References:
[1] Pascal Paillier, "Public-Key Cryptosystems Based on Composite Degree Residuosity Classes," EUROCRYPT'99.
[2] Introduction to Paillier cryptosystem from Wikipedia.

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/**
* This program is free software: you can redistribute it and/or modify it
* Software Foundation, either version 3 of the License, or (at your option)
* any later version.
*
* This program is distributed in the hope that it will be useful, but WITHOUT
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
* FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for
* more details.
*
* You should have received a copy of the GNU General Public License along with
* this program. If not, see <http://www.gnu.org/licenses/>.
*/

import java.math.*;
import java.util.*;

/**
* Paillier Cryptosystem <br><br>
* References: <br>
* [1] Pascal Paillier, "Public-Key Cryptosystems Based on Composite Degree Residuosity Classes," EUROCRYPT'99.
* URL: <a href="http://www.gemplus.com/smart/rd/publications/pdf/Pai99pai.pdf">http://www.gemplus.com/smart/rd/publications/pdf/Pai99pai.pdf</a><br>
*
* [2] Paillier cryptosystem from Wikipedia.
* URL: <a href="http://en.wikipedia.org/wiki/Paillier_cryptosystem">http://en.wikipedia.org/wiki/Paillier_cryptosystem</a>
* @author Kun Liu (kunliu1@cs.umbc.edu)
* @version 1.0
*/
public class Paillier {

/**
* p and q are two large primes.
* lambda = lcm(p-1, q-1) = (p-1)*(q-1)/gcd(p-1, q-1).
*/
private BigInteger p, q, lambda;
/**
* n = p*q, where p and q are two large primes.
*/
public BigInteger n;
/**
* nsquare = n*n
*/
public BigInteger nsquare;
/**
* a random integer in Z*_{n^2} where gcd (L(g^lambda mod n^2), n) = 1.
*/
private BigInteger g;
/**
* number of bits of modulus
*/
private int bitLength;

/**
* Constructs an instance of the Paillier cryptosystem.
* @param bitLengthVal number of bits of modulus
* @param certainty The probability that the new BigInteger represents a prime number will exceed (1 - 2^(-certainty)). The execution time of this constructor is proportional to the value of this parameter.
*/
public Paillier(int bitLengthVal, int certainty) {
KeyGeneration(bitLengthVal, certainty);
}

/**
* Constructs an instance of the Paillier cryptosystem with 512 bits of modulus and at least 1-2^(-64) certainty of primes generation.
*/
public Paillier() {
KeyGeneration(512, 64);
}

/**
* Sets up the public key and private key.
* @param bitLengthVal number of bits of modulus.
* @param certainty The probability that the new BigInteger represents a prime number will exceed (1 - 2^(-certainty)). The execution time of this constructor is proportional to the value of this parameter.
*/
public void KeyGeneration(int bitLengthVal, int certainty) {
bitLength = bitLengthVal;
/*Constructs two randomly generated positive BigIntegers that are probably prime, with the specified bitLength and certainty.*/
p = new BigInteger(bitLength / 2, certainty, new Random());
q = new BigInteger(bitLength / 2, certainty, new Random());

n = p.multiply(q);
nsquare = n.multiply(n);

g = new BigInteger("2");
lambda = p.subtract(BigInteger.ONE).multiply(q.subtract(BigInteger.ONE)).divide(
p.subtract(BigInteger.ONE).gcd(q.subtract(BigInteger.ONE)));
/* check whether g is good.*/
if (g.modPow(lambda, nsquare).subtract(BigInteger.ONE).divide(n).gcd(n).intValue() != 1) {
System.out.println("g is not good. Choose g again.");
System.exit(1);
}
}

/**
* Encrypts plaintext m. ciphertext c = g^m * r^n mod n^2. This function explicitly requires random input r to help with encryption.
* @param m plaintext as a BigInteger
* @param r random plaintext to help with encryption
* @return ciphertext as a BigInteger
*/
public BigInteger Encryption(BigInteger m, BigInteger r) {
return g.modPow(m, nsquare).multiply(r.modPow(n, nsquare)).mod(nsquare);
}

/**
* Encrypts plaintext m. ciphertext c = g^m * r^n mod n^2. This function automatically generates random input r (to help with encryption).
* @param m plaintext as a BigInteger
* @return ciphertext as a BigInteger
*/
public BigInteger Encryption(BigInteger m) {
BigInteger r = new BigInteger(bitLength, new Random());
return g.modPow(m, nsquare).multiply(r.modPow(n, nsquare)).mod(nsquare);

}

/**
* Decrypts ciphertext c. plaintext m = L(c^lambda mod n^2) * u mod n, where u = (L(g^lambda mod n^2))^(-1) mod n.
* @param c ciphertext as a BigInteger
* @return plaintext as a BigInteger
*/
public BigInteger Decryption(BigInteger c) {
BigInteger u = g.modPow(lambda, nsquare).subtract(BigInteger.ONE).divide(n).modInverse(n);
return c.modPow(lambda, nsquare).subtract(BigInteger.ONE).divide(n).multiply(u).mod(n);
}

/**
* main function
* @param str intput string
*/
public static void main(String[] str) {
/* instantiating an object of Paillier cryptosystem*/
Paillier paillier = new Paillier();
/* instantiating two plaintext msgs*/
BigInteger m1 = new BigInteger("20");
BigInteger m2 = new BigInteger("60");
/* encryption*/
BigInteger em1 = paillier.Encryption(m1);
BigInteger em2 = paillier.Encryption(m2);
/* printout encrypted text*/
System.out.println(em1);
System.out.println(em2);
/* printout decrypted text */
System.out.println(paillier.Decryption(em1).toString());
System.out.println(paillier.Decryption(em2).toString());

/* test homomorphic properties -> D(E(m1)*E(m2) mod n^2) = (m1 + m2) mod n */
BigInteger product_em1em2 = em1.multiply(em2).mod(paillier.nsquare);
System.out.println("original sum: " + sum_m1m2.toString());
System.out.println("decrypted sum: " + paillier.Decryption(product_em1em2).toString());

/* test homomorphic properties -> D(E(m1)^m2 mod n^2) = (m1*m2) mod n */
BigInteger expo_em1m2 = em1.modPow(m2, paillier.nsquare);
BigInteger prod_m1m2 = m1.multiply(m2).mod(paillier.n);
System.out.println("original product: " + prod_m1m2.toString());
System.out.println("decrypted product: " + paillier.Decryption(expo_em1m2).toString());

}
}

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