RSA 加密原理

公鑰＝(E , N)
私鑰＝(D, N)

對外，我們只暴露公鑰。

示例

1、找出質數 P 、Q

P = 3  
Q = 11

2、計算公共模數

N = P * Q = 3 * 11 = 33
N = 33

3、 尤拉函式

φ(N) = (P-1)(Q-1) = 2 * 10 = 20
φ(N) = 20

4、計算公鑰E

1 < E < φ(N)
1 <E < 20

E 的取值範圍 {3, 7, 9, 11, 13, 17, 19}
E的取值必須是整數, E 和 φ(N) 必須是互質數
為了測試，我們取最小的值 E =3

5、計算私鑰D

E * D % φ(N) = 1
3 * D  % 20 = 1   

根據上面可計算出 D = 7

6、公鑰加密

我們這裡為了演示，就加密一個比較小的數字 M = 2

公式：C ＝ M^E mod N

M = 2
E = 3
N = 33

C = 2³ % 33 = 8

明文 “2” 經過 RSA 加密後變成了密文 “8”

7、私鑰解密

M ＝C^D mod N

C = 8
D = 7
N = 33

M = 8⁷ % 33
8 * 8 * 8 * 8 * 8 * 8 * 8=2097152
8 * 8 * 8 * 8 * 8 * 8 * 8 % 33 = 2

密文 “8” 經過 RSA 解密後變成了明文 2。

公鑰加密 - 私鑰解密流程圖

私鑰加密 - 公鑰解密流程圖

JDK 自帶的 RSA 演算法 示例

import java.security.KeyFactory;
import java.security.KeyPair;
import java.security.KeyPairGenerator;
import java.security.NoSuchAlgorithmException;
import java.security.PrivateKey;
import java.security.PublicKey;
import java.security.interfaces.RSAPrivateKey;
import java.security.interfaces.RSAPublicKey;
import java.security.spec.PKCS8EncodedKeySpec;
import java.security.spec.X509EncodedKeySpec;

import javax.crypto.Cipher;

import org.apache.commons.codec.binary.Base64;

/**
 * 非對稱加密 唯一廣泛接受並實現 資料加密&數字簽名 公鑰加密、私鑰解密 私鑰加密、公鑰解密
 * 
 * @author jjs
 *
 */
public class RSADemo {

    private static String src = "infcn";

    private static RSAPublicKey rsaPublicKey;
    private static RSAPrivateKey rsaPrivateKey;

    static {
        // 1、初始化金鑰
        KeyPairGenerator keyPairGenerator;
        try {
            keyPairGenerator = KeyPairGenerator.getInstance("RSA");
            keyPairGenerator.initialize(512);// 64的整倍數
            KeyPair keyPair = keyPairGenerator.generateKeyPair();
            rsaPublicKey = (RSAPublicKey) keyPair.getPublic();
            rsaPrivateKey = (RSAPrivateKey) keyPair.getPrivate();
            System.out.println("Public Key : " + Base64.encodeBase64String(rsaPublicKey.getEncoded()));
            System.out.println("Private Key : " + Base64.encodeBase64String(rsaPrivateKey.getEncoded()));
        } catch (NoSuchAlgorithmException e) {
            e.printStackTrace();
        }
    }

    /**
     * 公鑰加密，私鑰解密
     * @author jijs
     */
    public static void pubEn2PriDe() {
        //公鑰加密
        X509EncodedKeySpec x509EncodedKeySpec = new X509EncodedKeySpec(rsaPublicKey.getEncoded());
        KeyFactory keyFactory = KeyFactory.getInstance("RSA");
        PublicKey publicKey = keyFactory.generatePublic(x509EncodedKeySpec);
        Cipher cipher = Cipher.getInstance("RSA");
        cipher.init(Cipher.ENCRYPT_MODE, publicKey);
        byte[] result = cipher.doFinal(src.getBytes());
        System.out.println("公鑰加密，私鑰解密 --加密: " + Base64.encodeBase64String(result));

        //私鑰解密
        PKCS8EncodedKeySpec pkcs8EncodedKeySpec = new PKCS8EncodedKeySpec(rsaPrivateKey.getEncoded());
        keyFactory = KeyFactory.getInstance("RSA");
        PrivateKey privateKey = keyFactory.generatePrivate(pkcs8EncodedKeySpec);
        cipher = Cipher.getInstance("RSA");
        cipher.init(Cipher.DECRYPT_MODE, privateKey);
        result = cipher.doFinal(result);
        System.out.println("公鑰加密，私鑰解密 --解密: " + new String(result));
    }


    /**
     * 私鑰加密，公鑰解密
     * @author jijs
     */
    public static void priEn2PubDe() {

        //私鑰加密
        PKCS8EncodedKeySpec pkcs8EncodedKeySpec = new PKCS8EncodedKeySpec(rsaPrivateKey.getEncoded());
        KeyFactory keyFactory = KeyFactory.getInstance("RSA");
        PrivateKey privateKey = keyFactory.generatePrivate(pkcs8EncodedKeySpec);
        Cipher cipher = Cipher.getInstance("RSA");
        cipher.init(Cipher.ENCRYPT_MODE, privateKey);
        byte[] result = cipher.doFinal(src.getBytes());
        System.out.println("私鑰加密，公鑰解密 --加密 : " + Base64.encodeBase64String(result));

        //公鑰解密
        X509EncodedKeySpec x509EncodedKeySpec = new X509EncodedKeySpec(rsaPublicKey.getEncoded());
        keyFactory = KeyFactory.getInstance("RSA");
        PublicKey publicKey = keyFactory.generatePublic(x509EncodedKeySpec);
        cipher = Cipher.getInstance("RSA");
        cipher.init(Cipher.DECRYPT_MODE, publicKey);
        result = cipher.doFinal(result);
        System.out.println("私鑰加密，公鑰解密   --解密: " + new String(result));
    }

    public static void main(String[] args) {
        pubEn2PriDe();  //公鑰加密，私鑰解密
        priEn2PubDe();  //私鑰加密，公鑰解密
    }
}