation of Shamir Secret Sharing made my friend. It's his code:
import java.math.BigInteger;
import java.security.SecureRandom;
import java.util.Random;
public final class Shamir
{
public static SecretShare[] split(final BigInteger secret, int needed, int available, BigInteger prime, Random random)
{
System.out.println("Prime Number: " + prime);
final BigInteger[] coeff = new BigInteger[needed];
coeff[0] = secret;
for (int i = 1; i < needed; i++)
{
BigInteger r;
while (true)
{
r = new BigInteger(prime.bitLength(), random);
if (r.compareTo(BigInteger.ZERO) > 0 && r.compareTo(prime) < 0)
{
break;
}
}
coeff[i] = r;
}
final SecretShare[] shares = new SecretShare[available];
for (int x = 1; x <= available; x++)
{
BigInteger accum = secret;
for (int exp = 1; exp < needed; exp++)
{
accum = accum.add(coeff[exp].multiply(BigInteger.valueOf(x).pow(exp).mod(prime))).mod(prime);
}
shares[x - 1] = new SecretShare(x, accum);
System.out.println("Share " + shares[x - 1]);
}
return shares;
}
public static BigInteger combine(final SecretShare[] shares, final BigInteger prime)
{
BigInteger accum = BigInteger.ZERO;
for(int formula = 0; formula < shares.length; formula++)
{
BigInteger numerator = BigInteger.ONE;
BigInteger denominator = BigInteger.ONE;
for(int count = 0; count < shares.length; count++)
{
if(formula == count)
continue; // If not the same value
int startposition = shares[formula].getNumber();
int nextposition = shares[count].getNumber();
numerator = numerator.multiply(BigInteger.valueOf(nextposition).negate()).mod(prime); // (numerator * -nextposition) % prime;
denominator = denominator.multiply(BigInteger.valueOf(startposition - nextposition)).mod(prime); // (denominator * (startposition - nextposition)) % prime;
}
BigInteger value = shares[formula].getShare();
BigInteger tmp = value.multiply(numerator) . multiply(modInverse(denominator, prime));
accum = prime.add(accum).add(tmp) . mod(prime); // (prime + accum + (value * numerator * modInverse(denominator))) % prime;
}
System.out.println("The secret is: " + accum + "\n");
return accum;
}
private static BigInteger[] gcdD(BigInteger a, BigInteger b)
{
if (b.compareTo(BigInteger.ZERO) == 0)
return new BigInteger[] {a, BigInteger.ONE, BigInteger.ZERO};
else
{
BigInteger n = a.divide(b);
BigInteger c = a.mod(b);
BigInteger[] r = gcdD(b, c);
return new BigInteger[] {r[0], r[2], r[1].subtract(r[2].multiply(n))};
}
}
private static BigInteger modInverse(BigInteger k, BigInteger prime)
{
k = k.mod(prime);
BigInteger r = (k.compareTo(BigInteger.ZERO) == -1) ? (gcdD(prime, k.negate())[2]).negate() : gcdD(prime,k)[2];
return prime.add(r).mod(prime);
}
public static void main(final String[] args)
{
final int CERTAINTY = 256;
final SecureRandom random = new SecureRandom();
final BigInteger secret = new BigInteger("123");
// prime number must be longer then secret number
final BigInteger prime = new BigInteger(secret.bitLength() + 1, CERTAINTY, random);
// 2 - at least 2 secret parts are needed to view secret
// 5 - there are 5 persons that get secret parts
final SecretShare[] shares = Shamir.split(secret, 2, 5, prime, random);
// we can use any combination of 2 or more parts of secret
SecretShare[] sharesToViewSecret = new SecretShare[] {shares[0],shares[1]}; // 0 & 1
BigInteger result = Shamir.combine(sharesToViewSecret, prime);
sharesToViewSecret = new SecretShare[] {shares[1],shares[4]}; // 1 & 4
result = Shamir.combine(sharesToViewSecret, prime);
sharesToViewSecret = new SecretShare[] {shares[0],shares[1],shares[3]}; // 0 & 1 & 3
result = Shamir.combine(sharesToViewSecret, prime);
}
}
SecretShare.java:
import java.math.BigInteger;
public class SecretShare
{
public SecretShare(final int number, final BigInteger share)
{
this.number = number;
this.share = share;
}
public int getNumber()
{
return number;
}
public BigInteger getShare()
{
return share;
}
@Override
public String toString()
{
return "SecretShare [num=" + number + ", share=" + share + "]";
}
private final int number;
private final BigInteger share;
}
回答3:
https://github.com/codahale/jsecretsharing
// build 10 shares, of which 2 are required to recover the secret
ShareBuilder builder = new ShareBuilder("I am Batman. Seriously.".getBytes(), 2, 512); List shares = builder.build(10);
// takes 2 shares, recovers secret List someShares = new ArrayList(); someShares.add(shares.get(2)); someShares.add(shares.get(7)); ShareCombiner combiner = new ShareCombiner(someShares); System.err.println(new String(combiner.combine()));
// omg I'm batman
来源：https://stackoverflow.com/questions/19327651/java-implementation-of-shamirs-secret-sharing