java实现同态加密算法的实例代码

什么是同态加密？

同态加密是上世纪七十年代就被提出的一个开放问题，旨在不暴露数据的情况下完成对数据的处理，关注的是数据处理安全。

想象一下这样一个场景，作为一名满怀理想的楼二代，你每天过着枯燥乏味的收租生活，希望摆脱世俗的枷锁、铜臭的苟且去追求诗与远方。

你需要雇一个代理人去承担收租的粗活，但又不希望其窥探你每月躺赚的收入。于是，你请高人打造了一套装备，既能保证代理人顺利完成收租，又不会泄露收入信息。

这套装备包括信封、胶水、皮夹和神奇剪刀，每一样东西都有奇特的功能：

信封一旦用胶水密封，只有神奇剪刀才能拆开。
不论信封里装了多少钱，信封的大小和重量都不会发生改变。
把多个信封放在皮夹里后，信封会在不拆开的情况下两两合并，最后变成一个信封，里面装的钱正好是合并前所有信封金额的总和。

你把信封和胶水分发给所有租客，把皮夹交给代理人。

到了约定交租的日子，租客把租金放到信封里密封后交给代理人；代理人收齐信封，放到皮夹中，最后得到一个装满所有租金的信封，再转交给你；你使用神奇剪刀拆开，拿到租金。

在这个场景中，信封的a、b两个性质其实就是公钥加密的特性，即使用公钥加密得到的密文只有掌握私钥的人能够解密，并且密文不会泄露明文的语义信息；而c则代表加法同态的特性，两个密文可以进行计算，得到的结果解密后正好是两个原始明文的和。

原理：

paillier加密算法步骤：密钥生成、加密、解密

1、密钥生成

1.1 随机选择两个大质数p和q满足gcd(pq,(p-1)(q-1)) =1。这个属性保证两个质数长度相等。

1.2 计算n=pq和λ=lcm(p-1,q-1)

1.3 选择随机整数g(g ∈ Z n 2 ∗ g∈Z_{n^2}^*g∈Zn2∗)，使得满足n整除g的阶。

1.4 公钥为(N,g)

1.5 私钥为λ

g c d ( L ( g λ m o d n 2 ) , n ) = 1 gcd(L(g^λ mod n^2),n)=1gcd(L(gλmodn2),n)=1

2、加密

2.1 选择随机数r ∈ Z n r∈Z_nr∈Zn

2.2 计算密文

c = E ( m , r ) = g m r n m o d n 2 , r ∈ Z n c = E(m,r) = g^m r^n mod n^2 ,r∈Z_nc=E(m,r)=gmrnmodn2,r∈Zn,其中m为加密信息。

3、解密

m = D ( c , λ ) = ( L ( c λ m o d n 2 ) / L ( g λ m o d n 2 ) ) m o d n , 其中 L ( u ) = u − 1 / N m= D(c,λ)=(L(c^λ mod n^2)/L(g^λ mod n^2)) mod n,其中 L(u)=u-1/Nm=D(c,λ)=(L(cλmodn2)/L(gλmodn2))modn,其中L(u)=u−1/N

java实现：

package com;
 
/**
 * This program is free software: you can redistribute it and/or modify it
 * under the terms of the GNU General Public License as published by the Free
 * 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" rel="external nofollow" >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" rel="external nofollow" >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);
 }
 
 /**
  * sum of (cipher) em1 and em2
  *
  * @param em1
  * @param em2
  * @return
  */
 public BigInteger cipher_add(BigInteger em1, BigInteger em2) {
  return em1.multiply(em2).mod(nsquare);
 }
 
 /**
  * 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
   */
  // m1+m2,求明文数值的和
  BigInteger sum_m1m2 = m1.add(m2).mod(paillier.n);
  System.out.println("original sum: " + sum_m1m2.toString());
  // em1+em2，求密文数值的乘
  BigInteger product_em1em2 = em1.multiply(em2).mod(paillier.nsquare);
  System.out.println("encrypted sum: " + product_em1em2.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 */
  // m1*m2,求明文数值的乘
  BigInteger prod_m1m2 = m1.multiply(m2).mod(paillier.n);
  System.out.println("original product: " + prod_m1m2.toString());
  // em1的m2次方，再mod paillier.nsquare
  BigInteger expo_em1m2 = em1.modPow(m2, paillier.nsquare);
  System.out.println("encrypted product: " + expo_em1m2.toString());
  System.out.println("decrypted product: " + paillier.Decryption(expo_em1m2).toString());
 
  //sum test
  System.out.println("--------------------------------");
  Paillier p = new Paillier();
  BigInteger t1 = new BigInteger("21");System.out.println(t1.toString());
  BigInteger t2 = new BigInteger("50");System.out.println(t2.toString());
  BigInteger t3 = new BigInteger("50");System.out.println(t3.toString());
  BigInteger et1 = p.Encryption(t1);System.out.println(et1.toString());
  BigInteger et2 = p.Encryption(t2);System.out.println(et2.toString());
  BigInteger et3 = p.Encryption(t3);System.out.println(et3.toString());
  BigInteger sum = new BigInteger("1");
  sum = p.cipher_add(sum, et1);
  sum = p.cipher_add(sum, et2);
  sum = p.cipher_add(sum, et3);
  System.out.println("sum: "+sum.toString());
  System.out.println("decrypted sum: "+p.Decryption(sum).toString());
  System.out.println("--------------------------------");
 }
}

参考：https://mp.weixin.qq.com/s?__biz=MzA3MTI5Njg4Mw==&mid=2247486135&idx=1&sn=8c9431012aef19bbdefdcd673a783c34&chksm=9f2ef8aba85971bdfb623e8303b103fd70ac2a5ad802668388233ca930d1b0cd77fb02d4b0f2&scene=21#wechat_redirect

https://www.csee.umbc.edu/~kunliu1/research/Paillier.html

总结

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