【密码学】RSA加解密原理及其Java实现算法

密钥生成

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RSA的密钥通过如下步骤生成：

1. 选取两个不同的质数p和q

   • 为了安全考虑，p和q应该随机选取，并且具有相似的数量级。如果p和q仅仅通过几个数字使得长度不同，那么分解因子更加困难

2. 计算n = pq

   • n作为公钥和私钥的模数。n的比特数就是密钥的长度

3. 计算n的欧拉函数φ(n) = (p-1)(q-1)

4. 选取一个整数e，1 < e < φ(n)，e与φ(n)互质

5. 计算e模φ(n)的逆d

   • d就是使得ed ≡ 1(mod φ(n))
   • e一般是一个短比特长度、小海明权重的数，这样使得加密更加有效。e一般选取2^16 + 1 = 65537。如果e比较小（比如3），那么在某些情况显得很不安全。
   • e作为公钥发布
   • d作为私钥保存

加解密过程

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加密

如果Bob想要发送消息M给Alice

他首先把M转换为整数m(0 < m < n)，然后用Alice的公钥e计算密文c，表达式为：

如果使用平方-乘算法可以使计算非常快，即使是500-bit的整数

然后Bob把c发送给Alice

解密

Alice用私钥d把密文c恢复为整数m

然后Alice把整数m恢复为明文M

例子

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选择p和q

p = 61q = 53

计算n = pq

n = 61 * 53 = 3233

计算φ(n)

φ(3233) = 3120

选择e

e = 17

计算d

d = 2753

加密

65^17 ≡ 2790 (mod 3233)

解密

2790^2753 ≡ 65 (mod 3233)

算法

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源代码

下面算法是使用Java的BigInteger类，主要是为了阐释RSA的加解密过程，算法不具有代表性，仅供参考

import java.math.BigInteger;import java.util.Random;public class RSAs {    private static BigInteger n; // large prime    private static BigInteger e; // public key    private static BigInteger b; // private key    private static BigInteger p; // prime    private static BigInteger q; // prime    public static void main(String[] args) {        String plaintext = "The quick brown fox jumps over the lazy dog";        RSAs rsas = new RSAs();        rsas.giveKey();        BigInteger[] encrypt = rsas.encrypt(plaintext);        String decrypt = rsas.decrypt(encrypt);        System.out.println("\nplaintext:" + plaintext + "\n\nencrpyt:");        for (int i = 0; i < encrypt.length; ++i) {            System.out.println(encrypt[i]);        }        System.out.println("\ndecrypt:" + decrypt);    }    // RSA encryption    public BigInteger[] encrypt(String plaintext) {        BigInteger[] encrypt = new BigInteger[plaintext.length()];        BigInteger m, c;        for (int i = 0; i < plaintext.length(); ++i) {            m = BigInteger.valueOf(plaintext.charAt(i));            c = m.modPow(e, n);            encrypt[i] = c;        }        return encrypt;    }    // RSA decryption    public String decrypt(BigInteger[] encrypt) {        StringBuffer plaintext = new StringBuffer();        BigInteger m, c;        for (int i = 0; i < encrypt.length; ++i) {            c = encrypt[i];            m = c.modPow(b, n);            plaintext.append((char) m.intValue());        }        return plaintext.toString();    }    // give public key and private key    public void giveKey() {        // get p,q,n,e,b        producePQ();        n = p.multiply(q);        BigInteger eulerN = p.subtract(new BigInteger("1")).multiply(q.subtract(new BigInteger("1")));        produceEB(eulerN);        System.out.println("n:" + n + "\np:" + p + "\nq:" + q + "\ne:" + e + "\nb:" + b);    }    // large prime p and q generation    public void producePQ() {        p = BigInteger.probablePrime(32, new Random());        q = BigInteger.probablePrime(32, new Random());        while (p.equals(q)) {            p = BigInteger.probablePrime(32, new Random());            q = BigInteger.probablePrime(32, new Random());        }    }    // produce public key e,private key b    public void produceEB(BigInteger eulerN) {        e = BigInteger.probablePrime((int) (Math.random() * 63 + 2), new Random());        while (e.compareTo(eulerN) != -1 | eulerN.divide(e).equals(0)) {            e = BigInteger.probablePrime((int) (Math.random() * 63 + 2), new Random());        }        //e = BigInteger.valueOf(65537);        b = e.modInverse(eulerN);    }}

运行结果

n:7722718897492023517p:2362511213q:3268860209e:11182711b:7692894186493341255plaintext:The quick brown fox jumps over the lazy dogencrpyt:460677377892844302231910653542017758692063576957539994540324066101898781055046430765699281330616724493591560186042592828770772456814118024883408945721037207707714312482840324066101898781055072847910235050690201666173794082244580743509746308901414877173124234753816002862511757604585560324066101898781055026005102734154614957435097463089014148570658665461456630932406610189878105505828898437482186613672449359156018604210000311860047525311594082521594404755641933813051444079632406610189878105507435097463089014148235250470560124312220635769575399945401666173794082244580324066101898781055033821947450664659593191065354201775869206357695753999454032406610189878105506165260746857331774868384751760782699745245957974217151927654902873831066043240661018987810550426189291042634827074350974630890141483115531137912443469decrypt:The quick brown fox jumps over the lazy dog