1．CRC校驗原理1.1 CRC的基本概念：
本文引用地址：http://www.104case.com/article/201612/330194.htmCRC是英文Cyclical Redundancy Check的縮寫，翻譯成中文通常稱作循環(huán)冗余校驗或簡稱為CRC校驗。它是數(shù)據(jù)傳輸領(lǐng)域中最常用的一種差錯校驗方法，其特點是傳輸數(shù)據(jù)和CRC校驗值的長度可以任意選定。在當今手機、計算機和數(shù)碼產(chǎn)品普及的信息數(shù)字化時代，CRC校驗無處不在。CRC分為多種標準，例如：CRC -12碼通常用來傳送6-bit字符串。CRC-16及CRC-CCITT碼則用是來傳送8-bit字符，其中CRC-16多為美國采用，而CRC-CCITT多為歐洲國家所采用。而CRC-32碼大都被應用在Point-to-Point的同步傳輸中。更多的CRC校驗標準類型參見附表：CRC校驗標準類型。
1.2 CRC校驗的基本原理：
在代數(shù)編碼理論中，一個數(shù)值可以表示為一個多項式。例如：一個十進制數(shù)值2892，可以用多項式表示為2x3+ 8x2+ 9x + 2(x=10)。同理，一個二進制數(shù)值1010101對應的多項式為x6+ x4+ x2+ 1(x=2)。
生成CRC碼的基本原理是：設(shè)被校驗的數(shù)據(jù)為K位，校驗碼為R位，碼字長度為N(=K+R)，則對于CRC碼集中的任一碼字，存在且僅存在一個R次多項式g(x)，使得
V(x)=A(x)g(x)=xRm(x) + r(x);
其中: m(x)為被校驗數(shù)據(jù)的K-1次多項式
r(x)為校驗碼的R-1次多項式
g(x)稱為生成多項式：g(x)=g0+ g1x1+ g2x2+ ... + g(R-1)x(R-1)+ gRxR
發(fā)送方通過指定的g(x)計算出CRC校驗碼，接收方則通過該g(x)來驗證收到的CRC校驗碼。綜上所述，一個完整的CRC校驗過程是：
發(fā)送方：根據(jù)要傳送的K位原始數(shù)據(jù)(二進制碼序列)，以標準指定的多項式計算出一個R位校驗碼(CRC碼)，附在原始數(shù)據(jù)后邊，構(gòu)成一個新的二進制碼序列共K + R位，然后發(fā)送出去。
接收方：將接收到的數(shù)據(jù)除以與發(fā)送方相同的多項式值，如果能夠除盡，則正確，否則證明出錯。還有另外一種處理，就是接收方用發(fā)送方相同的方法計算出接收到數(shù)據(jù)的CRC校驗值，再與發(fā)送方發(fā)來的校驗值比較，相同則正確，否則證明出錯。
2. CRC校驗碼的計算步驟：
例如：有一個要發(fā)送的7位二進制數(shù)1011001；對應的m(x)=x6+ x4+ x3+ 1。設(shè)CRC校驗碼取4位并設(shè)g(x)=x4+ x + 1，則該多項式對應的值是10011。根據(jù)CRC規(guī)則，為保證被除數(shù)夠除，首先需將要發(fā)送的數(shù)擴大2R即24倍(左移4位)，得到10110010000，對應的xRm(x)=x10+ x8+ x7+ x4。
CRC校驗碼的生成本質(zhì)其實就是采用模2除法取余數(shù)，該除法的簡捷計算就是將除數(shù)和被除數(shù)按位做異或(相同為0，不同為1。0^0=0; 0^1=1; 1^0=1; 1^1=0)運算。需要注意的是，進行異或運算時除數(shù)必須和被除數(shù)最高有效位對齊。下面是將10110010000除以11001手工計算的演示，得到余數(shù)為1010，該值即是數(shù)據(jù)1011001的CRC校驗值。
10110010000將1011001左移4位
10011多項式值與被除數(shù)最高有效位對齊
=00101010000第1次異或結(jié)果
10011多項式值與被除數(shù)最高有效位對齊
=00001100000第2次異或結(jié)果
10011多項式值與被除數(shù)最高有效位對齊
=00000101100第3次異或結(jié)果
10011多項式值與被除數(shù)最高有效位對齊
=00000001010第4次異或結(jié)果
為了簡便計算機程序求解CRC，在實際應用中通常把多項式值的最高位舍掉，并且將參加計算的數(shù)據(jù)高低位顛倒后再計算。前面的演算數(shù)據(jù)顛倒后的運算情況如下：
00001001101
11001
00001010100
11001
00000110000
11001
00110100000
11001
01010000000
提請注意：只有真正讀懂以上手工演算的步驟和規(guī)律，才能理解下節(jié)內(nèi)容。
3. CRC校驗的程序設(shè)計
下面以最常用的CRC-16-IBM校驗標準為例來說明CRC校驗碼的生成過程。 因為CRC-16碼由兩個字節(jié)構(gòu)成，所以首先要準備一個16位的CRC寄存器，并將每一位都置1。具體的計算步驟為：
① 設(shè)置CRC寄存器，并給其賦值FFFFH。
② 將被校驗數(shù)據(jù)的第一個字節(jié)(8Bit)與CRC寄存器的低8位進行異或，結(jié)果存CRC寄存器。
③CRC寄存器的值向右移一位，最高位(MSB)補零，檢查移出的最低位(LSB)是否為1。
④ 如果LSB為0，重復第三步；若LSB為1，CRC寄存器與CRC16多項式值A(chǔ)001H相異或。
⑤ 重復第③與第④步，直到該字節(jié)的8次移位全部完成。
⑥ 如果被校驗數(shù)據(jù)有多個字節(jié)，則重復第②至第⑤步直到所有數(shù)據(jù)全部處理完。
⑦ 最終CRC寄存器的內(nèi)容即為CRC值。
3.1用查表法求CRC函數(shù)的程序
查表法省去了將被校驗數(shù)據(jù)移位并與多項式異或計算的步驟，因此比計算法更快捷，但程序編碼量偏大。
3.1.1用VC編寫的程序
unsigned short CRC16(puchMsg, usDataLen)
unsigned char * puchMsg ;
unsigned short usDataLen ;
{
unsigned char uchCRCHi = 0xFF ;
unsigned char uchCRCLo = 0xFF ;
unsigned uIndex ;
while (usDataLen--)
{
uIndex = uchCRCLo ^ * puchMsg++ ;
uchCRCLo = uchCRCHi ^ auchCRCLo [uIndex] ;
uchCRCHi = auchCRCHi [uIndex] ;
}
return (uchCRCLo << 8 | uchCRCHi) ;
}
static char auchCRCHi[] = {
0x00, 0xC0, 0xC1, 0x01, 0xC3, 0x03, 0x02, 0xC2, 0xC6, 0x06,
0x07, 0xC7, 0x05, 0xC5, 0xC4, 0x04, 0xCC, 0x0C, 0x0D, 0xCD,
0x0F, 0xCF, 0xCE, 0x0E, 0x0A, 0xCA, 0xCB, 0x0B, 0xC9, 0x09,
0x08, 0xC8, 0xD8, 0x18, 0x19, 0xD9, 0x1B, 0xDB, 0xDA, 0x1A,
0x1E, 0xDE, 0xDF, 0x1F, 0xDD, 0x1D, 0x1C, 0xDC, 0x14, 0xD4,
0xD5, 0x15, 0xD7, 0x17, 0x16, 0xD6, 0xD2, 0x12, 0x13, 0xD3,
0x11, 0xD1, 0xD0, 0x10, 0xF0, 0x30, 0x31, 0xF1, 0x33, 0xF3,
0xF2, 0x32, 0x36, 0xF6, 0xF7, 0x37, 0xF5, 0x35, 0x34, 0xF4,
0x3C, 0xFC, 0xFD, 0x3D, 0xFF, 0x3F, 0x3E, 0xFE, 0xFA, 0x3A,
0x3B, 0xFB, 0x39, 0xF9, 0xF8, 0x38, 0x28, 0xE8, 0xE9, 0x29,
0xEB, 0x2B, 0x2A, 0xEA, 0xEE, 0x2E, 0x2F, 0xEF, 0x2D, 0xED,
0xEC, 0x2C, 0xE4, 0x24, 0x25, 0xE5, 0x27, 0xE7, 0xE6, 0x26,
0x22, 0xE2, 0xE3, 0x23, 0xE1, 0x21, 0x20, 0xE0, 0xA0, 0x60,
0x61, 0xA1, 0x63, 0xA3, 0xA2, 0x62, 0x66, 0xA6, 0xA7, 0x67,
0xA5, 0x65, 0x64, 0xA4, 0x6C, 0xAC, 0xAD, 0x6D, 0xAF, 0x6F,
0x6E, 0xAE, 0xAA, 0x6A, 0x6B, 0xAB, 0x69, 0xA9, 0xA8, 0x68,
0x78, 0xB8, 0xB9, 0x79, 0xBB, 0x7B, 0x7A, 0xBA, 0xBE, 0x7E,
0x7F, 0xBF, 0x7D, 0xBD, 0xBC, 0x7C, 0xB4, 0x74, 0x75, 0xB5,
0x77, 0xB7, 0xB6, 0x76, 0x72, 0xB2, 0xB3, 0x73, 0xB1, 0x71,
0x70, 0xB0, 0x50, 0x90, 0x91, 0x51, 0x93, 0x53, 0x52, 0x92,
0x96, 0x56, 0x57, 0x97, 0x55, 0x95, 0x94, 0x54, 0x9C, 0x5C,
0x5D, 0x9D, 0x5F, 0x9F, 0x9E, 0x5E, 0x5A, 0x9A, 0x9B, 0x5B,
0x99, 0x59, 0x58, 0x98, 0x88, 0x48, 0x49, 0x89, 0x4B, 0x8B,
0x8A, 0x4A, 0x4E, 0x8E, 0x8F, 0x4F, 0x8D, 0x4D, 0x4C, 0x8C,
0x44, 0x84, 0x85, 0x45, 0x87, 0x47, 0x46, 0x86, 0x82, 0x42,
0x43, 0x83, 0x41, 0x81, 0x80, 0x40 } ;
static unsigned char auchCRCLo[] = {
0x00, 0xC1, 0x81, 0x40, 0x01, 0xC0, 0x80, 0x41, 0x01, 0xC0,
0x80, 0x41, 0x00, 0xC1, 0x81, 0x40, 0x01, 0xC0, 0x80, 0x41,
0x00, 0xC1, 0x81, 0x40, 0x00, 0xC1, 0x81, 0x40, 0x01, 0xC0,
0x80, 0x41, 0x01, 0xC0, 0x80, 0x41, 0x00, 0xC1, 0x81, 0x40,
0x00, 0xC1, 0x81, 0x40, 0x01, 0xC0, 0x80, 0x41, 0x00, 0xC1,
0x81, 0x40, 0x01, 0xC0, 0x80, 0x41, 0x01, 0xC0, 0x80, 0x41,
0x00, 0xC1, 0x81, 0x40, 0x01, 0xC0, 0x80, 0x41, 0x00, 0xC1,
0x81, 0x40, 0x00, 0xC1, 0x81, 0x40, 0x01, 0xC0, 0x80, 0x41,
0x00, 0xC1, 0x81, 0x40, 0x01, 0xC0, 0x80, 0x41, 0x01, 0xC0,
0x80, 0x41, 0x00, 0xC1, 0x81, 0x40, 0x00, 0xC1, 0x81, 0x40,
0x01, 0xC0, 0x80, 0x41, 0x01, 0xC0, 0x80, 0x41, 0x00, 0xC1,
0x81, 0x40, 0x01, 0xC0, 0x80, 0x41, 0x00, 0xC1, 0x81, 0x40,
0x00, 0xC1, 0x81, 0x40, 0x01, 0xC0, 0x80, 0x41, 0x01, 0xC0,
0x80, 0x41, 0x00, 0xC1, 0x81, 0x40, 0x00, 0xC1, 0x81, 0x40,
0x01, 0xC0, 0x80, 0x41, 0x00, 0xC1, 0x81, 0x40, 0x01, 0xC0,
0x80, 0x41, 0x01, 0xC0, 0x80, 0x41, 0x00, 0xC1, 0x81, 0x40,
0x00, 0xC1, 0x81, 0x40, 0x01, 0xC0, 0x80, 0x41, 0x01, 0xC0,
0x80, 0x41, 0x00, 0xC1, 0x81, 0x40, 0x01, 0xC0, 0x80, 0x41,
0x00, 0xC1, 0x81, 0x40, 0x00, 0xC1, 0x81, 0x40, 0x01, 0xC0,
0x80, 0x41, 0x00, 0xC1, 0x81, 0x40, 0x01, 0xC0, 0x80, 0x41,
0x01, 0xC0, 0x80, 0x41, 0x00, 0xC1, 0x81, 0x40, 0x01, 0xC0,
0x80, 0x41, 0x00, 0xC1, 0x81, 0x40, 0x00, 0xC1, 0x81, 0x40,
0x01, 0xC0, 0x80, 0x41, 0x01, 0xC0, 0x80, 0x41, 0x00, 0xC1,
0x81, 0x40, 0x00, 0xC1, 0x81, 0x40, 0x01, 0xC0, 0x80, 0x41,
0x00, 0xC1, 0x81, 0x40, 0x01, 0xC0, 0x80, 0x41, 0x01, 0xC0,
0x80, 0x41, 0x00, 0xC1, 0x81, 0x40 } ;
3.1.2用VB編寫的程序
Private Function CRC16(data() As Byte) As String
Dim CRC16Hi As Byte
Dim CRC16Lo As Byte
CRC16Hi = &HFF
CRC16Lo = &HFF
Dim i As Integer
Dim iIndex As Long
For i = 0 To UBound(data)
iIndex = CRC16Lo Xor data(i)
CRC16Lo = CRC16Hi Xor GetCRCLo(iIndex)’低位處理
CRC16Hi = GetCRCHi(iIndex)’高位處理
Next i
Dim ReturnData(1) As Byte
ReturnData(0) = CRC16Hi’CRC高位
ReturnData(1) = CRC16Lo’CRC低位
CRC16 = ReturnData
End Function
Function GetCRCHi(Ind As Long) As Byte’CRC高位字節(jié)值表
GetCRCHi = Choose(Ind + 1,_
&H00, &HC0, &HC1, &H01, &HC3, &H03, &H02, &HC2, &HC6, &H06,_
&H07, &HC7, &H05, &HC5, &HC4, &H04, &HCC, &H0C, &H0D, &HCD,_
&H0F, &HCF, &HCE, &H0E, &H0A, &HCA, &HCB, &H0B, &HC9, &H09,_
&H08, &HC8, &HD8, &H18, &H19, &HD9, &H1B, &HDB, &HDA, &H1A,_
&H1E, &HDE, &HDF, &H1F, &HDD, &H1D, &H1C, &HDC, &H14, &HD4,_
&HD5, &H15, &HD7, &H17, &H16, &HD6, &HD2, &H12, &H13, &HD3,_
&H11, &HD1, &HD0, &H10, &HF0, &H30, &H31, &HF1, &H33, &HF3,_
&HF2, &H32, &H36, &HF6, &HF7, &H37, &HF5, &H35, &H34, &HF4,_
&H3C, &HFC, &HFD, &H3D, &HFF, &H3F, &H3E, &HFE, &HFA, &H3A,_
&H3B, &HFB, &H39, &HF9, &HF8, &H38, &H28, &HE8, &HE9, &H29,_
&HEB, &H2B, &H2A, &HEA, &HEE, &H2E, &H2F, &HEF, &H2D, &HED,_
&HEC, &H2C, &HE4, &H24, &H25, &HE5, &H27, &HE7, &HE6, &H26,_
&H22, &HE2, &HE3, &H23, &HE1, &H21, &H20, &HE0, &HA0, &H60,_
&H61, &HA1, &H63, &HA3, &HA2, &H62, &H66, &HA6, &HA7, &H67,_
&HA5, &H65, &H64, &HA4, &H6C, &HAC, &HAD, &H6D, &HAF, &H6F,_
&H6E, &HAE, &HAA, &H6A, &H6B, &HAB, &H69, &HA9, &HA8, &H68,_
&H78, &HB8, &HB9, &H79, &HBB, &H7B, &H7A, &HBA, &HBE, &H7E,_
&H7F, &HBF, &H7D, &HBD, &HBC, &H7C, &HB4, &H74, &H75, &HB5,_
&H77, &HB7, &HB6, &H76, &H72, &HB2, &HB3, &H73, &HB1, &H71,_
&H70, &HB0, &H50, &H90, &H91, &H51, &H93, &H53, &H52, &H92,_
&H96, &H56, &H57, &H97, &H55, &H95, &H94, &H54, &H9C, &H5C,_
&H5D, &H9D, &H5F, &H9F, &H9E, &H5E, &H5A, &H9A, &H9B, &H5B,_
&H99, &H59, &H58, &H98, &H88, &H48, &H49, &H89, &H4B, &H8B,_
&H8A, &H4A, &H4E, &H8E, &H8F, &H4F, &H8D, &H4D, &H4C, &H8C,_
&H44, &H84, &H85, &H45, &H87, &H47, &H46, &H86, &H82, &H42,_
&H43, &H83, &H41, &H81, &H80, &H40)
End Function
Function GetCRCLo(Ind As Long) As Byte’CRC低位字節(jié)值表
GetCRCLo = Choose(Ind + 1,_
&H00, &HC1, &H81, &H40, &H01, &HC0, &H80, &H41, &H01, &HC0,_
&H80, &H41, &H00, &HC1, &H81, &H40, &H01, &HC0, &H80, &H41,_
&H00, &HC1, &H81, &H40, &H00, &HC1, &H81, &H40, &H01, &HC0,_
&H80, &H41, &H01, &HC0, &H80, &H41, &H00, &HC1, &H81, &H40,_
&H00, &HC1, &H81, &H40, &H01, &HC0, &H80, &H41, &H00, &HC1,_
&H81, &H40, &H01, &HC0, &H80, &H41, &H01, &HC0, &H80, &H41,_
&H00, &HC1, &H81, &H40, &H01, &HC0, &H80, &H41, &H00, &HC1,_
&H81, &H40, &H00, &HC1, &H81, &H40, &H01, &HC0, &H80, &H41,_
&H00, &HC1, &H81, &H40, &H01, &HC0, &H80, &H41, &H01, &HC0,_
&H80, &H41, &H00, &HC1, &H81, &H40, &H00, &HC1, &H81, &H40,_
&H01, &HC0, &H80, &H41, &H01, &HC0, &H80, &H41, &H00, &HC1,_
&H81, &H40, &H01, &HC0, &H80, &H41, &H00, &HC1, &H81, &H40,_
&H00, &HC1, &H81, &H40, &H01, &HC0, &H80, &H41, &H01, &HC0,_
&H80, &H41, &H00, &HC1, &H81, &H40, &H00, &HC1, &H81, &H40,_
&H01, &HC0, &H80, &H41, &H00, &HC1, &H81, &H40, &H01, &HC0,_
&H80, &H41, &H01, &HC0, &H80, &H41, &H00, &HC1, &H81, &H40,_
&H00, &HC1, &H81, &H40, &H01, &HC0, &H80, &H41, &H01, &HC0,_
&H80, &H41, &H00, &HC1, &H81, &H40, &H01, &HC0, &H80, &H41,_
&H00, &HC1, &H81, &H40, &H00, &HC1, &H81, &H40, &H01, &HC0,_
&H80, &H41, &H00, &HC1, &H81, &H40, &H01, &HC0, &H80, &H41,_
&H01, &HC0, &H80, &H41, &H00, &HC1, &H81, &H40, &H01, &HC0,_
&H80, &H41, &H00, &HC1, &H81, &H40, &H00, &HC1, &H81, &H40,_
&H01, &HC0, &H80, &H41, &H01, &HC0, &H80, &H41, &H00, &HC1,_
&H81, &H40, &H00, &HC1, &H81, &H40, &H01, &HC0, &H80, &H41,_
&H00, &HC1, &H81, &H40, &H01, &HC0, &H80, &H41, &H01, &HC0,_
&H80, &H41, &H00, &HC1, &H81, &H40)
End Function
3.2用計算法求CRC函數(shù)的程序
計算法的優(yōu)點是程序編碼量小，雖然比查表法速度偏慢，但當被校驗的數(shù)據(jù)量不大時并不明顯。
3.2.1用VC編寫的程序
unsigned short CRC16(unsigned char buf[], int n)
{
unsigned short crc = 0xffff ;
for(i = 0; i < n; i++)
{
crc ^ = buf[i] ;
for(b = 0; b < 8; b++)
{
bool f = crc & 1 ;
crc >> = 1 ;
if(f)
crc ^ = 0xa001 ;
}
}
return(crc) ;
}
3.2.2用VB編寫的程序
Function CRC16(Data() As Byte, No As Integer, CRC16Lo As Byte, CRC16Hi As Byte) As String
Dim CL As Byte, CH As Byte, SaveLo As Byte, SaveHi As Byte
CRC16Hi = &HFF為16位CRC校驗寄存器賦初始值FFFF
CRC16Lo = &HFF
CH = &HA0為16位CRC校驗多項式賦初始值A(chǔ)001
CL = &H1
For i = 1 To No
CRC16Lo = CRC16Lo Xor Data(i - 1)每一個數(shù)據(jù)與CRC寄存器進行異或
For j = 1 To 8
SaveHi = CRC16Hi
SaveLo = CRC16Lo
CRC16Hi = CRC16Hi 2高位右移一位
CRC16Lo = CRC16Lo 2低位右移一位
If ((SaveHi And &H1) = &H1) Then如果高位字節(jié)最右一位為1
CRC16Lo = CRC16Lo Or &H80則低位字節(jié)最左一位補1，否則自動補0
End If
If ((SaveLo And &H1) = &H1) Then如低位字節(jié)最右一位為1，則與多項式值異或
CRC16Hi = CRC16Hi Xor CH
CRC16Lo = CRC16Lo Xor CL
End If
Next j
Next i
End Function
參考文獻：
①GB/T 19582.1-2008《基于Modbus協(xié)議的工業(yè)自動化網(wǎng)絡(luò)規(guī)范》
②《Modbus協(xié)議中文版》可在互聯(lián)網(wǎng)上下載
附表：CRC校驗標準類型
CRC校驗名 | 多項式(標準引用名) | 對應的16進制值或反值注：已去掉最高位的1 |
CRC-1 | x + 1 (most hardware; also known as parity bit) | 01或01 (01) |
CRC-4-ITU | x4+x+ 1 (ITU G.704, p. 12) | 03或0C (09) |
CRC-5-ITU | x5+x4+x2+ 1 (ITU G.704, p. 9) | 15或15 (1A) |
CRC-5-USB | x5+x2+ 1 (USB token packets) | 05或14 (12) |
CRC-6-ITU | x6+x+ 1 (ITU G.704, p. 3) | 03或30 (21) |
CRC-7 | x7+x3+ 1 (telecom systems, MMC,SD) | 09或48 (44) |
CRC-8-ATM | x8+x2+x+ 1 (ATM HEC) | 07或E0 (83) |
CRC-8-CCITT | x8+x7+x3+x2+ 1 (1-Wire bus) | 8D或B1 (C6) |
CRC-8-Dallas/Maxim | x8+x5+x4+ 1 (1-Wire bus) | 31或8C (98) |
CRC-8 | x8+x7+x6+x4+x2+ 1 | D5或AB (EA) |
CRC-8-SAE J1850 | x8+x4+x3+x2+ 1 | 1D或B8 (8E) |
CRC-10 | x10+x9+x5+x4+x+ 1 | 233或331 (319) |
CRC-11 | x11+x9+x8+x7+x+ 1 (FlexRay) | 385或50E (5C2) |
CRC-12 | x12+x11+x3+x2+x+ 1 (telecom systems ) | 80F或F01 (C07) |
CRC-15-CAN | x15+x14+x10+x8+x7+x4+x3+ 1 | 4599或4CD1 (62CC) |
CRC-16-Fletcher | Not a CRC; see Fletchers checksum | Used in Adler-32 A & B CRCs |
CRC-16-CCITT | x16+x12+x5+ 1 (CRC-CCITT, G.hn PHY headers, 802.15.4, X.25, V.41, CDMA, Bluetooth, XMODEM, HDLC,PPP, IrDA, BACnet, MMC,SD) | 1021或8408 (8810) |
CRC-16-DNP | x16+x13+x12+x11+x10+x8+x6+x5+x2+ 1 (DNP, IEC 870, M-Bus) | 3D65或A6BC (9EB2) |
CRC-16-IBM | x16+x15+x2+ 1 (CRC-16, SDLC, USB, many others) | 8005或A001 (C002) |
CRC-24-Radix-64 | x24+x23+x18+x17+x14+x11+x10+x7+x6+x5+x4+x3+x+ 1 (FlexRay) | 864CFB或DF3261 (C3267D) |
CRC-30 | x30+x29+x21+x20+x15+x13+x12+x11+x8+x7+x6+x2+x+ 1 (CDMA) | 2030B9C7或38E74301 (30185CE3) |
CRC-32-Adler | Not a CRC; see Adler-32 | See Adler-32 |
CRC-32-IEEE 802.3 | x32+x26+x23+x22+x16+x12+x11+x10+x8+x7+x5+x4+x2+x+ 1 (V.42, MPEG-2, PNG [10], POSIX cksum) | 04C11DB7或EDB88320 (82608EDB) |
CRC-32C (Castagnoli) | x32+ x28+ x27+ x26+ x25+ x23+ x22+ x20+ x19+ x18+ x14+ x13+ x11+ x10+ x9+ x8+ x6+ 1 (G.hn payload) | 1EDC6F41或82F63B78 (8F6E37A0) |
CRC-32K (Koopman) | x32+x30+x29+x28+x26+x20+x19+x17+x16+x15+x11+x10+x7+x6+x4+x2+x+ 1 | 741B8CD7或EB31D82E (BA0DC66B) |
CRC-64-ISO | x64+x4+x3+ x + 1 (HDLC-ISO 3309) | 000000000000001B或D800000000000000 (800000000000000D) |
CRC-64-ECMA-182 | x64+ x62+ x57+ x55+ x54+ x53+ x52+ x47+ x46+ x45+ x40+ x39+ x38+ x37+ x35+ x33+ x32+ x31+ x29+ x27+ x24+ x23+ x22+ x21+ x19+ x17+ x13+ x12+ x10+ x9+ x7+ x4+ x + 1 (As described in ECMA-182 p.51) | 42F0E1EBA9EA3693或C96C5795D7870F42 (A17870F5D4F51B49) |
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