RSA 加密原理
| 步骤 | 说明 | 描述 | 备注 |
|---|---|---|---|
| 1 | 找出质数 | P 、Q | - |
| 2 | 计算公共模数 | N = P * Q | - |
| 3 | 欧拉函数 | φ(N) = (P-1)(Q-1) | - |
| 4 | 计算公钥E | 1 < E < φ(N) | E的取值必须是整数E 和 φ(N) 必须是互质数 |
| 5 | 计算私钥D | E * D % φ(N) = 1 | - |
| 6 | 加密 | C ＝ M E mod N | C：密文 M：明文 |
| 7 | 解密 | M ＝C D mod N | C：密文 M：明文 |
公钥＝(E , N)
私钥＝(D, N)
对外，我们只暴露公钥。
示例
1、找出质数 P 、Q
P = 3
Q = 11
2、计算公共模数
N = P * Q = 3 * 11 = 33
N = 33
3、 欧拉函数
φ(N) = (P-1)(Q-1) = 2 * 10 = 20
φ(N) = 20
4、计算公钥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
E * D % φ(N) = 1
3 * D % 20 = 1
根据上面可计算出 D = 7
6、公钥加密
我们这里为了演示，就加密一个比较小的数字 M = 2
公式：C ＝ ME mod N
M = 2
E = 3
N = 33
C = 23 % 33 = 8
明文 “2” 经过 RSA 加密后变成了密文 “8”
7、私钥解密
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
密文 “8” 经过 RSA 解密后变成了明文 2。
JDK 自带的 RSA 算法
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
}
}
