RSA asymmetric encryption principle

RSA encryption principle

step	Explanation	description	Remark
1	To find a prime number	P and Q	-
2	Calculating public modulus	N = P * Q	-
3	Euler function	φ(N) = (P-1)(Q-1)	-
4	Calculate a public key E	1 < E < φ(N)	E value of E must be an integer and φ (N) must be co-prime
5	Compute the private key D	E * D% φ (N) = 1	-
6	encryption	C ＝ M E mod N	C: ciphertext M: plain
7	Decryption	M ＝C D mod N	C: ciphertext M: plain

Public Key = (E, N)
private key = (D, N)

Externally, we only expose the public key.

Examples

1, to find a prime number P, Q

P = 3  
Q = 11

2, the public modulus calculated

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

3, Euler function

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

4, calculates the public key E

1 < E < φ(N)
1 <E < 20

In the range of E. 3 {,. 7,. 9,. 11, 13 is,. 17,. 19}
E values must be an integer, E and φ (N) must be co-prime
For testing, we take a minimum value. 3 = E
. 3 and φ (N) = 20 relatively prime, the condition

5, the private key D is calculated

E * D % φ(N) = 1
3 * D  % 20 = 1  

According to the above calculated D = 7

6, public key encryption

We are here to demonstrate, to encrypt a relatively small number M = 2

Official: C = ME mod N

M = 2
E = 3
N = 33

C = 23 % 33 = 8

Clear "2" after the RSA encryption into ciphertext "8"

7, the private key to decrypt

M ＝CD 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

Ciphertext "8" After RSA decryption plaintext into 2.

JDK comes with the RSA algorithm

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
    }
}

Personal blog
Tencent cloud community
Nuggets
Jane book
Public number: