一、什么是CRC校驗算法
最近在學網絡時在以太網的數據幀的末尾有一個叫CRC校驗碼的東西，遂不解。于是便一起學習一下，什么是CRC校驗碼。
CRC就是循環冗余校驗碼（Cyclic Redundancy Check），是數據通信領域常見的差錯校驗碼，特征是信息字段和校驗字段的長度可以任意的選定。

循環冗余檢查（CRC）是一種數據傳輸檢錯功能，對數據進行多項式計算，并將得到的結果附在幀的后面，接受設備也執行類似的算法來保證數據傳輸的正確性和完整性。
二、CRC校驗算法的算法的原理
CRC校驗算法的原理就是：原始幀數據發送之前，在n個bit位的原始數據后面加上通過特定運算得到的K位校驗序列，組成新的幀來發送給接收端。接收端會根據原始數據后的校驗序列再次進行特定的運算，若正確則接受，若結果錯誤則丟棄。

把上面的K位校驗碼序列就稱為：FCS
CRC校驗算法原理的示圖如下：

我們把特定的運算就稱為異或運算。
這樣看來CRC校驗算法也就是把原始數據通過異或運算得到FCS，接收端根據原始數據再次運算如果相等那么就接收。那么CRC算法的核心就是如何得到FCS。

假設要發送的數據是M，M里面有K個數據，現在要計算冗余碼。冗余碼的計算方法如下：
1、用二進制模2運算來進行2^n*M也就是M左移了n位，也即是在M的后面加上了n個0，現在M的長度就是K+n
2、用M去除收發雙方事先商定的長度為n+1的除數p，得到余數是R
3、這個R就是FCS（幀檢驗序列），將這個FCS序列加到M后面發出去就行了。
最后接收端對數據進行CRC校驗，若余數為R就表示這個幀沒有錯，就接受。若R不為0表示這個幀出錯就丟棄。
一般在數據傳輸之前，發送端與接收端會相互約定好一個除數（也是一個二進制序列，用來進行模2算法）。這個除數就是生成多項式。這個多項式的最高位和最低位必須為1。
常見的生成多項式為：
CRC8=X8+X5+X4+1（100110001）
CRC-CCITT=X16+X12+X5+1（1001000000100001）
CRC16=X16+X15+X5+1（11000000000100001）
CRC12=X12+X11+X3+X2+1（1100000001101）
CRC32=X32+X26+X23+X22+X16+X12+X11+X10+X8+X7+X5+X4+X2+X1+1
（100000100110000010001110110011111）
給個栗子吧：
M= 101001，p = 1101，n = 3
M是要發送的數據，p是除數，n是在M后面差錯檢測的n位冗余碼
發送端：
M=(2^n*M)，所以M=101001000
用M除以p：

得到的余數是FCS，將其加到M的后面就是要發送的幀
M = 101001000 + FCS = 101001001

接收端：
接收到的每一幀都要進行差錯檢驗，假設收到的101001001，p=1101，具體如下：

我們可以看到最后的余數R=0，沒有出錯，所以信息是被接收的。
三、CRC算法的編程實現
下面我們通過一個栗子來說明是如何實現CRC校驗的，生成多項式為：100110001（簡記0x31）,也就是CRC-8
計算步驟如下：
（1） 將CRC寄存器（8-bits，比生成多項式少1bit）賦初值0
（2） 在待傳輸信息流后面加入8個0
（3） While (數據未處理完)
（4） Begin
（5） If (CRC寄存器首位是1)
（6） reg = reg XOR 0x31
（7） CRC寄存器左移一位，讀入一個新的數據于CRC寄存器的0 bit的位置。
（8） End
（9） CRC寄存器就是我們所要求的余數。
程序實現的示圖：

代碼展示：（代碼來自參考文章的鏈接里面）
代碼是C++實現了CRC8、CRC16和CRC32，代碼參考如下：

#ifndef CRCCOMPUTE_H #define CRCCOMPUTE_H #include <stdint.h> template <typename TYPE> class CRC { public: CRC(); CRC(TYPE polynomial, TYPE init_remainder, TYPE final_xor_value); void build(TYPE polynomial, TYPE init_remainder, TYPE final_xor_value); /** * Compute the CRC checksum of a binary message block. * @para message, 用來計算的數據 * @para nBytes, 數據的長度 */ TYPE crcCompute(char * message, unsigned int nBytes); TYPE crcCompute(char * message, unsigned int nBytes, bool reinit); protected: TYPE m_polynomial; TYPE m_initial_remainder; TYPE m_final_xor_value; TYPE m_remainder; TYPE crcTable[256]; int m_width; int m_topbit; /** * Initialize the CRC lookup table. * This table is used by crcCompute() to make CRC computation faster. */ void crcInit(void); }; template <typename TYPE> CRC<TYPE>::CRC() { m_width = 8 * sizeof(TYPE); m_topbit = 1 << (m_width - 1); } template <typename TYPE> CRC<TYPE>::CRC(TYPE polynomial, TYPE init_remainder, TYPE final_xor_value) { m_width = 8 * sizeof(TYPE); m_topbit = 1 << (m_width - 1); m_polynomial = polynomial; m_initial_remainder = init_remainder; m_final_xor_value = final_xor_value; crcInit(); } template <typename TYPE> void CRC<TYPE>::build(TYPE polynomial, TYPE init_remainder, TYPE final_xor_value) { m_polynomial = polynomial; m_initial_remainder = init_remainder; m_final_xor_value = final_xor_value; crcInit(); } template <typename TYPE> TYPE CRC<TYPE>::crcCompute(char * message, unsigned int nBytes) { unsigned int offset; unsigned char byte; TYPE remainder = m_initial_remainder; /* Divide the message by the polynomial, a byte at a time. */ for( offset = 0; offset < nBytes; offset++) { byte = (remainder >> (m_width - 8)) ^ message[offset]; remainder = crcTable[byte] ^ (remainder << 8); } /* The final remainder is the CRC result. */ return (remainder ^ m_final_xor_value); } template <typename TYPE> TYPE CRC<TYPE>::crcCompute(char * message, unsigned int nBytes, bool reinit) { unsigned int offset; unsigned char byte; if(reinit) { m_remainder = m_initial_remainder; } /* Divide the message by the polynomial, a byte at a time. */ for( offset = 0; offset < nBytes; offset++) { byte = (m_remainder >> (m_width - 8)) ^ message[offset]; m_remainder = crcTable[byte] ^ (m_remainder << 8); } /* The final remainder is the CRC result. */ return (m_remainder ^ m_final_xor_value); } class CRC8 : public CRC<uint8_t> { public: enum CRC8_TYPE {eCRC8, eAUTOSAR, eCDMA2000, eDARC, eDVB_S2, eEBU, eAES, eGSM_A, eGSM_B, eI_CODE, eITU, eLTE, eMAXIM, eOPENSAFETY, eROHC, eSAE_J1850, eWCDMA}; CRC8(CRC8_TYPE type); CRC8(uint8_t polynomial, uint8_t init_remainder, uint8_t final_xor_value) :CRC<uint8_t>(polynomial, init_remainder, final_xor_value){} }; class CRC16 : public CRC<uint16_t> { public: enum CRC16_TYPE {eCCITT, eKERMIT, eCCITT_FALSE, eIBM, eARC, eLHA, eSPI_FUJITSU, eBUYPASS, eVERIFONE, eUMTS, eCDMA2000, eCMS, eDDS_110, eDECT_R, eDECT_X, eDNP, eEN_13757, eGENIBUS, eEPC, eDARC, eI_CODE, eGSM, eLJ1200, eMAXIM, eMCRF4XX, eOPENSAFETY_A, eOPENSAFETY_B, ePROFIBUS, eIEC_61158_2, eRIELLO, eT10_DIF, eTELEDISK, eTMS37157, eUSB, eCRC_A, eMODBUS, eX_25, eCRC_B, eISO_HDLC, eIBM_SDLC, eXMODEM, eZMODEM, eACORN, eLTE}; CRC16(CRC16_TYPE type); CRC16(uint16_t polynomial, uint16_t init_remainder, uint16_t final_xor_value) :CRC<uint16_t>(polynomial, init_remainder, final_xor_value){} }; class CRC32 : public CRC<uint32_t> { public: enum CRC32_TYPE {eADCCP, ePKZIP, eCRC32, eAAL5, eDECT_B, eB_CRC32, eBZIP2, eAUTOSAR, eCRC32C, eCRC32D, eMPEG2, ePOSIX, eCKSUM, eCRC32Q, eJAMCRC, eXFER}; CRC32(CRC32_TYPE type); }; #endif // CRCCOMPUTE_H

#include "crcCompute.h" template <typename TYPE> void CRC<TYPE>::crcInit(void) { TYPE remainder; TYPE dividend; int bit; /* Perform binary long division, a bit at a time. */ for(dividend = 0; dividend < 256; dividend++) { /* Initialize the remainder. */ remainder = dividend << (m_width - 8); /* Shift and XOR with the polynomial. */ for(bit = 0; bit < 8; bit++) { /* Try to divide the current data bit. */ if(remainder & m_topbit) { remainder = (remainder << 1) ^ m_polynomial; } else { remainder = remainder << 1; } } /* Save the result in the table. */ crcTable[dividend] = remainder; } } CRC8::CRC8(CRC8_TYPE type) { switch (type) { case eCRC8: m_polynomial = 0x07; //http://reveng.sourceforge.net/crc-catalogue/all.htm m_initial_remainder = 0x00; m_final_xor_value = 0x00; break; case eAUTOSAR: m_polynomial = 0x2f; m_initial_remainder = 0xff; m_final_xor_value = 0xff; break; case eCDMA2000: m_polynomial = 0x9b; m_initial_remainder = 0xFF; m_final_xor_value = 0x00; break; case eDARC: m_polynomial = 0x39; m_initial_remainder = 0x00; m_final_xor_value = 0x00; break; case eDVB_S2: m_polynomial = 0xd5; m_initial_remainder = 0x00; m_final_xor_value = 0x00; break; case eEBU: case eAES: m_polynomial = 0x1d; m_initial_remainder = 0xFF; m_final_xor_value = 0x00; break; case eGSM_A: m_polynomial = 0x1d; m_initial_remainder = 0x00; m_final_xor_value = 0x00; break; case eGSM_B: m_polynomial = 0x49; m_initial_remainder = 0x00; m_final_xor_value = 0xFF; break; case eI_CODE: m_polynomial = 0x1d; m_initial_remainder = 0xFD; m_final_xor_value = 0x00; break; case eITU: m_polynomial = 0x07; m_initial_remainder = 0x00; m_final_xor_value = 0x55; break; case eLTE: m_polynomial = 0x9b; m_initial_remainder = 0x00; m_final_xor_value = 0x00; break; case eMAXIM: m_polynomial = 0x31; m_initial_remainder = 0x00; m_final_xor_value = 0x00; break; case eOPENSAFETY: m_polynomial = 0x2f; m_initial_remainder = 0x00; m_final_xor_value = 0x00; break; case eROHC: m_polynomial = 0x07; m_initial_remainder = 0xff; m_final_xor_value = 0x00; break; case eSAE_J1850: m_polynomial = 0x1d; m_initial_remainder = 0xff; m_final_xor_value = 0xff; break; case eWCDMA: m_polynomial = 0x9b; m_initial_remainder = 0x00; m_final_xor_value = 0x00; break; default: m_polynomial = 0x07; m_initial_remainder = 0x00; m_final_xor_value = 0x00; break; } crcInit(); } CRC16::CRC16(CRC16_TYPE type) { switch (type) { case eCCITT_FALSE: case eMCRF4XX: m_polynomial = 0x1021; m_initial_remainder = 0xFFFF; m_final_xor_value = 0x0000; break; case eIBM: case eARC: case eLHA: case eBUYPASS: case eVERIFONE: case eUMTS: m_polynomial = 0x8005; m_initial_remainder = 0x0000; m_final_xor_value = 0x0000; break; case eSPI_FUJITSU: m_polynomial = 0x1021; m_initial_remainder = 0x1d0f; m_final_xor_value = 0x0000; break; case eCCITT: case eKERMIT: case eXMODEM: case eZMODEM: case eACORN: case eLTE: m_polynomial = 0x1021; m_initial_remainder = 0x0000; m_final_xor_value = 0x0000; break; case eCDMA2000: m_polynomial = 0xc867; m_initial_remainder = 0xffff; m_final_xor_value = 0x0000; break; case eCMS: case eMODBUS: m_polynomial = 0x8005; m_initial_remainder = 0xffff; m_final_xor_value = 0x0000; break; case eDDS_110: m_polynomial = 0x8005; m_initial_remainder = 0x800d; m_final_xor_value = 0x0000; break; case eDECT_R: m_polynomial = 0x0589; m_initial_remainder = 0x0000; m_final_xor_value = 0x0001; break; case eDECT_X: m_polynomial = 0x0589; m_initial_remainder = 0x0000; m_final_xor_value = 0x0000; break; case eDNP: case eEN_13757: m_polynomial = 0x3d65; m_initial_remainder = 0x0000; m_final_xor_value = 0xffff; break; case eGENIBUS: case eEPC: case eDARC: case eI_CODE: case eX_25: case eCRC_B: case eISO_HDLC: case eIBM_SDLC: m_polynomial = 0x1021; m_initial_remainder = 0xffff; m_final_xor_value = 0xffff; break; case eGSM: m_polynomial = 0x1021; m_initial_remainder = 0x0000; m_final_xor_value = 0xffff; break; case eLJ1200: m_polynomial = 0x6f63; m_initial_remainder = 0x0000; m_final_xor_value = 0x0000; break; case eMAXIM: m_polynomial = 0x8005; m_initial_remainder = 0x0000; m_final_xor_value = 0xffff; break; case eOPENSAFETY_A: m_polynomial = 0x5935; m_initial_remainder = 0x0000; m_final_xor_value = 0x0000; break; case eOPENSAFETY_B: m_polynomial = 0x755b; m_initial_remainder = 0x0000; m_final_xor_value = 0x0000; break; case ePROFIBUS: case eIEC_61158_2: m_polynomial = 0x1dcf; m_initial_remainder = 0xffff; m_final_xor_value = 0xffff; break; case eRIELLO: m_polynomial = 0x1021; m_initial_remainder = 0xb2aa; m_final_xor_value = 0x0000; break; case eT10_DIF: m_polynomial = 0x8bb7; m_initial_remainder = 0x0000; m_final_xor_value = 0x0000; break; case eTELEDISK: m_polynomial = 0xa097; m_initial_remainder = 0x0000; m_final_xor_value = 0x0000; break; case eTMS37157: m_polynomial = 0x1021; m_initial_remainder = 0x89ec; m_final_xor_value = 0x0000; break; case eUSB: m_polynomial = 0x8005; m_initial_remainder = 0xffff; m_final_xor_value = 0xffff; break; case eCRC_A: m_polynomial = 0x1021; m_initial_remainder = 0xc6c6; m_final_xor_value = 0x0000; break; default: m_polynomial = 0x8005; m_initial_remainder = 0x0000; m_final_xor_value = 0x0000; break; } crcInit(); } CRC32::CRC32(CRC32_TYPE type) { switch (type) { case eADCCP: case ePKZIP: case eCRC32: case eBZIP2: case eAAL5: case eDECT_B: case eB_CRC32: m_polynomial = 0x04c11db7; m_initial_remainder = 0xFFFFFFFF; m_final_xor_value = 0xFFFFFFFF; break; case eAUTOSAR: m_polynomial = 0xf4acfb13; m_initial_remainder = 0xFFFFFFFF; m_final_xor_value = 0xFFFFFFFF; break; case eCRC32C: m_polynomial = 0x1edc6f41; m_initial_remainder = 0xFFFFFFFF; m_final_xor_value = 0xFFFFFFFF; break; case eCRC32D: m_polynomial = 0xa833982b; m_initial_remainder = 0xFFFFFFFF; m_final_xor_value = 0xFFFFFFFF; break; case eMPEG2: case eJAMCRC: m_polynomial = 0x04c11db7; m_initial_remainder = 0xFFFFFFFF; m_final_xor_value = 0x00000000; break; case ePOSIX: case eCKSUM: m_polynomial = 0x04c11db7; m_initial_remainder = 0x00000000; m_final_xor_value = 0xFFFFFFFF; break; case eCRC32Q: m_polynomial = 0x814141ab; m_initial_remainder = 0x00000000; m_final_xor_value = 0x00000000; break; case eXFER: m_polynomial = 0x000000af; m_initial_remainder = 0x00000000; m_final_xor_value = 0x00000000; break; default: m_polynomial = 0x04C11DB7; m_initial_remainder = 0xFFFFFFFF; m_final_xor_value = 0xFFFFFFFF; break; } crcInit(); }

#include <iostream> #include <stdio.h> #include "crcCompute.h" using namespace std; int main(int argc, char *argv[]) { CRC16 crc16(CRC16::eCCITT_FALSE); char data1[] = {'1', '2', '3', '4', '5', '6', '7', '8', '9'}; char data2[] = {'5', '6', '7', '8', '9'}; unsigned short c1, c2; c1 = crc16.crcCompute(data1, 9); c2 = crc16.crcCompute(data1, 4, true); c2 = crc16.crcCompute(data2, 5, false); printf("%04x\n", c1); printf("%04x\n", c2); return 0; }

參考文章：
http://blog.csdn.net/liyuanbhu/article/details/7882789

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