RSA 非对称加密原理（小白也能看懂哦~）

川长思鸟来 2022-03-31 05:08 259阅读 0赞

RSA 加密原理

<table> 
 <thead> 
 <tr> 
 <th> 步骤 </th> 
 <th> 说明 </th> 
 <th> 描述 </th> 
 <th> 备注 </th> 
 </tr> 
 </thead> 
 <tbody> 
 <tr> 
 <td> 1 </td> 
 <td> 找出质数 </td> 
 <td> P 、Q </td> 
 <td> - </td> 
 </tr> 
 <tr> 
 <td> 2 </td> 
 <td> 计算公共模数 </td> 
 <td> N = P * Q </td> 
 <td> - </td> 
 </tr> 
 <tr> 
 <td> 3 </td> 
 <td> 欧拉函数 </td> 
 <td> φ(N) = (P-1)(Q-1) </td> 
 <td> - </td> 
 </tr> 
 <tr> 
 <td> 4 </td> 
 <td> 计算公钥E </td> 
 <td> 1 &lt; E &lt; φ(N) </td> 
 <td> E的取值必须是整数 E 和 φ(N) 必须是互质数 </td> 
 </tr> 
 <tr> 
 <td> 5 </td> 
 <td> 计算私钥D </td> 
 <td> E * D % φ(N) = 1 </td> 
 <td> - </td> 
 </tr> 
 <tr> 
 <td> 6 </td> 
 <td> 加密 </td> 
 <td> C ＝ M E mod N </td> 
 <td> C：密文 M：明文 </td> 
 </tr> 
 <tr> 
 <td> 7 </td> 
 <td> 解密 </td> 
 <td> M ＝C D mod N </td> 
 <td> C：密文 M：明文 </td> 
 </tr> 
 </tbody> 
</table>

公钥＝(E , N) 私钥＝(D, N)

对外，我们只暴露公钥。

## 示例 ##

1.找出两个质数

P = 3  
Q = 11

2.计算公共模数

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

N = 33 //公共取模数

3.欧拉函数求质数最大范围

φ(N) = (P-1)(Q-1) = 2 \* 10 = 20 //求质数取数范围  
φ(N) = 20

4.计算公钥E

// E 质素范围内取公钥E 
1 < E < φ(N) 
1 <E < 20

E 的取值范围 \{3, 7, 9, 11, 13, 17, 19\}  
E的取值必须是整数, E 和 φ(N) 必须是互质数  
为了测试，我们取最小的值 E =3  
3 和 φ(N) =20 互为质数，满足条件

5.计算私钥D

// 通过欧拉公式计算出  
E \* D % φ(N) = 1  
3 \* D  % 20 = 1

e.g.

3\*7 % 20 = 1

根据上面可计算出  D = 7

6.公钥加密

公式：C ＝ ME mod N

M = 2  //被加密的数字  
E = 3  //公钥  
N = 33 //公共取模数

C = 23 % 33

C = 2\*2\*2%33 = 8  //加密后的值

7.私钥解密

M ＝CD mod N

C = 8  //密文  
D = 7 //秘钥  
N = 33 //公共取模数

M = 87 % 33

8 \* 8 \* 8 \* 8 \* 8 \* 8 \* 8=2097152  
8 \* 8 \* 8 \* 8 \* 8 \* 8 \* 8 % 33 = 2 //解密后的原文

## 公钥加密 - 私钥解密流程图 ##

![1620][]

公钥加密 - 私钥解密流程图

## 私钥加密 - 公钥解密流程图 ##

![1620 1][]

私钥加密 - 公钥解密流程图

## JDK 自带的 RSA 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;
    
    /**
     * 非对称加密 唯一广泛接受并实现 数据加密&数字签名 公钥加密、私钥解密 私钥加密、公钥解密
     * 
     * @author jjs
     *
     */
    public class RSADemo {
    
        private static String src = "infcn";
    
        private static RSAPublicKey rsaPublicKey;
        private static RSAPrivateKey rsaPrivateKey;
    
        static {
            // 1、初始化密钥
            KeyPairGenerator keyPairGenerator;
            try {
                keyPairGenerator = KeyPairGenerator.getInstance("RSA");
                keyPairGenerator.initialize(512);// 64的整倍数
                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();
            }
        }
        
        /**
         * 公钥加密，私钥解密
         * @author jijs
         */
        public static void pubEn2PriDe() {
            //公钥加密
            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("公钥加密，私钥解密 --加密: " + Base64.encodeBase64String(result));
    
            //私钥解密
            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("公钥加密，私钥解密 --解密: " + new String(result));
        }
        
    
        /**
         * 私钥加密，公钥解密
         * @author jijs
         */
        public static void priEn2PubDe() {
    
            //私钥加密
            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("私钥加密，公钥解密 --加密 : " + Base64.encodeBase64String(result));
    
            //公钥解密
            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("私钥加密，公钥解密   --解密: " + new String(result));
        }
    
        public static void main(String[] args) {
            pubEn2PriDe();  //公钥加密，私钥解密
            priEn2PubDe();  //私钥加密，公钥解密
        }
    }

参考文献：

[https://cloud.tencent.com/developer/article/1130051][https_cloud.tencent.com_developer_article_1130051]

[1620]: /images/20220331/378a7c8f8acc46e9a4549e1130239b23.png
[1620 1]: /images/20220331/a7b283c99dc942449ae98eef42125b5c.png
[https_cloud.tencent.com_developer_article_1130051]: https://cloud.tencent.com/developer/article/1130051