摘要：作為新一代的加密標準，AES 旨在取代 DES（請看《DES加密算法的C++實現》），以適應當今分布式開放網絡對數據加密安全性的要求。本文在分析了 AES 加密原理的基礎上著重說明了算法實現的具體步驟，并用 C++ 實現了對文件的加密和解密。
一、AES 介紹
AES（高級加密標準，Advanced Encryption Standard），在密碼學中又稱 Rijndael 加密法，是美國聯邦政府采用的一種分組加密標準。這個標準用來替代原先的 DES，目前已經廣為全世界所使用，成為對稱密鑰算法中最流行的算法之一。
在 AES 出現之前，最常用的對稱密鑰算法是 DES 加密算法，它在 1977 年被公布成為美國政府的商用加密標準。DES 的主要問題是密鑰長度較短，漸漸不適合于分布式開放網絡對數據加密安全性的要求。因此，1998年美國政府決定不再繼續延用 DES 作為聯邦加密標準，并發起了征集 AES 候選算法的活動。征集活動對 AES 的基本要求是： 比三重DES快、至少與三重DES一樣安全、數據分組長度為128比特、密鑰長度為128/192/256比特。
經過三年多的甄選，比利時的密碼學家所設計的 Rijndael 算法最終脫穎而出，成為新一代的高級加密標準，并于 2001 年由美國國家標準與技術研究院（NIST）發布于 FIPS PUB 197。
二、AES 算法原理
AES算法（即 Rijndael 算法）是一個對稱分組密碼算法。數據分組長度必須是 128 bits，使用的密鑰長度為 128，192 或 256 bits。對于三種不同密鑰長度的 AES 算法，分別稱為“AES-128”、“AES-192”、“AES-256”。（Rijndael 的設計還可以處理其它的分組長度和密鑰長度，但 AES 標準中沒有采用）
下圖是 AES 加密解密的整體流程圖：
這里我們需要知道3個符號：Nb—— 狀態 State 包含的列（32-bit 字）的個數，也就是說 Nb=4；Nk—— 密鑰包含的 32-bit 字的個數，也就是說 Nk=4，6 或 8；Nr—— 加密的輪數，對于不同密鑰長度，輪數不一樣，具體如下圖所示：
下面分為密鑰擴展、分組加密、分組解密三個部分來講 AES 算法，我會盡可能地簡明扼要，若還有不懂的，請自行 Google。
1）密鑰擴展
AES 算法通過密鑰擴展程序（Key Expansion）將用戶輸入的密鑰 K 擴展生成 Nb(Nr+1)個字，存放在一個線性數組w[Nb*(Nr+1)]中。具體如下：
注意，上面的第四步說明適合于 AES-128 和 AES-192，詳細的偽代碼如下：
密鑰擴展程序的 C++ 代碼（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 輪加密 const int Nk = 4; // Nk 表示輸入密鑰的 word 個數 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} }; // 輪常數，密鑰擴展中用到。（AES-128只需要10輪） word Rcon[10] = {0x01000000, 0x02000000, 0x04000000, 0x08000000, 0x10000000, 0x20000000, 0x40000000, 0x80000000, 0x1b000000, 0x36000000}; /** * 將4個 byte 轉換為一個 word */ word Word(byte& k1, byte& k2, byte& k3, byte& k4) { word result(0x00000000); word temp; temp = kto_ulong(); // K1 temp <<= 24; result |= temp; temp = kto_ulong(); // K2 temp <<= 16; result |= temp; temp = kto_ulong(); // K3 temp <<= 8; result |= temp; temp = kto_ulong(); // K4 result |= temp; return result; } /** * 按字節 循環左移一位 * 即把[a0, a1, a2, a3]變成[a1, a2, a3, a0] */ word RotWord(word& rw) { word high = rw << 8; word low = rw >> 24; return high | low; } /** * 對輸入word中的每一個字節進行S-盒變換 */ 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; } /** * 密鑰擴展函數 - 對128位密鑰進行擴展得到 w[4*(Nr+1)] */ void KeyExpansion(byte key[4*Nk], word w[4*(Nr+1)]) { word temp; int i = 0; // w[]的前4個就是輸入的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]; // 記錄前一個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); // 測試 for(int i=0; i<4*(Nr+1); ++i) cout << "w[" << dec << i << "] = " << hex << w[i]to_ulong() << endl; return 0; }
測試輸出結果：
2）加密
根據 AES 加密的整體流程圖（本文開頭），偽代碼如下：
從偽代碼描述中可以看出，AES 加密時涉及到的子程序有SubBytes()、ShiftRows()、MixColumns()和AddRoundKey()。下面我們一個一個進行介紹：
① S盒變換-SubBytes()
在密鑰擴展部分已經講過了，S盒是一個 16 行 16 列的表，表中每個元素都是一個字節。S盒變換很簡單：函數SubBytes()接受一個 4x4 的字節矩陣作為輸入，對其中的每個字節，前四位組成十六進制數 x 作為行號，后四位組成的十六進制數 y 作為列號，查找表中對應的值替換原來位置上的字節。
② 行變換-ShiftRows()
行變換也很簡單，它僅僅是將矩陣的每一行以字節為單位循環移位：第一行不變，第二行左移一位，第三行左移兩位，第四行左移三位。如下圖所示：
③ 列變換-MixColumns()
函數MixColumns()同樣接受一個 4x4 的字節矩陣作為輸入，并對矩陣進行逐列變換，變換方式如下：
注意公式中用到的乘法是伽羅華域（GF，有限域）上的乘法，高級加密標準文檔 fips-197 上有講，如果還是不懂，請自行Google。
④ 與擴展密鑰的異或-AddRoundKey()
擴展密鑰只參與了這一步。根據當前加密的輪數，用w[]中的 4 個擴展密鑰與矩陣的 4 個列進行按位異或。如下圖：
好了，到這里 AES 加密的各個部分就講完了。算法實現的 C++ 源碼在文章后面第三部分。
3）解密
根據 AES 解密的整體流程圖（本文開頭），偽代碼如下：
從偽代碼可以看出，我們需要分別實現 S 盒變換、行變換和列變換的逆變換InvShiftRows()、InvSubBytes()和InvMixColumns()。下面就簡單的講一下這三個逆變換：
① 逆行變換-InvShiftRows()
上面講到ShiftRows()是對矩陣的每一行進行循環左移，所以InvShiftRows()是對矩陣每一行進行循環右移。
② 逆 S 盒變換-InvSubBytes()
與 S 盒變換一樣，也是查表，查表的方式也一樣，只不過查的是另外一個置換表（S-Box的逆表）。
③ 逆列變換-InvMixColumns()
與列變換的方式一樣，只不過計算公式的系數矩陣發生了變化。如下圖：
好了，AES 解密到這里也講完了。只要寫出三個逆變換的函數，然后根據偽代碼就很容易實現 AES 解密算法了。
三、C++實現
下面我用 C++ 實現 AES 的加密和解密算法，并實現了對文件的加密和解密。這里我使用 C++ STL 的bitset定義了兩個類型：byte和word。需要提到的是，對于有限域上的乘法，我們既可以通過查表（6個結果表），也可以寫一個函數來實現。當然，查表的效率會更高，但考慮到貼代碼，這里我就用一個函數來實現的。
下面是 AES-128 對一個 128 位數據加密和解密的源代碼：
/************************************************************************* > File Name: AEScpp > Author: SongLee > E-mail: lisongshine@qqcom > Created Time: 2014年12月12日 星期五 20時15分50秒 > Personal Blog: http://songleegithubcom ************************************************************************/ #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 輪加密 const int Nk = 4; // Nk 表示輸入密鑰的 word 個數 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} }; // 輪常數，密鑰擴展中用到。（AES-128只需要10輪） word Rcon[10] = {0x01000000, 0x02000000, 0x04000000, 0x08000000, 0x10000000, 0x20000000, 0x40000000, 0x80000000, 0x1b000000, 0x36000000}; /**********************************************************************/ /* */ /* AES算法實現 */ /* */ /**********************************************************************/ /******************************下面是加密的變換函數**********************/ /** * S盒變換 - 前4位為行號，后4位為列號 */ 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]; } } /** * 行變換 - 按字節循環移位 */ void ShiftRows(byte mtx[4*4]) { // 第二行循環左移一位 byte temp = mtx[4]; for(int i=0; i<3; ++i) mtx[i+4] = mtx[i+5]; mtx[7] = temp; // 第三行循環左移兩位 for(int i=0; i<2; ++i) { temp = mtx[i+8]; mtx[i+8] = mtx[i+10]; mtx[i+10] = temp; } // 第四行循環左移三位 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; } /** * 列變換 */ 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]); } } /** * 輪密鑰加變換 - 將每一列與擴展密鑰進行異或 */ 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(kto_ulong()); mtx[i+4] = mtx[i+4] ^ byte(kto_ulong()); mtx[i+8] = mtx[i+8] ^ byte(kto_ulong()); mtx[i+12] = mtx[i+12] ^ byte(kto_ulong()); } } /**************************下面是解密的逆變換函數***********************/ /** * 逆S盒變換 */ 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]; } } /** * 逆行變換 - 以字節為單位循環右移 */ void InvShiftRows(byte mtx[4*4]) { // 第二行循環右移一位 byte temp = mtx[7]; for(int i=3; i>0; --i) mtx[i+4] = mtx[i+3]; mtx[4] = temp; // 第三行循環右移兩位 for(int i=0; i<2; ++i) { temp = mtx[i+8]; mtx[i+8] = mtx[i+10]; mtx[i+10] = temp; } // 第四行循環右移三位 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]); } } /******************************下面是密鑰擴展部分***********************/ /** * 將4個 byte 轉換為一個 word */ word Word(byte& k1, byte& k2, byte& k3, byte& k4) { word result(0x00000000); word temp; temp = kto_ulong(); // K1 temp <<= 24; result |= temp; temp = kto_ulong(); // K2 temp <<= 16; result |= temp; temp = kto_ulong(); // K3 temp <<= 8; result |= temp; temp = kto_ulong(); // K4 result |= temp; return result; } /** * 按字節 循環左移一位 * 即把[a0, a1, a2, a3]變成[a1, a2, a3, a0] */ word RotWord(word& rw) { word high = rw << 8; word low = rw >> 24; return high | low; } /** * 對輸入word中的每一個字節進行S-盒變換 */ 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; } /** * 密鑰擴展函數 - 對128位密鑰進行擴展得到 w[4*(Nr+1)] */ void KeyExpansion(byte key[4*Nk], word w[4*(Nr+1)]) { word temp; int i = 0; // w[]的前4個就是輸入的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]; // 記錄前一個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數**************************/ /** * 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); } /**********************************************************************/ /* */ /* 測試 */ /* */ /**********************************************************************/ 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}; // 輸出密鑰 cout << "密鑰是："; for(int i=0; i<16; ++i) cout << hex << key[i]to_ulong() << " "; cout << endl; word w[4*(Nr+1)]; KeyExpansion(key, w); // 輸出待加密的明文 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，輸出密文 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，輸出明文 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; } 測試用例如下圖：
測試結果截圖：
可見，測試結果和預期輸出相同，表明對數據的加密和解密成功?。。?/p>
下面我們來寫 AES 對文件的加密和解密，在對 128 位的數據加解密成功以后，對文件的加解密就很簡單了！只需要每次讀 128 位，加密以后，將 128 位的密文寫入另外一個文件…如此循環，直到文件尾。下面是對一張圖片進行 AES 加密和解密的測試代碼（效率先不管了，有時間我再優化）：
//#include <fstream> typedef bitset<8> byte; typedef bitset<32> word; /** * 將一個char字符數組轉化為二進制 * 存到一個 byte 數組中 */ 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); } /** * 將連續的128位分成16組，存到一個 byte 數組中 */ 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] = tempto_ulong(); } } /** * 將16個 byte 合并成連續的128位 */ bitset<128> mergeByte(byte in[16]) { bitset<128> res; resreset(); // 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, keyStrc_str()); // 密鑰擴展 word w[4*(Nr+1)]; KeyExpansion(key, w); bitset<128> data; byte plain[16]; // 將文件 flowerjpg 加密到 ciphertxt 中 ifstream in; ofstream out; inopen("D://flowerjpg", ios::binary); outopen("D://ciphertxt", ios::binary); while(inread((char*)&data, sizeof(data))) { divideToByte(plain, data); encrypt(plain, w); data = mergeByte(plain); outwrite((char*)&data, sizeof(data)); datareset(); // 置0 } inclose(); outclose(); // 解密 ciphertxt，并寫入圖片 flowerjpg inopen("D://ciphertxt", ios::binary); outopen("D://flowerjpg", ios::binary); while(inread((char*)&data, sizeof(data))) { divideToByte(plain, data); decrypt(plain, w); data = mergeByte(plain); outwrite((char*)&data, sizeof(data)); datareset(); // 置0 } inclose(); outclose(); return 0; } 有限域 GF(28) 上的乘法改用查表的方式實現，AES的加密速度馬上提升 80% 以上，所以建議最好使用查表的方式。下面是 AES 算法中用到的 6 個乘法結果表：
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 }; 以上就是本文的全部內容，希望對大家的學習有所幫助，也希望大家多多支持VEVB武林網。
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