Java使用RSA加密解密签名及校验
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2022-06-14 21:03:49
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项目要用到非对称加密解密签名校验什么的,于是从网上参考写一个RSA进行加密解密签名及校验的实例,听说是来源于《Java加密解密的艺术》。
RSA加密解密类RSAEncrypt.java:
package com.bijian.study; import java.io.BufferedReader; import java.io.BufferedWriter; import java.io.FileReader; import java.io.FileWriter; import java.io.IOException; import java.security.InvalidKeyException; import java.security.KeyFactory; import java.security.KeyPair; import java.security.KeyPairGenerator; import java.security.NoSuchAlgorithmException; import java.security.SecureRandom; import java.security.interfaces.RSAPrivateKey; import java.security.interfaces.RSAPublicKey; import java.security.spec.InvalidKeySpecException; import java.security.spec.PKCS8EncodedKeySpec; import java.security.spec.X509EncodedKeySpec; import javax.crypto.BadPaddingException; import javax.crypto.Cipher; import javax.crypto.IllegalBlockSizeException; import javax.crypto.NoSuchPaddingException; import org.apache.commons.codec.binary.Base64; public class RSAEncrypt { /** * 字节数据转字符串专用集合 */ private static final char[] HEX_CHAR = { '0', '1', '2', '3', '4', '5', '6', '7', '8', '9', 'a', 'b', 'c', 'd', 'e', 'f' }; /** * 随机生成密钥对 */ public static void genKeyPair(String filePath) { // KeyPairGenerator类用于生成公钥和私钥对,基于RSA算法生成对象 KeyPairGenerator keyPairGen = null; try { keyPairGen = KeyPairGenerator.getInstance("RSA"); } catch (NoSuchAlgorithmException e) { // TODO Auto-generated catch block e.printStackTrace(); } // 初始化密钥对生成器,密钥大小为96-1024位 keyPairGen.initialize(1024, new SecureRandom()); // 生成一个密钥对,保存在keyPair中 KeyPair keyPair = keyPairGen.generateKeyPair(); // 得到私钥 RSAPrivateKey privateKey = (RSAPrivateKey) keyPair.getPrivate(); // 得到公钥 RSAPublicKey publicKey = (RSAPublicKey) keyPair.getPublic(); try { // 得到公钥字符串 String publicKeyString = Base64.encodeBase64String(publicKey.getEncoded()); // 得到私钥字符串 String privateKeyString = Base64.encodeBase64String(privateKey.getEncoded()); // 将密钥对写入到文件 FileWriter pubfw = new FileWriter(filePath + "/publicKey.keystore"); FileWriter prifw = new FileWriter(filePath + "/privateKey.keystore"); BufferedWriter pubbw = new BufferedWriter(pubfw); BufferedWriter pribw = new BufferedWriter(prifw); pubbw.write(publicKeyString); pribw.write(privateKeyString); pubbw.flush(); pubbw.close(); pubfw.close(); pribw.flush(); pribw.close(); prifw.close(); } catch (Exception e) { e.printStackTrace(); } } /** * 从文件中输入流中加载公钥 * * @param in * 公钥输入流 * @throws Exception * 加载公钥时产生的异常 */ public static String loadPublicKeyByFile(String path) throws Exception { try { BufferedReader br = new BufferedReader(new FileReader(path + "/publicKey.keystore")); String readLine = null; StringBuilder sb = new StringBuilder(); while ((readLine = br.readLine()) != null) { sb.append(readLine); } br.close(); return sb.toString(); } catch (IOException e) { throw new Exception("公钥数据流读取错误"); } catch (NullPointerException e) { throw new Exception("公钥输入流为空"); } } /** * 从字符串中加载公钥 * * @param publicKeyStr * 公钥数据字符串 * @throws Exception * 加载公钥时产生的异常 */ public static RSAPublicKey loadPublicKeyByStr(String publicKeyStr) throws Exception { try { byte[] buffer = Base64.decodeBase64(publicKeyStr); KeyFactory keyFactory = KeyFactory.getInstance("RSA"); X509EncodedKeySpec keySpec = new X509EncodedKeySpec(buffer); return (RSAPublicKey) keyFactory.generatePublic(keySpec); } catch (NoSuchAlgorithmException e) { throw new Exception("无此算法"); } catch (InvalidKeySpecException e) { throw new Exception("公钥非法"); } catch (NullPointerException e) { throw new Exception("公钥数据为空"); } } /** * 从文件中加载私钥 * * @param keyFileName * 私钥文件名 * @return 是否成功 * @throws Exception */ public static String loadPrivateKeyByFile(String path) throws Exception { try { BufferedReader br = new BufferedReader(new FileReader(path + "/privateKey.keystore")); String readLine = null; StringBuilder sb = new StringBuilder(); while ((readLine = br.readLine()) != null) { sb.append(readLine); } br.close(); return sb.toString(); } catch (IOException e) { throw new Exception("私钥数据读取错误"); } catch (NullPointerException e) { throw new Exception("私钥输入流为空"); } } public static RSAPrivateKey loadPrivateKeyByStr(String privateKeyStr) throws Exception { try { byte[] buffer = Base64.decodeBase64(privateKeyStr); PKCS8EncodedKeySpec keySpec = new PKCS8EncodedKeySpec(buffer); KeyFactory keyFactory = KeyFactory.getInstance("RSA"); return (RSAPrivateKey) keyFactory.generatePrivate(keySpec); } catch (NoSuchAlgorithmException e) { throw new Exception("无此算法"); } catch (InvalidKeySpecException e) { throw new Exception("私钥非法"); } catch (NullPointerException e) { throw new Exception("私钥数据为空"); } } /** * 公钥加密过程 * * @param publicKey * 公钥 * @param plainTextData * 明文数据 * @return * @throws Exception * 加密过程中的异常信息 */ public static byte[] encrypt(RSAPublicKey publicKey, byte[] plainTextData) throws Exception { if (publicKey == null) { throw new Exception("加密公钥为空, 请设置"); } Cipher cipher = null; try { // 使用默认RSA cipher = Cipher.getInstance("RSA"); // cipher= Cipher.getInstance("RSA", new BouncyCastleProvider()); cipher.init(Cipher.ENCRYPT_MODE, publicKey); byte[] output = cipher.doFinal(plainTextData); return output; } catch (NoSuchAlgorithmException e) { throw new Exception("无此加密算法"); } catch (NoSuchPaddingException e) { e.printStackTrace(); return null; } catch (InvalidKeyException e) { throw new Exception("加密公钥非法,请检查"); } catch (IllegalBlockSizeException e) { throw new Exception("明文长度非法"); } catch (BadPaddingException e) { throw new Exception("明文数据已损坏"); } } /** * 私钥加密过程 * * @param privateKey * 私钥 * @param plainTextData * 明文数据 * @return * @throws Exception * 加密过程中的异常信息 */ public static byte[] encrypt(RSAPrivateKey privateKey, byte[] plainTextData) throws Exception { if (privateKey == null) { throw new Exception("加密私钥为空, 请设置"); } Cipher cipher = null; try { // 使用默认RSA cipher = Cipher.getInstance("RSA"); cipher.init(Cipher.ENCRYPT_MODE, privateKey); byte[] output = cipher.doFinal(plainTextData); return output; } catch (NoSuchAlgorithmException e) { throw new Exception("无此加密算法"); } catch (NoSuchPaddingException e) { e.printStackTrace(); return null; } catch (InvalidKeyException e) { throw new Exception("加密私钥非法,请检查"); } catch (IllegalBlockSizeException e) { throw new Exception("明文长度非法"); } catch (BadPaddingException e) { throw new Exception("明文数据已损坏"); } } /** * 私钥解密过程 * * @param privateKey * 私钥 * @param cipherData * 密文数据 * @return 明文 * @throws Exception * 解密过程中的异常信息 */ public static byte[] decrypt(RSAPrivateKey privateKey, byte[] cipherData) throws Exception { if (privateKey == null) { throw new Exception("解密私钥为空, 请设置"); } Cipher cipher = null; try { // 使用默认RSA cipher = Cipher.getInstance("RSA"); // cipher= Cipher.getInstance("RSA", new BouncyCastleProvider()); cipher.init(Cipher.DECRYPT_MODE, privateKey); byte[] output = cipher.doFinal(cipherData); return output; } catch (NoSuchAlgorithmException e) { throw new Exception("无此解密算法"); } catch (NoSuchPaddingException e) { e.printStackTrace(); return null; } catch (InvalidKeyException e) { throw new Exception("解密私钥非法,请检查"); } catch (IllegalBlockSizeException e) { throw new Exception("密文长度非法"); } catch (BadPaddingException e) { throw new Exception("密文数据已损坏"); } } /** * 公钥解密过程 * * @param publicKey * 公钥 * @param cipherData * 密文数据 * @return 明文 * @throws Exception * 解密过程中的异常信息 */ public static byte[] decrypt(RSAPublicKey publicKey, byte[] cipherData) throws Exception { if (publicKey == null) { throw new Exception("解密公钥为空, 请设置"); } Cipher cipher = null; try { // 使用默认RSA cipher = Cipher.getInstance("RSA"); // cipher= Cipher.getInstance("RSA", new BouncyCastleProvider()); cipher.init(Cipher.DECRYPT_MODE, publicKey); byte[] output = cipher.doFinal(cipherData); return output; } catch (NoSuchAlgorithmException e) { throw new Exception("无此解密算法"); } catch (NoSuchPaddingException e) { e.printStackTrace(); return null; } catch (InvalidKeyException e) { throw new Exception("解密公钥非法,请检查"); } catch (IllegalBlockSizeException e) { throw new Exception("密文长度非法"); } catch (BadPaddingException e) { throw new Exception("密文数据已损坏"); } } /** * 字节数据转十六进制字符串 * * @param data * 输入数据 * @return 十六进制内容 */ public static String byteArrayToString(byte[] data) { StringBuilder stringBuilder = new StringBuilder(); for (int i = 0; i < data.length; i++) { // 取出字节的高四位 作为索引得到相应的十六进制标识符 注意无符号右移 stringBuilder.append(HEX_CHAR[(data[i] & 0xf0) >>> 4]); // 取出字节的低四位 作为索引得到相应的十六进制标识符 stringBuilder.append(HEX_CHAR[(data[i] & 0x0f)]); if (i < data.length - 1) { stringBuilder.append(' '); } } return stringBuilder.toString(); } }
签名及校验类RSASignature.java:
package com.bijian.study; import java.security.KeyFactory; import java.security.PrivateKey; import java.security.PublicKey; import java.security.spec.PKCS8EncodedKeySpec; import java.security.spec.X509EncodedKeySpec; import org.apache.commons.codec.binary.Base64; /** * RSA签名验签类 */ public class RSASignature { /** * 签名算法 */ public static final String SIGN_ALGORITHMS = "SHA1WithRSA"; /** * RSA签名 * * @param content * 待签名数据 * @param privateKey * 商户私钥 * @param encode * 字符集编码 * @return 签名值 */ public static String sign(String content, String privateKey, String encode) { try { PKCS8EncodedKeySpec priPKCS8 = new PKCS8EncodedKeySpec( Base64.decodeBase64(privateKey)); KeyFactory keyf = KeyFactory.getInstance("RSA"); PrivateKey priKey = keyf.generatePrivate(priPKCS8); java.security.Signature signature = java.security.Signature .getInstance(SIGN_ALGORITHMS); signature.initSign(priKey); signature.update(content.getBytes(encode)); byte[] signed = signature.sign(); return Base64.encodeBase64String(signed); } catch (Exception e) { e.printStackTrace(); } return null; } public static String sign(String content, String privateKey) { try { PKCS8EncodedKeySpec priPKCS8 = new PKCS8EncodedKeySpec( Base64.decodeBase64(privateKey)); KeyFactory keyf = KeyFactory.getInstance("RSA"); PrivateKey priKey = keyf.generatePrivate(priPKCS8); java.security.Signature signature = java.security.Signature .getInstance(SIGN_ALGORITHMS); signature.initSign(priKey); signature.update(content.getBytes()); byte[] signed = signature.sign(); return Base64.encodeBase64String(signed); } catch (Exception e) { e.printStackTrace(); } return null; } /** * RSA验签名检查 * * @param content * 待签名数据 * @param sign * 签名值 * @param publicKey * 分配给开发商公钥 * @param encode * 字符集编码 * @return 布尔值 */ public static boolean doCheck(String content, String sign, String publicKey, String encode) { try { KeyFactory keyFactory = KeyFactory.getInstance("RSA"); byte[] encodedKey = Base64.decodeBase64(publicKey); PublicKey pubKey = keyFactory .generatePublic(new X509EncodedKeySpec(encodedKey)); java.security.Signature signature = java.security.Signature .getInstance(SIGN_ALGORITHMS); signature.initVerify(pubKey); signature.update(content.getBytes(encode)); boolean bverify = signature.verify(Base64.decodeBase64(sign)); return bverify; } catch (Exception e) { e.printStackTrace(); } return false; } public static boolean doCheck(String content, String sign, String publicKey) { try { KeyFactory keyFactory = KeyFactory.getInstance("RSA"); byte[] encodedKey = Base64.decodeBase64(publicKey); PublicKey pubKey = keyFactory .generatePublic(new X509EncodedKeySpec(encodedKey)); java.security.Signature signature = java.security.Signature .getInstance(SIGN_ALGORITHMS); signature.initVerify(pubKey); signature.update(content.getBytes()); boolean bverify = signature.verify(Base64.decodeBase64(sign)); return bverify; } catch (Exception e) { e.printStackTrace(); } return false; } }
再来一个Base64的类,当然你也可以用commons-codec-1.10.jar
Base64.java
public final class Base64 { static private final int BASELENGTH = 128; static private final int LOOKUPLENGTH = 64; static private final int TWENTYFOURBITGROUP = 24; static private final int EIGHTBIT = 8; static private final int SIXTEENBIT = 16; static private final int FOURBYTE = 4; static private final int SIGN = -128; static private final char PAD = '='; static private final boolean fDebug = false; static final private byte[] base64Alphabet = new byte[BASELENGTH]; static final private char[] lookUpBase64Alphabet = new char[LOOKUPLENGTH]; static { for (int i = 0; i < BASELENGTH; ++i) { base64Alphabet[i] = -1; } for (int i = 'Z'; i >= 'A'; i--) { base64Alphabet[i] = (byte) (i - 'A'); } for (int i = 'z'; i >= 'a'; i--) { base64Alphabet[i] = (byte) (i - 'a' + 26); } for (int i = '9'; i >= '0'; i--) { base64Alphabet[i] = (byte) (i - '0' + 52); } base64Alphabet['+'] = 62; base64Alphabet['/'] = 63; for (int i = 0; i <= 25; i++) { lookUpBase64Alphabet[i] = (char) ('A' + i); } for (int i = 26, j = 0; i <= 51; i++, j++) { lookUpBase64Alphabet[i] = (char) ('a' + j); } for (int i = 52, j = 0; i <= 61; i++, j++) { lookUpBase64Alphabet[i] = (char) ('0' + j); } lookUpBase64Alphabet[62] = (char) '+'; lookUpBase64Alphabet[63] = (char) '/'; } private static boolean isWhiteSpace(char octect) { return (octect == 0x20 || octect == 0xd || octect == 0xa || octect == 0x9); } private static boolean isPad(char octect) { return (octect == PAD); } private static boolean isData(char octect) { return (octect < BASELENGTH && base64Alphabet[octect] != -1); } /** * Encodes hex octects into Base64 * * @param binaryData Array containing binaryData * @return Encoded Base64 array */ public static String encode(byte[] binaryData) { if (binaryData == null) { return null; } int lengthDataBits = binaryData.length * EIGHTBIT; if (lengthDataBits == 0) { return ""; } int fewerThan24bits = lengthDataBits % TWENTYFOURBITGROUP; int numberTriplets = lengthDataBits / TWENTYFOURBITGROUP; int numberQuartet = fewerThan24bits != 0 ? numberTriplets + 1 : numberTriplets; char encodedData[] = null; encodedData = new char[numberQuartet * 4]; byte k = 0, l = 0, b1 = 0, b2 = 0, b3 = 0; int encodedIndex = 0; int dataIndex = 0; if (fDebug) { System.out.println("number of triplets = " + numberTriplets); } for (int i = 0; i < numberTriplets; i++) { b1 = binaryData[dataIndex++]; b2 = binaryData[dataIndex++]; b3 = binaryData[dataIndex++]; if (fDebug) { System.out.println("b1= " + b1 + ", b2= " + b2 + ", b3= " + b3); } l = (byte) (b2 & 0x0f); k = (byte) (b1 & 0x03); byte val1 = ((b1 & SIGN) == 0) ? (byte) (b1 >> 2) : (byte) ((b1) >> 2 ^ 0xc0); byte val2 = ((b2 & SIGN) == 0) ? (byte) (b2 >> 4) : (byte) ((b2) >> 4 ^ 0xf0); byte val3 = ((b3 & SIGN) == 0) ? (byte) (b3 >> 6) : (byte) ((b3) >> 6 ^ 0xfc); if (fDebug) { System.out.println("val2 = " + val2); System.out.println("k4 = " + (k << 4)); System.out.println("vak = " + (val2 | (k << 4))); } encodedData[encodedIndex++] = lookUpBase64Alphabet[val1]; encodedData[encodedIndex++] = lookUpBase64Alphabet[val2 | (k << 4)]; encodedData[encodedIndex++] = lookUpBase64Alphabet[(l << 2) | val3]; encodedData[encodedIndex++] = lookUpBase64Alphabet[b3 & 0x3f]; } // form integral number of 6-bit groups if (fewerThan24bits == EIGHTBIT) { b1 = binaryData[dataIndex]; k = (byte) (b1 & 0x03); if (fDebug) { System.out.println("b1=" + b1); System.out.println("b1<<2 = " + (b1 >> 2)); } byte val1 = ((b1 & SIGN) == 0) ? (byte) (b1 >> 2) : (byte) ((b1) >> 2 ^ 0xc0); encodedData[encodedIndex++] = lookUpBase64Alphabet[val1]; encodedData[encodedIndex++] = lookUpBase64Alphabet[k << 4]; encodedData[encodedIndex++] = PAD; encodedData[encodedIndex++] = PAD; } else if (fewerThan24bits == SIXTEENBIT) { b1 = binaryData[dataIndex]; b2 = binaryData[dataIndex + 1]; l = (byte) (b2 & 0x0f); k = (byte) (b1 & 0x03); byte val1 = ((b1 & SIGN) == 0) ? (byte) (b1 >> 2) : (byte) ((b1) >> 2 ^ 0xc0); byte val2 = ((b2 & SIGN) == 0) ? (byte) (b2 >> 4) : (byte) ((b2) >> 4 ^ 0xf0); encodedData[encodedIndex++] = lookUpBase64Alphabet[val1]; encodedData[encodedIndex++] = lookUpBase64Alphabet[val2 | (k << 4)]; encodedData[encodedIndex++] = lookUpBase64Alphabet[l << 2]; encodedData[encodedIndex++] = PAD; } return new String(encodedData); } /** * Decodes Base64 data into octects * * @param encoded string containing Base64 data * @return Array containind decoded data. */ public static byte[] decode(String encoded) { if (encoded == null) { return null; } char[] base64Data = encoded.toCharArray(); // remove white spaces int len = removeWhiteSpace(base64Data); if (len % FOURBYTE != 0) { return null;//should be divisible by four } int numberQuadruple = (len / FOURBYTE); if (numberQuadruple == 0) { return new byte[0]; } byte decodedData[] = null; byte b1 = 0, b2 = 0, b3 = 0, b4 = 0; char d1 = 0, d2 = 0, d3 = 0, d4 = 0; int i = 0; int encodedIndex = 0; int dataIndex = 0; decodedData = new byte[(numberQuadruple) * 3]; for (; i < numberQuadruple - 1; i++) { if (!isData((d1 = base64Data[dataIndex++])) || !isData((d2 = base64Data[dataIndex++])) || !isData((d3 = base64Data[dataIndex++])) || !isData((d4 = base64Data[dataIndex++]))) { return null; }//if found "no data" just return null b1 = base64Alphabet[d1]; b2 = base64Alphabet[d2]; b3 = base64Alphabet[d3]; b4 = base64Alphabet[d4]; decodedData[encodedIndex++] = (byte) (b1 << 2 | b2 >> 4); decodedData[encodedIndex++] = (byte) (((b2 & 0xf) << 4) | ((b3 >> 2) & 0xf)); decodedData[encodedIndex++] = (byte) (b3 << 6 | b4); } if (!isData((d1 = base64Data[dataIndex++])) || !isData((d2 = base64Data[dataIndex++]))) { return null;//if found "no data" just return null } b1 = base64Alphabet[d1]; b2 = base64Alphabet[d2]; d3 = base64Data[dataIndex++]; d4 = base64Data[dataIndex++]; if (!isData((d3)) || !isData((d4))) {//Check if they are PAD characters if (isPad(d3) && isPad(d4)) { if ((b2 & 0xf) != 0)//last 4 bits should be zero { return null; } byte[] tmp = new byte[i * 3 + 1]; System.arraycopy(decodedData, 0, tmp, 0, i * 3); tmp[encodedIndex] = (byte) (b1 << 2 | b2 >> 4); return tmp; } else if (!isPad(d3) && isPad(d4)) { b3 = base64Alphabet[d3]; if ((b3 & 0x3) != 0)//last 2 bits should be zero { return null; } byte[] tmp = new byte[i * 3 + 2]; System.arraycopy(decodedData, 0, tmp, 0, i * 3); tmp[encodedIndex++] = (byte) (b1 << 2 | b2 >> 4); tmp[encodedIndex] = (byte) (((b2 & 0xf) << 4) | ((b3 >> 2) & 0xf)); return tmp; } else { return null; } } else { //No PAD e.g 3cQl b3 = base64Alphabet[d3]; b4 = base64Alphabet[d4]; decodedData[encodedIndex++] = (byte) (b1 << 2 | b2 >> 4); decodedData[encodedIndex++] = (byte) (((b2 & 0xf) << 4) | ((b3 >> 2) & 0xf)); decodedData[encodedIndex++] = (byte) (b3 << 6 | b4); } return decodedData; } /** * remove WhiteSpace from MIME containing encoded Base64 data. * * @param data the byte array of base64 data (with WS) * @return the new length */ private static int removeWhiteSpace(char[] data) { if (data == null) { return 0; } // count characters that's not whitespace int newSize = 0; int len = data.length; for (int i = 0; i < len; i++) { if (!isWhiteSpace(data[i])) { data[newSize++] = data[i]; } } return newSize; } }
最后是一个MainTest.java
package com.bijian.study; import org.apache.commons.codec.binary.Base64; public class MainTest { public static void main(String[] args) throws Exception { String filepath="D:/temp/"; //RSAEncrypt.genKeyPair(filepath); System.out.println("--------------公钥加密私钥解密过程-------------------"); String plainText="ihep_公钥加密私钥解密"; //公钥加密过程 byte[] cipherData=RSAEncrypt.encrypt(RSAEncrypt.loadPublicKeyByStr(RSAEncrypt.loadPublicKeyByFile(filepath)), plainText.getBytes()); String cipher=Base64.encodeBase64String(cipherData); //私钥解密过程 byte[] res=RSAEncrypt.decrypt(RSAEncrypt.loadPrivateKeyByStr(RSAEncrypt.loadPrivateKeyByFile(filepath)), Base64.decodeBase64(cipher)); String restr=new String(res); System.out.println("原文:"+plainText); System.out.println("加密:"+cipher); System.out.println("解密:"+restr); System.out.println(); System.out.println("--------------私钥加密公钥解密过程-------------------"); plainText="ihep_私钥加密公钥解密"; //私钥加密过程 cipherData=RSAEncrypt.encrypt(RSAEncrypt.loadPrivateKeyByStr(RSAEncrypt.loadPrivateKeyByFile(filepath)),plainText.getBytes()); cipher=Base64.encodeBase64String(cipherData); //公钥解密过程 res=RSAEncrypt.decrypt(RSAEncrypt.loadPublicKeyByStr(RSAEncrypt.loadPublicKeyByFile(filepath)), Base64.decodeBase64(cipher)); restr=new String(res); System.out.println("原文:"+plainText); System.out.println("加密:"+cipher); System.out.println("解密:"+restr); System.out.println(); System.out.println("---------------私钥签名过程------------------"); String content="ihep_这是用于签名的原始数据"; String signstr=RSASignature.sign(content,RSAEncrypt.loadPrivateKeyByFile(filepath)); System.out.println("签名原串:"+content); System.out.println("签名串:"+signstr); System.out.println(); System.out.println("---------------公钥校验签名------------------"); System.out.println("签名原串:"+content); System.out.println("签名串:"+signstr); System.out.println("验签结果:"+RSASignature.doCheck(content, signstr, RSAEncrypt.loadPublicKeyByFile(filepath))); System.out.println(); } }
把生产密钥对的代码注释打开,运行生成密钥对如下:
文章来源:https://blog.csdn.net/wangqiuyun/article/details/42143957