摘要：作为新一代的加密标准，AES 旨在取代 DES（请看《DES加密算法的C++实现》），以适应当今分布式开放网络对数据加密安全性的要求。本文在分析了 AES 加密原理的基础上着重说明了算法实现的具体步骤，并用 C++ 实现了对文件的加密和解密。
一、AES 介绍
AES（高级加密标准，Advanced Encryption Standard），在密码学中又称 Rijndael 加密法，是美国联邦政府采用的一种分组加密标准。这个标准用来替代原先的 DES，目前已经广为全世界所使用，成为对称密钥算法中最流行的算法之一。
在 AES 出现之前，最常用的对称密钥算法是 DES 加密算法，它在 1977 年被公布成为美国政府的商用加密标准。DES 的主要问题是密钥长度较短，渐渐不适合于分布式开放网络对数据加密安全性的要求。因此，1998年美国政府决定不再继续延用 DES 作为联邦加密标准，并发起了征集 AES 候选算法的活动。征集活动对 AES 的基本要求是： 比三重DES快、至少与三重DES一样安全、数据分组长度为128比特、密钥长度为128/192/256比特。
经过三年多的甄选，比利时的密码学家所设计的 Rijndael 算法最终脱颖而出，成为新一代的高级加密标准，并于 2001 年由美国国家标准与技术研究院（NIST）发布于 FIPS PUB 197。
二、AES 算法原理
AES算法（即 Rijndael 算法）是一个对称分组密码算法。数据分组长度必须是 128 bits，使用的密钥长度为 128，192 或 256 bits。对于三种不同密钥长度的 AES 算法，分别称为“AES-128”、“AES-192”、“AES-256”。（Rijndael 的设计还可以处理其它的分组长度和密钥长度，但 AES 标准中没有采用）
下图是 AES 加密解密的整体流程图：
这里我们需要知道3个符号：Nb—— 状态 State 包含的列（32-bit 字）的个数，也就是说 Nb=4；Nk——
密钥包含的 32-bit 字的个数，也就是说 Nk=4，6 或 8；Nr—— 加密的轮数，对于不同密钥长度，轮数不一样，具体如下图所示：
下面分为密钥扩展、分组加密、分组解密三个部分来讲 AES 算法，我会尽可能地简明扼要，若还有不懂的，请自行 Google。
1）密钥扩展
AES 算法通过密钥扩展程序（Key Expansion）将用户输入的密钥 K 扩展生成 Nb(Nr+1)个字，存放在一个线性数组w[Nb*(Nr+1)]中。具体如下：
-
位置变换函数
RotWord()，接受一个字 [a0, a1, a2, a3] 作为输入，循环左移一个字节后输出 [a1, a2, a3, a0]。 -
S盒变换函数
SubWord()，接受一个字 [a0, a1, a2, a3] 作为输入。S盒是一个16x16的表，其中每一个元素是一个字节。对于输入的每一个字节，前四位组成十六进制数 x 作为行号，后四位组成的十六进制数 y 作为列号，查找表中对应的值。最后函数输出 4 个新字节组成的 32-bit 字。 -
轮常数
Rcon[]，如何计算的就不说了，直接把它当做常量数组。 -
扩展密钥数组
w[]的前 Nk 个元素就是外部密钥 K，以后的元素w[i]等于它前一个元素w[i-1]与前第 Nk 个元素w[i-Nk]的异或，即w[i] = w[i-1] XOR w[i-Nk]；但若 i 为 Nk 的倍数，则w[i] = w[i-Nk] XOR SubWord(RotWord(w[i-1])) XOR Rcon[i/Nk-1]。
注意，上面的第四步说明适合于 AES-128 和 AES-192，详细的伪代码如下：
密钥扩展程序的 C++ 代码（AES-128）：
#include <iostream>
#include <bitset>
using namespace std;
typedef bitset<8> byte;
typedef bitset<32> word;
const int Nr = 10; // AES-128需要 10 轮加密
const int Nk = 4; // Nk 表示输入密钥的 word 个数
byte S_Box[16][16] = {
{0x63, 0x7C, 0x77, 0x7B, 0xF2, 0x6B, 0x6F, 0xC5, 0x30, 0x01, 0x67, 0x2B, 0xFE, 0xD7, 0xAB, 0x76},
{0xCA, 0x82, 0xC9, 0x7D, 0xFA, 0x59, 0x47, 0xF0, 0xAD, 0xD4, 0xA2, 0xAF, 0x9C, 0xA4, 0x72, 0xC0},
{0xB7, 0xFD, 0x93, 0x26, 0x36, 0x3F, 0xF7, 0xCC, 0x34, 0xA5, 0xE5, 0xF1, 0x71, 0xD8, 0x31, 0x15},
{0x04, 0xC7, 0x23, 0xC3, 0x18, 0x96, 0x05, 0x9A, 0x07, 0x12, 0x80, 0xE2, 0xEB, 0x27, 0xB2, 0x75},
{0x09, 0x83, 0x2C, 0x1A, 0x1B, 0x6E, 0x5A, 0xA0, 0x52, 0x3B, 0xD6, 0xB3, 0x29, 0xE3, 0x2F, 0x84},
{0x53, 0xD1, 0x00, 0xED, 0x20, 0xFC, 0xB1, 0x5B, 0x6A, 0xCB, 0xBE, 0x39, 0x4A, 0x4C, 0x58, 0xCF},
{0xD0, 0xEF, 0xAA, 0xFB, 0x43, 0x4D, 0x33, 0x85, 0x45, 0xF9, 0x02, 0x7F, 0x50, 0x3C, 0x9F, 0xA8},
{0x51, 0xA3, 0x40, 0x8F, 0x92, 0x9D, 0x38, 0xF5, 0xBC, 0xB6, 0xDA, 0x21, 0x10, 0xFF, 0xF3, 0xD2},
{0xCD, 0x0C, 0x13, 0xEC, 0x5F, 0x97, 0x44, 0x17, 0xC4, 0xA7, 0x7E, 0x3D, 0x64, 0x5D, 0x19, 0x73},
{0x60, 0x81, 0x4F, 0xDC, 0x22, 0x2A, 0x90, 0x88, 0x46, 0xEE, 0xB8, 0x14, 0xDE, 0x5E, 0x0B, 0xDB},
{0xE0, 0x32, 0x3A, 0x0A, 0x49, 0x06, 0x24, 0x5C, 0xC2, 0xD3, 0xAC, 0x62, 0x91, 0x95, 0xE4, 0x79},
{0xE7, 0xC8, 0x37, 0x6D, 0x8D, 0xD5, 0x4E, 0xA9, 0x6C, 0x56, 0xF4, 0xEA, 0x65, 0x7A, 0xAE, 0x08},
{0xBA, 0x78, 0x25, 0x2E, 0x1C, 0xA6, 0xB4, 0xC6, 0xE8, 0xDD, 0x74, 0x1F, 0x4B, 0xBD, 0x8B, 0x8A},
{0x70, 0x3E, 0xB5, 0x66, 0x48, 0x03, 0xF6, 0x0E, 0x61, 0x35, 0x57, 0xB9, 0x86, 0xC1, 0x1D, 0x9E},
{0xE1, 0xF8, 0x98, 0x11, 0x69, 0xD9, 0x8E, 0x94, 0x9B, 0x1E, 0x87, 0xE9, 0xCE, 0x55, 0x28, 0xDF},
{0x8C, 0xA1, 0x89, 0x0D, 0xBF, 0xE6, 0x42, 0x68, 0x41, 0x99, 0x2D, 0x0F, 0xB0, 0x54, 0xBB, 0x16}
};
// 轮常数，密钥扩展中用到。（AES-128只需要10轮）
word Rcon[10] = {0x01000000, 0x02000000, 0x04000000, 0x08000000, 0x10000000,
0x20000000, 0x40000000, 0x80000000, 0x1b000000, 0x36000000};
/**
* 将4个 byte 转换为一个 word.
*/
word Word(byte& k1, byte& k2, byte& k3, byte& k4)
{
word result(0x00000000);
word temp;
temp = k1.to_ulong(); // K1
temp <<= 24;
result |= temp;
temp = k2.to_ulong(); // K2
temp <<= 16;
result |= temp;
temp = k3.to_ulong(); // K3
temp <<= 8;
result |= temp;
temp = k4.to_ulong(); // K4
result |= temp;
return result;
}
/**
* 按字节 循环左移一位
* 即把[a0, a1, a2, a3]变成[a1, a2, a3, a0]
*/
word RotWord(word& rw)
{
word high = rw << 8;
word low = rw >> 24;
return high | low;
}
/**
* 对输入word中的每一个字节进行S-盒变换
*/
word SubWord(word& sw)
{
word temp;
for(int i=0; i<32; i+=8)
{
int row = sw[i+7]*8 + sw[i+6]*4 + sw[i+5]*2 + sw[i+4];
int col = sw[i+3]*8 + sw[i+2]*4 + sw[i+1]*2 + sw[i];
byte val = S_Box[row][col];
for(int j=0; j<8; ++j)
temp[i+j] = val[j];
}
return temp;
}
/**
* 密钥扩展函数 - 对128位密钥进行扩展得到 w[4*(Nr+1)]
*/
void KeyExpansion(byte key[4*Nk], word w[4*(Nr+1)])
{
word temp;
int i = 0;
// w[]的前4个就是输入的key
while(i < Nk)
{
w[i] = Word(key[4*i], key[4*i+1], key[4*i+2], key[4*i+3]);
++i;
}
i = Nk;
while(i < 4*(Nr+1))
{
temp = w[i-1]; // 记录前一个word
if(i % Nk == 0)
w[i] = w[i-Nk] ^ SubWord(RotWord(temp)) ^ Rcon[i/Nk-1];
else
w[i] = w[i-Nk] ^ temp;
++i;
}
}
int main()
{
byte key[16] = {0x2b, 0x7e, 0x15, 0x16,
0x28, 0xae, 0xd2, 0xa6,
0xab, 0xf7, 0x15, 0x88,
0x09, 0xcf, 0x4f, 0x3c};
word w[4*(Nr+1)];
cout << "KEY IS: ";
for(int i=0; i<16; ++i)
cout << hex << key[i].to_ulong() << " ";
cout << endl;
KeyExpansion(key, w);
// 测试
for(int i=0; i<4*(Nr+1); ++i)
cout << "w[" << dec << i << "] = " << hex << w[i].to_ulong() << endl;
return 0;
}
测试输出结果：
2）加密
根据 AES 加密的整体流程图（本文开头），伪代码如下：
从伪代码描述中可以看出，AES 加密时涉及到的子程序有SubBytes()、ShiftRows()、MixColumns()和AddRoundKey()。下面我们一个一个进行介绍：
① S盒变换-SubBytes()
在密钥扩展部分已经讲过了，S盒是一个 16 行 16 列的表，表中每个元素都是一个字节。S盒变换很简单：函数SubBytes()接受一个
4x4 的字节矩阵作为输入，对其中的每个字节，前四位组成十六进制数 x 作为行号，后四位组成的十六进制数 y 作为列号，查找表中对应的值替换原来位置上的字节。
② 行变换-ShiftRows()
行变换也很简单，它仅仅是将矩阵的每一行以字节为单位循环移位：第一行不变，第二行左移一位，第三行左移两位，第四行左移三位。如下图所示：
③ 列变换-MixColumns()
函数MixColumns()同样接受一个 4x4 的字节矩阵作为输入，并对矩阵进行逐列变换，变换方式如下：
注意公式中用到的乘法是伽罗华域（GF，有限域）上的乘法，高级加密标准文档 fips-197 上有讲，如果还是不懂，请自行Google。
④ 与扩展密钥的异或-AddRoundKey()
扩展密钥只参与了这一步。根据当前加密的轮数，用w[]中的 4 个扩展密钥与矩阵的 4 个列进行按位异或。如下图：
好了，到这里 AES 加密的各个部分就讲完了。算法实现的 C++ 源码在文章后面第三部分。
3）解密
根据 AES 解密的整体流程图（本文开头），伪代码如下：
从伪代码可以看出，我们需要分别实现 S 盒变换、行变换和列变换的逆变换InvShiftRows()、InvSubBytes()和InvMixColumns()。下面就简单的讲一下这三个逆变换：
① 逆行变换-InvShiftRows()
上面讲到ShiftRows()是对矩阵的每一行进行循环左移，所以InvShiftRows()是对矩阵每一行进行循环右移。
② 逆 S 盒变换-InvSubBytes()
与 S 盒变换一样，也是查表，查表的方式也一样，只不过查的是另外一个置换表（S-Box的逆表）。
③ 逆列变换-InvMixColumns()
与列变换的方式一样，只不过计算公式的系数矩阵发生了变化。如下图：
好了，AES 解密到这里也讲完了。只要写出三个逆变换的函数，然后根据伪代码就很容易实现 AES 解密算法了。
三、C++实现
下面我用 C++ 实现 AES 的加密和解密算法，并实现了对文件的加密和解密。这里我使用 C++ STL 的bitset定义了两个类型：byte和word。需要提到的是，对于有限域上的乘法，我们既可以通过查表（6个结果表），也可以写一个函数来实现。当然，查表的效率会更高，但考虑到贴代码，这里我就用一个函数来实现的。
下面是 AES-128 对一个 128 位数据加密和解密的源代码：
/*************************************************************************
> File Name: AES.cpp
> Author: SongLee
> E-mail: lisong.shine@qq.com
> Created Time: 2014年12月12日 星期五 20时15分50秒
> Personal Blog: http://songlee24.github.com
************************************************************************/
#include <iostream>
#include <bitset>
#include <string>
using namespace std;
typedef bitset<8> byte;
typedef bitset<32> word;
const int Nr = 10; // AES-128需要 10 轮加密
const int Nk = 4; // Nk 表示输入密钥的 word 个数
byte S_Box[16][16] = {
{0x63, 0x7C, 0x77, 0x7B, 0xF2, 0x6B, 0x6F, 0xC5, 0x30, 0x01, 0x67, 0x2B, 0xFE, 0xD7, 0xAB, 0x76},
{0xCA, 0x82, 0xC9, 0x7D, 0xFA, 0x59, 0x47, 0xF0, 0xAD, 0xD4, 0xA2, 0xAF, 0x9C, 0xA4, 0x72, 0xC0},
{0xB7, 0xFD, 0x93, 0x26, 0x36, 0x3F, 0xF7, 0xCC, 0x34, 0xA5, 0xE5, 0xF1, 0x71, 0xD8, 0x31, 0x15},
{0x04, 0xC7, 0x23, 0xC3, 0x18, 0x96, 0x05, 0x9A, 0x07, 0x12, 0x80, 0xE2, 0xEB, 0x27, 0xB2, 0x75},
{0x09, 0x83, 0x2C, 0x1A, 0x1B, 0x6E, 0x5A, 0xA0, 0x52, 0x3B, 0xD6, 0xB3, 0x29, 0xE3, 0x2F, 0x84},
{0x53, 0xD1, 0x00, 0xED, 0x20, 0xFC, 0xB1, 0x5B, 0x6A, 0xCB, 0xBE, 0x39, 0x4A, 0x4C, 0x58, 0xCF},
{0xD0, 0xEF, 0xAA, 0xFB, 0x43, 0x4D, 0x33, 0x85, 0x45, 0xF9, 0x02, 0x7F, 0x50, 0x3C, 0x9F, 0xA8},
{0x51, 0xA3, 0x40, 0x8F, 0x92, 0x9D, 0x38, 0xF5, 0xBC, 0xB6, 0xDA, 0x21, 0x10, 0xFF, 0xF3, 0xD2},
{0xCD, 0x0C, 0x13, 0xEC, 0x5F, 0x97, 0x44, 0x17, 0xC4, 0xA7, 0x7E, 0x3D, 0x64, 0x5D, 0x19, 0x73},
{0x60, 0x81, 0x4F, 0xDC, 0x22, 0x2A, 0x90, 0x88, 0x46, 0xEE, 0xB8, 0x14, 0xDE, 0x5E, 0x0B, 0xDB},
{0xE0, 0x32, 0x3A, 0x0A, 0x49, 0x06, 0x24, 0x5C, 0xC2, 0xD3, 0xAC, 0x62, 0x91, 0x95, 0xE4, 0x79},
{0xE7, 0xC8, 0x37, 0x6D, 0x8D, 0xD5, 0x4E, 0xA9, 0x6C, 0x56, 0xF4, 0xEA, 0x65, 0x7A, 0xAE, 0x08},
{0xBA, 0x78, 0x25, 0x2E, 0x1C, 0xA6, 0xB4, 0xC6, 0xE8, 0xDD, 0x74, 0x1F, 0x4B, 0xBD, 0x8B, 0x8A},
{0x70, 0x3E, 0xB5, 0x66, 0x48, 0x03, 0xF6, 0x0E, 0x61, 0x35, 0x57, 0xB9, 0x86, 0xC1, 0x1D, 0x9E},
{0xE1, 0xF8, 0x98, 0x11, 0x69, 0xD9, 0x8E, 0x94, 0x9B, 0x1E, 0x87, 0xE9, 0xCE, 0x55, 0x28, 0xDF},
{0x8C, 0xA1, 0x89, 0x0D, 0xBF, 0xE6, 0x42, 0x68, 0x41, 0x99, 0x2D, 0x0F, 0xB0, 0x54, 0xBB, 0x16}
};
byte Inv_S_Box[16][16] = {
{0x52, 0x09, 0x6A, 0xD5, 0x30, 0x36, 0xA5, 0x38, 0xBF, 0x40, 0xA3, 0x9E, 0x81, 0xF3, 0xD7, 0xFB},
{0x7C, 0xE3, 0x39, 0x82, 0x9B, 0x2F, 0xFF, 0x87, 0x34, 0x8E, 0x43, 0x44, 0xC4, 0xDE, 0xE9, 0xCB},
{0x54, 0x7B, 0x94, 0x32, 0xA6, 0xC2, 0x23, 0x3D, 0xEE, 0x4C, 0x95, 0x0B, 0x42, 0xFA, 0xC3, 0x4E},
{0x08, 0x2E, 0xA1, 0x66, 0x28, 0xD9, 0x24, 0xB2, 0x76, 0x5B, 0xA2, 0x49, 0x6D, 0x8B, 0xD1, 0x25},
{0x72, 0xF8, 0xF6, 0x64, 0x86, 0x68, 0x98, 0x16, 0xD4, 0xA4, 0x5C, 0xCC, 0x5D, 0x65, 0xB6, 0x92},
{0x6C, 0x70, 0x48, 0x50, 0xFD, 0xED, 0xB9, 0xDA, 0x5E, 0x15, 0x46, 0x57, 0xA7, 0x8D, 0x9D, 0x84},
{0x90, 0xD8, 0xAB, 0x00, 0x8C, 0xBC, 0xD3, 0x0A, 0xF7, 0xE4, 0x58, 0x05, 0xB8, 0xB3, 0x45, 0x06},
{0xD0, 0x2C, 0x1E, 0x8F, 0xCA, 0x3F, 0x0F, 0x02, 0xC1, 0xAF, 0xBD, 0x03, 0x01, 0x13, 0x8A, 0x6B},
{0x3A, 0x91, 0x11, 0x41, 0x4F, 0x67, 0xDC, 0xEA, 0x97, 0xF2, 0xCF, 0xCE, 0xF0, 0xB4, 0xE6, 0x73},
{0x96, 0xAC, 0x74, 0x22, 0xE7, 0xAD, 0x35, 0x85, 0xE2, 0xF9, 0x37, 0xE8, 0x1C, 0x75, 0xDF, 0x6E},
{0x47, 0xF1, 0x1A, 0x71, 0x1D, 0x29, 0xC5, 0x89, 0x6F, 0xB7, 0x62, 0x0E, 0xAA, 0x18, 0xBE, 0x1B},
{0xFC, 0x56, 0x3E, 0x4B, 0xC6, 0xD2, 0x79, 0x20, 0x9A, 0xDB, 0xC0, 0xFE, 0x78, 0xCD, 0x5A, 0xF4},
{0x1F, 0xDD, 0xA8, 0x33, 0x88, 0x07, 0xC7, 0x31, 0xB1, 0x12, 0x10, 0x59, 0x27, 0x80, 0xEC, 0x5F},
{0x60, 0x51, 0x7F, 0xA9, 0x19, 0xB5, 0x4A, 0x0D, 0x2D, 0xE5, 0x7A, 0x9F, 0x93, 0xC9, 0x9C, 0xEF},
{0xA0, 0xE0, 0x3B, 0x4D, 0xAE, 0x2A, 0xF5, 0xB0, 0xC8, 0xEB, 0xBB, 0x3C, 0x83, 0x53, 0x99, 0x61},
{0x17, 0x2B, 0x04, 0x7E, 0xBA, 0x77, 0xD6, 0x26, 0xE1, 0x69, 0x14, 0x63, 0x55, 0x21, 0x0C, 0x7D}
};
// 轮常数，密钥扩展中用到。（AES-128只需要10轮）
word Rcon[10] = {0x01000000, 0x02000000, 0x04000000, 0x08000000, 0x10000000,
0x20000000, 0x40000000, 0x80000000, 0x1b000000, 0x36000000};
/**********************************************************************/
/* */
/* AES算法实现 */
/* */
/**********************************************************************/
/******************************下面是加密的变换函数**********************/
/**
* S盒变换 - 前4位为行号，后4位为列号
*/
void SubBytes(byte mtx[4*4])
{
for(int i=0; i<16; ++i)
{
int row = mtx[i][7]*8 + mtx[i][6]*4 + mtx[i][5]*2 + mtx[i][4];
int col = mtx[i][3]*8 + mtx[i][2]*4 + mtx[i][1]*2 + mtx[i][0];
mtx[i] = S_Box[row][col];
}
}
/**
* 行变换 - 按字节循环移位
*/
void ShiftRows(byte mtx[4*4])
{
// 第二行循环左移一位
byte temp = mtx[4];
for(int i=0; i<3; ++i)
mtx[i+4] = mtx[i+5];
mtx[7] = temp;
// 第三行循环左移两位
for(int i=0; i<2; ++i)
{
temp = mtx[i+8];
mtx[i+8] = mtx[i+10];
mtx[i+10] = temp;
}
// 第四行循环左移三位
temp = mtx[15];
for(int i=3; i>0; --i)
mtx[i+12] = mtx[i+11];
mtx[12] = temp;
}
/**
* 有限域上的乘法 GF(2^8)
*/
byte GFMul(byte a, byte b) {
byte p = 0;
byte hi_bit_set;
for (int counter = 0; counter < 8; counter++) {
if ((b & byte(1)) != 0) {
p ^= a;
}
hi_bit_set = (byte) (a & byte(0x80));
a <<= 1;
if (hi_bit_set != 0) {
a ^= 0x1b; /* x^8 + x^4 + x^3 + x + 1 */
}
b >>= 1;
}
return p;
}
/**
* 列变换
*/
void MixColumns(byte mtx[4*4])
{
byte arr[4];
for(int i=0; i<4; ++i)
{
for(int j=0; j<4; ++j)
arr[j] = mtx[i+j*4];
mtx[i] = GFMul(0x02, arr[0]) ^ GFMul(0x03, arr[1]) ^ arr[2] ^ arr[3];
mtx[i+4] = arr[0] ^ GFMul(0x02, arr[1]) ^ GFMul(0x03, arr[2]) ^ arr[3];
mtx[i+8] = arr[0] ^ arr[1] ^ GFMul(0x02, arr[2]) ^ GFMul(0x03, arr[3]);
mtx[i+12] = GFMul(0x03, arr[0]) ^ arr[1] ^ arr[2] ^ GFMul(0x02, arr[3]);
}
}
/**
* 轮密钥加变换 - 将每一列与扩展密钥进行异或
*/
void AddRoundKey(byte mtx[4*4], word k[4])
{
for(int i=0; i<4; ++i)
{
word k1 = k[i] >> 24;
word k2 = (k[i] << 8) >> 24;
word k3 = (k[i] << 16) >> 24;
word k4 = (k[i] << 24) >> 24;
mtx[i] = mtx[i] ^ byte(k1.to_ulong());
mtx[i+4] = mtx[i+4] ^ byte(k2.to_ulong());
mtx[i+8] = mtx[i+8] ^ byte(k3.to_ulong());
mtx[i+12] = mtx[i+12] ^ byte(k4.to_ulong());
}
}
/**************************下面是解密的逆变换函数***********************/
/**
* 逆S盒变换
*/
void InvSubBytes(byte mtx[4*4])
{
for(int i=0; i<16; ++i)
{
int row = mtx[i][7]*8 + mtx[i][6]*4 + mtx[i][5]*2 + mtx[i][4];
int col = mtx[i][3]*8 + mtx[i][2]*4 + mtx[i][1]*2 + mtx[i][0];
mtx[i] = Inv_S_Box[row][col];
}
}
/**
* 逆行变换 - 以字节为单位循环右移
*/
void InvShiftRows(byte mtx[4*4])
{
// 第二行循环右移一位
byte temp = mtx[7];
for(int i=3; i>0; --i)
mtx[i+4] = mtx[i+3];
mtx[4] = temp;
// 第三行循环右移两位
for(int i=0; i<2; ++i)
{
temp = mtx[i+8];
mtx[i+8] = mtx[i+10];
mtx[i+10] = temp;
}
// 第四行循环右移三位
temp = mtx[12];
for(int i=0; i<3; ++i)
mtx[i+12] = mtx[i+13];
mtx[15] = temp;
}
void InvMixColumns(byte mtx[4*4])
{
byte arr[4];
for(int i=0; i<4; ++i)
{
for(int j=0; j<4; ++j)
arr[j] = mtx[i+j*4];
mtx[i] = GFMul(0x0e, arr[0]) ^ GFMul(0x0b, arr[1]) ^ GFMul(0x0d, arr[2]) ^ GFMul(0x09, arr[3]);
mtx[i+4] = GFMul(0x09, arr[0]) ^ GFMul(0x0e, arr[1]) ^ GFMul(0x0b, arr[2]) ^ GFMul(0x0d, arr[3]);
mtx[i+8] = GFMul(0x0d, arr[0]) ^ GFMul(0x09, arr[1]) ^ GFMul(0x0e, arr[2]) ^ GFMul(0x0b, arr[3]);
mtx[i+12] = GFMul(0x0b, arr[0]) ^ GFMul(0x0d, arr[1]) ^ GFMul(0x09, arr[2]) ^ GFMul(0x0e, arr[3]);
}
}
/******************************下面是密钥扩展部分***********************/
/**
* 将4个 byte 转换为一个 word.
*/
word Word(byte& k1, byte& k2, byte& k3, byte& k4)
{
word result(0x00000000);
word temp;
temp = k1.to_ulong(); // K1
temp <<= 24;
result |= temp;
temp = k2.to_ulong(); // K2
temp <<= 16;
result |= temp;
temp = k3.to_ulong(); // K3
temp <<= 8;
result |= temp;
temp = k4.to_ulong(); // K4
result |= temp;
return result;
}
/**
* 按字节 循环左移一位
* 即把[a0, a1, a2, a3]变成[a1, a2, a3, a0]
*/
word RotWord(word& rw)
{
word high = rw << 8;
word low = rw >> 24;
return high | low;
}
/**
* 对输入word中的每一个字节进行S-盒变换
*/
word SubWord(word& sw)
{
word temp;
for(int i=0; i<32; i+=8)
{
int row = sw[i+7]*8 + sw[i+6]*4 + sw[i+5]*2 + sw[i+4];
int col = sw[i+3]*8 + sw[i+2]*4 + sw[i+1]*2 + sw[i];
byte val = S_Box[row][col];
for(int j=0; j<8; ++j)
temp[i+j] = val[j];
}
return temp;
}
/**
* 密钥扩展函数 - 对128位密钥进行扩展得到 w[4*(Nr+1)]
*/
void KeyExpansion(byte key[4*Nk], word w[4*(Nr+1)])
{
word temp;
int i = 0;
// w[]的前4个就是输入的key
while(i < Nk)
{
w[i] = Word(key[4*i], key[4*i+1], key[4*i+2], key[4*i+3]);
++i;
}
i = Nk;
while(i < 4*(Nr+1))
{
temp = w[i-1]; // 记录前一个word
if(i % Nk == 0)
w[i] = w[i-Nk] ^ SubWord(RotWord(temp)) ^ Rcon[i/Nk-1];
else
w[i] = w[i-Nk] ^ temp;
++i;
}
}
/******************************下面是加密和解密函数**************************/
/**
* 加密
*/
void encrypt(byte in[4*4], word w[4*(Nr+1)])
{
word key[4];
for(int i=0; i<4; ++i)
key[i] = w[i];
AddRoundKey(in, key);
for(int round=1; round<Nr; ++round)
{
SubBytes(in);
ShiftRows(in);
MixColumns(in);
for(int i=0; i<4; ++i)
key[i] = w[4*round+i];
AddRoundKey(in, key);
}
SubBytes(in);
ShiftRows(in);
for(int i=0; i<4; ++i)
key[i] = w[4*Nr+i];
AddRoundKey(in, key);
}
/**
* 解密
*/
void decrypt(byte in[4*4], word w[4*(Nr+1)])
{
word key[4];
for(int i=0; i<4; ++i)
key[i] = w[4*Nr+i];
AddRoundKey(in, key);
for(int round=Nr-1; round>0; --round)
{
InvShiftRows(in);
InvSubBytes(in);
for(int i=0; i<4; ++i)
key[i] = w[4*round+i];
AddRoundKey(in, key);
InvMixColumns(in);
}
InvShiftRows(in);
InvSubBytes(in);
for(int i=0; i<4; ++i)
key[i] = w[i];
AddRoundKey(in, key);
}
/**********************************************************************/
/* */
/* 测试 */
/* */
/**********************************************************************/
int main()
{
byte key[16] = {0x2b, 0x7e, 0x15, 0x16,
0x28, 0xae, 0xd2, 0xa6,
0xab, 0xf7, 0x15, 0x88,
0x09, 0xcf, 0x4f, 0x3c};
byte plain[16] = {0x32, 0x88, 0x31, 0xe0,
0x43, 0x5a, 0x31, 0x37,
0xf6, 0x30, 0x98, 0x07,
0xa8, 0x8d, 0xa2, 0x34};
// 输出密钥
cout << "密钥是：";
for(int i=0; i<16; ++i)
cout << hex << key[i].to_ulong() << " ";
cout << endl;
word w[4*(Nr+1)];
KeyExpansion(key, w);
// 输出待加密的明文
cout << endl << "待加密的明文："<<endl;
for(int i=0; i<16; ++i)
{
cout << hex << plain[i].to_ulong() << " ";
if((i+1)%4 == 0)
cout << endl;
}
cout << endl;
// 加密，输出密文
encrypt(plain, w);
cout << "加密后的密文："<<endl;
for(int i=0; i<16; ++i)
{
cout << hex << plain[i].to_ulong() << " ";
if((i+1)%4 == 0)
cout << endl;
}
cout << endl;
// 解密，输出明文
decrypt(plain, w);
cout << "解密后的明文："<<endl;
for(int i=0; i<16; ++i)
{
cout << hex << plain[i].to_ulong() << " ";
if((i+1)%4 == 0)
cout << endl;
}
cout << endl;
return 0;
}测试用例如下图：
测试结果截图：
可见，测试结果和预期输出相同，表明对数据的加密和解密成功！！！
下面我们来写 AES 对文件的加密和解密，在对 128 位的数据加解密成功以后，对文件的加解密就很简单了！只需要每次读 128 位，加密以后，将 128 位的密文写入另外一个文件…..如此循环，直到文件尾。下面是对一张图片进行 AES 加密和解密的测试代码（效率先不管了，有时间我再优化）：
//#include <fstream>
typedef bitset<8> byte;
typedef bitset<32> word;
/**
* 将一个char字符数组转化为二进制
* 存到一个 byte 数组中
*/
void charToByte(byte out[16], const char s[16])
{
for(int i=0; i<16; ++i)
for(int j=0; j<8; ++j)
out[i][j]= ((s[i]>>j) & 1);
}
/**
* 将连续的128位分成16组，存到一个 byte 数组中
*/
void divideToByte(byte out[16], bitset<128>& data)
{
bitset<128> temp;
for(int i=0; i<16; ++i)
{
temp = (data << 8*i) >> 120;
out[i] = temp.to_ulong();
}
}
/**
* 将16个 byte 合并成连续的128位
*/
bitset<128> mergeByte(byte in[16])
{
bitset<128> res;
res.reset(); // 置0
bitset<128> temp;
for(int i=0; i<16; ++i)
{
temp = in[i].to_ulong();
temp <<= 8*(15-i);
res |= temp;
}
return res;
}
int main()
{
string keyStr = "abcdefghijklmnop";
byte key[16];
charToByte(key, keyStr.c_str());
// 密钥扩展
word w[4*(Nr+1)];
KeyExpansion(key, w);
bitset<128> data;
byte plain[16];
// 将文件 flower.jpg 加密到 cipher.txt 中
ifstream in;
ofstream out;
in.open("D://flower.jpg", ios::binary);
out.open("D://cipher.txt", ios::binary);
while(in.read((char*)&data, sizeof(data)))
{
divideToByte(plain, data);
encrypt(plain, w);
data = mergeByte(plain);
out.write((char*)&data, sizeof(data));
data.reset(); // 置0
}
in.close();
out.close();
// 解密 cipher.txt，并写入图片 flower1.jpg
in.open("D://cipher.txt", ios::binary);
out.open("D://flower1.jpg", ios::binary);
while(in.read((char*)&data, sizeof(data)))
{
divideToByte(plain, data);
decrypt(plain, w);
data = mergeByte(plain);
out.write((char*)&data, sizeof(data));
data.reset(); // 置0
}
in.close();
out.close();
return 0;
}（全文完）
更新 —— 2014.12.21
有限域 GF(28) 上的乘法改用查表的方式实现，AES的加密速度马上提升 80% 以上，所以建议最好使用查表的方式。下面是 AES 算法中用到的 6 个乘法结果表：
byte Mul_02[256] = {
0x00,0x02,0x04,0x06,0x08,0x0a,0x0c,0x0e,0x10,0x12,0x14,0x16,0x18,0x1a,0x1c,0x1e,
0x20,0x22,0x24,0x26,0x28,0x2a,0x2c,0x2e,0x30,0x32,0x34,0x36,0x38,0x3a,0x3c,0x3e,
0x40,0x42,0x44,0x46,0x48,0x4a,0x4c,0x4e,0x50,0x52,0x54,0x56,0x58,0x5a,0x5c,0x5e,
0x60,0x62,0x64,0x66,0x68,0x6a,0x6c,0x6e,0x70,0x72,0x74,0x76,0x78,0x7a,0x7c,0x7e,
0x80,0x82,0x84,0x86,0x88,0x8a,0x8c,0x8e,0x90,0x92,0x94,0x96,0x98,0x9a,0x9c,0x9e,
0xa0,0xa2,0xa4,0xa6,0xa8,0xaa,0xac,0xae,0xb0,0xb2,0xb4,0xb6,0xb8,0xba,0xbc,0xbe,
0xc0,0xc2,0xc4,0xc6,0xc8,0xca,0xcc,0xce,0xd0,0xd2,0xd4,0xd6,0xd8,0xda,0xdc,0xde,
0xe0,0xe2,0xe4,0xe6,0xe8,0xea,0xec,0xee,0xf0,0xf2,0xf4,0xf6,0xf8,0xfa,0xfc,0xfe,
0x1b,0x19,0x1f,0x1d,0x13,0x11,0x17,0x15,0x0b,0x09,0x0f,0x0d,0x03,0x01,0x07,0x05,
0x3b,0x39,0x3f,0x3d,0x33,0x31,0x37,0x35,0x2b,0x29,0x2f,0x2d,0x23,0x21,0x27,0x25,
0x5b,0x59,0x5f,0x5d,0x53,0x51,0x57,0x55,0x4b,0x49,0x4f,0x4d,0x43,0x41,0x47,0x45,
0x7b,0x79,0x7f,0x7d,0x73,0x71,0x77,0x75,0x6b,0x69,0x6f,0x6d,0x63,0x61,0x67,0x65,
0x9b,0x99,0x9f,0x9d,0x93,0x91,0x97,0x95,0x8b,0x89,0x8f,0x8d,0x83,0x81,0x87,0x85,
0xbb,0xb9,0xbf,0xbd,0xb3,0xb1,0xb7,0xb5,0xab,0xa9,0xaf,0xad,0xa3,0xa1,0xa7,0xa5,
0xdb,0xd9,0xdf,0xdd,0xd3,0xd1,0xd7,0xd5,0xcb,0xc9,0xcf,0xcd,0xc3,0xc1,0xc7,0xc5,
0xfb,0xf9,0xff,0xfd,0xf3,0xf1,0xf7,0xf5,0xeb,0xe9,0xef,0xed,0xe3,0xe1,0xe7,0xe5
};
byte Mul_03[256] = {
0x00,0x03,0x06,0x05,0x0c,0x0f,0x0a,0x09,0x18,0x1b,0x1e,0x1d,0x14,0x17,0x12,0x11,
0x30,0x33,0x36,0x35,0x3c,0x3f,0x3a,0x39,0x28,0x2b,0x2e,0x2d,0x24,0x27,0x22,0x21,
0x60,0x63,0x66,0x65,0x6c,0x6f,0x6a,0x69,0x78,0x7b,0x7e,0x7d,0x74,0x77,0x72,0x71,
0x50,0x53,0x56,0x55,0x5c,0x5f,0x5a,0x59,0x48,0x4b,0x4e,0x4d,0x44,0x47,0x42,0x41,
0xc0,0xc3,0xc6,0xc5,0xcc,0xcf,0xca,0xc9,0xd8,0xdb,0xde,0xdd,0xd4,0xd7,0xd2,0xd1,
0xf0,0xf3,0xf6,0xf5,0xfc,0xff,0xfa,0xf9,0xe8,0xeb,0xee,0xed,0xe4,0xe7,0xe2,0xe1,
0xa0,0xa3,0xa6,0xa5,0xac,0xaf,0xaa,0xa9,0xb8,0xbb,0xbe,0xbd,0xb4,0xb7,0xb2,0xb1,
0x90,0x93,0x96,0x95,0x9c,0x9f,0x9a,0x99,0x88,0x8b,0x8e,0x8d,0x84,0x87,0x82,0x81,
0x9b,0x98,0x9d,0x9e,0x97,0x94,0x91,0x92,0x83,0x80,0x85,0x86,0x8f,0x8c,0x89,0x8a,
0xab,0xa8,0xad,0xae,0xa7,0xa4,0xa1,0xa2,0xb3,0xb0,0xb5,0xb6,0xbf,0xbc,0xb9,0xba,
0xfb,0xf8,0xfd,0xfe,0xf7,0xf4,0xf1,0xf2,0xe3,0xe0,0xe5,0xe6,0xef,0xec,0xe9,0xea,
0xcb,0xc8,0xcd,0xce,0xc7,0xc4,0xc1,0xc2,0xd3,0xd0,0xd5,0xd6,0xdf,0xdc,0xd9,0xda,
0x5b,0x58,0x5d,0x5e,0x57,0x54,0x51,0x52,0x43,0x40,0x45,0x46,0x4f,0x4c,0x49,0x4a,
0x6b,0x68,0x6d,0x6e,0x67,0x64,0x61,0x62,0x73,0x70,0x75,0x76,0x7f,0x7c,0x79,0x7a,
0x3b,0x38,0x3d,0x3e,0x37,0x34,0x31,0x32,0x23,0x20,0x25,0x26,0x2f,0x2c,0x29,0x2a,
0x0b,0x08,0x0d,0x0e,0x07,0x04,0x01,0x02,0x13,0x10,0x15,0x16,0x1f,0x1c,0x19,0x1a
};
byte Mul_09[256] = {
0x00,0x09,0x12,0x1b,0x24,0x2d,0x36,0x3f,0x48,0x41,0x5a,0x53,0x6c,0x65,0x7e,0x77,
0x90,0x99,0x82,0x8b,0xb4,0xbd,0xa6,0xaf,0xd8,0xd1,0xca,0xc3,0xfc,0xf5,0xee,0xe7,
0x3b,0x32,0x29,0x20,0x1f,0x16,0x0d,0x04,0x73,0x7a,0x61,0x68,0x57,0x5e,0x45,0x4c,
0xab,0xa2,0xb9,0xb0,0x8f,0x86,0x9d,0x94,0xe3,0xea,0xf1,0xf8,0xc7,0xce,0xd5,0xdc,
0x76,0x7f,0x64,0x6d,0x52,0x5b,0x40,0x49,0x3e,0x37,0x2c,0x25,0x1a,0x13,0x08,0x01,
0xe6,0xef,0xf4,0xfd,0xc2,0xcb,0xd0,0xd9,0xae,0xa7,0xbc,0xb5,0x8a,0x83,0x98,0x91,
0x4d,0x44,0x5f,0x56,0x69,0x60,0x7b,0x72,0x05,0x0c,0x17,0x1e,0x21,0x28,0x33,0x3a,
0xdd,0xd4,0xcf,0xc6,0xf9,0xf0,0xeb,0xe2,0x95,0x9c,0x87,0x8e,0xb1,0xb8,0xa3,0xaa,
0xec,0xe5,0xfe,0xf7,0xc8,0xc1,0xda,0xd3,0xa4,0xad,0xb6,0xbf,0x80,0x89,0x92,0x9b,
0x7c,0x75,0x6e,0x67,0x58,0x51,0x4a,0x43,0x34,0x3d,0x26,0x2f,0x10,0x19,0x02,0x0b,
0xd7,0xde,0xc5,0xcc,0xf3,0xfa,0xe1,0xe8,0x9f,0x96,0x8d,0x84,0xbb,0xb2,0xa9,0xa0,
0x47,0x4e,0x55,0x5c,0x63,0x6a,0x71,0x78,0x0f,0x06,0x1d,0x14,0x2b,0x22,0x39,0x30,
0x9a,0x93,0x88,0x81,0xbe,0xb7,0xac,0xa5,0xd2,0xdb,0xc0,0xc9,0xf6,0xff,0xe4,0xed,
0x0a,0x03,0x18,0x11,0x2e,0x27,0x3c,0x35,0x42,0x4b,0x50,0x59,0x66,0x6f,0x74,0x7d,
0xa1,0xa8,0xb3,0xba,0x85,0x8c,0x97,0x9e,0xe9,0xe0,0xfb,0xf2,0xcd,0xc4,0xdf,0xd6,
0x31,0x38,0x23,0x2a,0x15,0x1c,0x07,0x0e,0x79,0x70,0x6b,0x62,0x5d,0x54,0x4f,0x46
};
byte Mul_0b[256] = {
0x00,0x0b,0x16,0x1d,0x2c,0x27,0x3a,0x31,0x58,0x53,0x4e,0x45,0x74,0x7f,0x62,0x69,
0xb0,0xbb,0xa6,0xad,0x9c,0x97,0x8a,0x81,0xe8,0xe3,0xfe,0xf5,0xc4,0xcf,0xd2,0xd9,
0x7b,0x70,0x6d,0x66,0x57,0x5c,0x41,0x4a,0x23,0x28,0x35,0x3e,0x0f,0x04,0x19,0x12,
0xcb,0xc0,0xdd,0xd6,0xe7,0xec,0xf1,0xfa,0x93,0x98,0x85,0x8e,0xbf,0xb4,0xa9,0xa2,
0xf6,0xfd,0xe0,0xeb,0xda,0xd1,0xcc,0xc7,0xae,0xa5,0xb8,0xb3,0x82,0x89,0x94,0x9f,
0x46,0x4d,0x50,0x5b,0x6a,0x61,0x7c,0x77,0x1e,0x15,0x08,0x03,0x32,0x39,0x24,0x2f,
0x8d,0x86,0x9b,0x90,0xa1,0xaa,0xb7,0xbc,0xd5,0xde,0xc3,0xc8,0xf9,0xf2,0xef,0xe4,
0x3d,0x36,0x2b,0x20,0x11,0x1a,0x07,0x0c,0x65,0x6e,0x73,0x78,0x49,0x42,0x5f,0x54,
0xf7,0xfc,0xe1,0xea,0xdb,0xd0,0xcd,0xc6,0xaf,0xa4,0xb9,0xb2,0x83,0x88,0x95,0x9e,
0x47,0x4c,0x51,0x5a,0x6b,0x60,0x7d,0x76,0x1f,0x14,0x09,0x02,0x33,0x38,0x25,0x2e,
0x8c,0x87,0x9a,0x91,0xa0,0xab,0xb6,0xbd,0xd4,0xdf,0xc2,0xc9,0xf8,0xf3,0xee,0xe5,
0x3c,0x37,0x2a,0x21,0x10,0x1b,0x06,0x0d,0x64,0x6f,0x72,0x79,0x48,0x43,0x5e,0x55,
0x01,0x0a,0x17,0x1c,0x2d,0x26,0x3b,0x30,0x59,0x52,0x4f,0x44,0x75,0x7e,0x63,0x68,
0xb1,0xba,0xa7,0xac,0x9d,0x96,0x8b,0x80,0xe9,0xe2,0xff,0xf4,0xc5,0xce,0xd3,0xd8,
0x7a,0x71,0x6c,0x67,0x56,0x5d,0x40,0x4b,0x22,0x29,0x34,0x3f,0x0e,0x05,0x18,0x13,
0xca,0xc1,0xdc,0xd7,0xe6,0xed,0xf0,0xfb,0x92,0x99,0x84,0x8f,0xbe,0xb5,0xa8,0xa3
};
byte Mul_0d[256] = {
0x00,0x0d,0x1a,0x17,0x34,0x39,0x2e,0x23,0x68,0x65,0x72,0x7f,0x5c,0x51,0x46,0x4b,
0xd0,0xdd,0xca,0xc7,0xe4,0xe9,0xfe,0xf3,0xb8,0xb5,0xa2,0xaf,0x8c,0x81,0x96,0x9b,
0xbb,0xb6,0xa1,0xac,0x8f,0x82,0x95,0x98,0xd3,0xde,0xc9,0xc4,0xe7,0xea,0xfd,0xf0,
0x6b,0x66,0x71,0x7c,0x5f,0x52,0x45,0x48,0x03,0x0e,0x19,0x14,0x37,0x3a,0x2d,0x20,
0x6d,0x60,0x77,0x7a,0x59,0x54,0x43,0x4e,0x05,0x08,0x1f,0x12,0x31,0x3c,0x2b,0x26,
0xbd,0xb0,0xa7,0xaa,0x89,0x84,0x93,0x9e,0xd5,0xd8,0xcf,0xc2,0xe1,0xec,0xfb,0xf6,
0xd6,0xdb,0xcc,0xc1,0xe2,0xef,0xf8,0xf5,0xbe,0xb3,0xa4,0xa9,0x8a,0x87,0x90,0x9d,
0x06,0x0b,0x1c,0x11,0x32,0x3f,0x28,0x25,0x6e,0x63,0x74,0x79,0x5a,0x57,0x40,0x4d,
0xda,0xd7,0xc0,0xcd,0xee,0xe3,0xf4,0xf9,0xb2,0xbf,0xa8,0xa5,0x86,0x8b,0x9c,0x91,
0x0a,0x07,0x10,0x1d,0x3e,0x33,0x24,0x29,0x62,0x6f,0x78,0x75,0x56,0x5b,0x4c,0x41,
0x61,0x6c,0x7b,0x76,0x55,0x58,0x4f,0x42,0x09,0x04,0x13,0x1e,0x3d,0x30,0x27,0x2a,
0xb1,0xbc,0xab,0xa6,0x85,0x88,0x9f,0x92,0xd9,0xd4,0xc3,0xce,0xed,0xe0,0xf7,0xfa,
0xb7,0xba,0xad,0xa0,0x83,0x8e,0x99,0x94,0xdf,0xd2,0xc5,0xc8,0xeb,0xe6,0xf1,0xfc,
0x67,0x6a,0x7d,0x70,0x53,0x5e,0x49,0x44,0x0f,0x02,0x15,0x18,0x3b,0x36,0x21,0x2c,
0x0c,0x01,0x16,0x1b,0x38,0x35,0x22,0x2f,0x64,0x69,0x7e,0x73,0x50,0x5d,0x4a,0x47,
0xdc,0xd1,0xc6,0xcb,0xe8,0xe5,0xf2,0xff,0xb4,0xb9,0xae,0xa3,0x80,0x8d,0x9a,0x97
};
byte Mul_0e[256] = {
0x00,0x0e,0x1c,0x12,0x38,0x36,0x24,0x2a,0x70,0x7e,0x6c,0x62,0x48,0x46,0x54,0x5a,
0xe0,0xee,0xfc,0xf2,0xd8,0xd6,0xc4,0xca,0x90,0x9e,0x8c,0x82,0xa8,0xa6,0xb4,0xba,
0xdb,0xd5,0xc7,0xc9,0xe3,0xed,0xff,0xf1,0xab,0xa5,0xb7,0xb9,0x93,0x9d,0x8f,0x81,
0x3b,0x35,0x27,0x29,0x03,0x0d,0x1f,0x11,0x4b,0x45,0x57,0x59,0x73,0x7d,0x6f,0x61,
0xad,0xa3,0xb1,0xbf,0x95,0x9b,0x89,0x87,0xdd,0xd3,0xc1,0xcf,0xe5,0xeb,0xf9,0xf7,
0x4d,0x43,0x51,0x5f,0x75,0x7b,0x69,0x67,0x3d,0x33,0x21,0x2f,0x05,0x0b,0x19,0x17,
0x76,0x78,0x6a,0x64,0x4e,0x40,0x52,0x5c,0x06,0x08,0x1a,0x14,0x3e,0x30,0x22,0x2c,
0x96,0x98,0x8a,0x84,0xae,0xa0,0xb2,0xbc,0xe6,0xe8,0xfa,0xf4,0xde,0xd0,0xc2,0xcc,
0x41,0x4f,0x5d,0x53,0x79,0x77,0x65,0x6b,0x31,0x3f,0x2d,0x23,0x09,0x07,0x15,0x1b,
0xa1,0xaf,0xbd,0xb3,0x99,0x97,0x85,0x8b,0xd1,0xdf,0xcd,0xc3,0xe9,0xe7,0xf5,0xfb,
0x9a,0x94,0x86,0x88,0xa2,0xac,0xbe,0xb0,0xea,0xe4,0xf6,0xf8,0xd2,0xdc,0xce,0xc0,
0x7a,0x74,0x66,0x68,0x42,0x4c,0x5e,0x50,0x0a,0x04,0x16,0x18,0x32,0x3c,0x2e,0x20,
0xec,0xe2,0xf0,0xfe,0xd4,0xda,0xc8,0xc6,0x9c,0x92,0x80,0x8e,0xa4,0xaa,0xb8,0xb6,
0x0c,0x02,0x10,0x1e,0x34,0x3a,0x28,0x26,0x7c,0x72,0x60,0x6e,0x44,0x4a,0x58,0x56,
0x37,0x39,0x2b,0x25,0x0f,0x01,0x13,0x1d,0x47,0x49,0x5b,0x55,0x7f,0x71,0x63,0x6d,
0xd7,0xd9,0xcb,0xc5,0xef,0xe1,0xf3,0xfd,0xa7,0xa9,0xbb,0xb5,0x9f,0x91,0x83,0x8d
};