CRC的基本原理詳解
阿新 • • 發佈：2019-02-08
顯然，每次內迴圈的行為取決於暫存器首位。由於異或運算滿足交換率和結合律，以及與0異unsigned short do_crc(unsigned char *message, unsigned int len) { int i, j; unsigned short crc_reg; crc_reg = (message[0] << 8) + message[1]; for (i = 0; i < len; i++) { if (i < len - 2) for (j = 0; j <= 7; j++) { if ((short)crc_reg < 0) crc_reg = ((crc_reg << 1) + (message[i + 2] >> (7 - i))) ^ 0x1021; else crc_reg = (crc_reg << 1) + (message[i + 2] >> (7 - i)); } else for (j = 0; j <= 7; j++) { if ((short)crc_reg < 0) crc_reg = (crc_reg << 1) ^ 0x1021; else crc_reg <<= 1; } } return crc_reg; }
或無影響，訊息可以不移入暫存器，而在每次內迴圈的時候，暫存器首位再與對應的訊息位
異或。改進的程式碼如下：
unsigned short do_crc(unsigned char *message, unsigned int len) { int i, j; unsigned short crc_reg = 0; unsigned short current; for (i = 0; i < len; i++) { current = message[i] << 8; for (j = 0; j < 8; j++) { if ((short)(crc_reg ^ current) < 0) crc_reg = (crc_reg << 1) ^ 0x1021; else crc_reg <<= 1; current <<= 1; } } return crc_reg; }
以上的討論中，訊息的每個位元組都是先傳輸MSB，CRC16-CCITT標準卻是按照先傳輸LSB，訊息
右移進暫存器來計算的。只需將程式碼改成判斷暫存器的LSB，將0x1021按位顛倒後(0x8408)與
暫存器異或即可，如下所示：
unsigned short do_crc(unsigned char *message, unsigned int len) { int i, j; unsigned short crc_reg = 0; unsigned short current; for (i = 0; i < len; i++) { current = message[i]; for (j = 0; j < 8; j++) { if ((crc_reg ^ current) & 0x0001) crc_reg = (crc_reg >> 1) ^ 0x8408; else crc_reg >>= 1; current >>= 1; } } return crc_reg; }
該演算法使用了兩層迴圈，對訊息逐位進行處理，這樣效率是很低的。為了提高時間效率，通
常的思想是以空間換時間。考慮到內迴圈只與當前的訊息位元組和crc_reg的低位元組有關，對該
演算法做以下等效轉換：
unsigned short do_crc(unsigned char *message, unsigned int len)
{
int i, j;
unsigned short crc_reg = 0;
unsigned char index;
unsigned short to_xor;
for (i = 0; i < len; i++)
{
index = (crc_reg ^ message[i]) & 0xff;
to_xor = index;
for (j = 0; j < 8; j++)
{
if (to_xor & 0x0001)
to_xor = (to_xor >> 1) ^ 0x8408;
else
to_xor >>= 1;
}
crc_reg = (crc_reg >> 8) ^ to_xor;
}
return crc_reg;
}
現在內迴圈只與index相關了，可以事先以陣列形式生成一個表crc16_ccitt_table，使得
to_xor = crc16_ccitt_table[index]，於是可以簡化為：
unsigned short do_crc(unsigned char *message, unsigned int len)
{
unsigned short crc_reg = 0;
while (len--)
crc_reg = (crc_reg >> 8) ^ crc16_ccitt_table[(crc_reg ^ *message++) & 0xff];
return crc_reg;
}
crc16_ccitt_table通過以下程式碼生成：
int main()
{
unsigned char index = 0;
unsigned short to_xor;
int i;
printf("unsigned short crc16_ccitt_table[256] =n{");
while (1)
{
if (!(index % 8))
printf("n");
to_xor = index;
for (i = 0; i < 8; i++)
{
if (to_xor & 0x0001)
to_xor = (to_xor >> 1) ^ 0x8408;
else
to_xor >>= 1;
}
printf("0xx", to_xor);
if (index == 255)
{
printf("n");
break;
}
else
{
printf(", ");
index++;
}
}
printf("};");
return 0;
}生成的表如下：
unsigned short crc16_ccitt_table[256] =
{
0x0000, 0x1189, 0x2312, 0x329b, 0x4624, 0x57ad, 0x6536, 0x74bf,
0x8c48, 0x9dc1, 0xaf5a, 0xbed3, 0xca6c, 0xdbe5, 0xe97e, 0xf8f7,
0x1081, 0x0108, 0x3393, 0x221a, 0x56a5, 0x472c, 0x75b7, 0x643e,
0x9cc9, 0x8d40, 0xbfdb, 0xae52, 0xdaed, 0xcb64, 0xf9ff, 0xe876,
0x2102, 0x308b, 0x0210, 0x1399, 0x6726, 0x76af, 0x4434, 0x55bd,
0xad4a, 0xbcc3, 0x8e58, 0x9fd1, 0xeb6e, 0xfae7, 0xc87c, 0xd9f5,
0x3183, 0x200a, 0x1291, 0x0318, 0x77a7, 0x662e, 0x54b5, 0x453c,
0xbdcb, 0xac42, 0x9ed9, 0x8f50, 0xfbef, 0xea66, 0xd8fd, 0xc974,
0x4204, 0x538d, 0x6116, 0x709f, 0x0420, 0x15a9, 0x2732, 0x36bb,
0xce4c, 0xdfc5, 0xed5e, 0xfcd7, 0x8868, 0x99e1, 0xab7a, 0xbaf3,
0x5285, 0x430c, 0x7197, 0x601e, 0x14a1, 0x0528, 0x37b3, 0x263a,
0xdecd, 0xcf44, 0xfddf, 0xec56, 0x98e9, 0x8960, 0xbbfb, 0xaa72,
0x6306, 0x728f, 0x4014, 0x519d, 0x2522, 0x34ab, 0x0630, 0x17b9,
0xef4e, 0xfec7, 0xcc5c, 0xddd5, 0xa96a, 0xb8e3, 0x8a78, 0x9bf1,
0x7387, 0x620e, 0x5095, 0x411c, 0x35a3, 0x242a, 0x16b1, 0x0738,
0xffcf, 0xee46, 0xdcdd, 0xcd54, 0xb9eb, 0xa862, 0x9af9, 0x8b70,
0x8408, 0x9581, 0xa71a, 0xb693, 0xc22c, 0xd3a5, 0xe13e, 0xf0b7,
0x0840, 0x19c9, 0x2b52, 0x3adb, 0x4e64, 0x5fed, 0x6d76, 0x7cff,
0x9489, 0x8500, 0xb79b, 0xa612, 0xd2ad, 0xc324, 0xf1bf, 0xe036,
0x18c1, 0x0948, 0x3bd3, 0x2a5a, 0x5ee5, 0x4f6c, 0x7df7, 0x6c7e,
0xa50a, 0xb483, 0x8618, 0x9791, 0xe32e, 0xf2a7, 0xc03c, 0xd1b5,
0x2942, 0x38cb, 0x0a50, 0x1bd9, 0x6f66, 0x7eef, 0x4c74, 0x5dfd,
0xb58b, 0xa402, 0x9699, 0x8710, 0xf3af, 0xe226, 0xd0bd, 0xc134,
0x39c3, 0x284a, 0x1ad1, 0x0b58, 0x7fe7, 0x6e6e, 0x5cf5, 0x4d7c,
0xc60c, 0xd785, 0xe51e, 0xf497, 0x8028, 0x91a1, 0xa33a, 0xb2b3,
0x4a44, 0x5bcd, 0x6956, 0x78df, 0x0c60, 0x1de9, 0x2f72, 0x3efb,
0xd68d, 0xc704, 0xf59f, 0xe416, 0x90a9, 0x8120, 0xb3bb, 0xa232,
0x5ac5, 0x4b4c, 0x79d7, 0x685e, 0x1ce1, 0x0d68, 0x3ff3, 0x2e7a,
0xe70e, 0xf687, 0xc41c, 0xd595, 0xa12a, 0xb0a3, 0x8238, 0x93b1,
0x6b46, 0x7acf, 0x4854, 0x59dd, 0x2d62, 0x3ceb, 0x0e70, 0x1ff9,
0xf78f, 0xe606, 0xd49d, 0xc514, 0xb1ab, 0xa022, 0x92b9, 0x8330,
0x7bc7, 0x6a4e, 0x58d5, 0x495c, 0x3de3, 0x2c6a, 0x1ef1, 0x0f78
};
這樣對於訊息unsigned char message[len]，校驗碼為：
unsigned short code = do_crc(message, len);
並且按以下方式傳送出去：
message[len] = code & 0x00ff;
message[len + 1] = (code >> 8) & 0x00ff;
接收端對收到的len + 2位元組執行do_crc，如果沒有差錯發生則結果應為0。
在一些傳輸協議中，傳送端並不指出訊息長度，而是採用結束標誌，考慮以下幾種差錯：
1)在訊息之前，增加1個或多個0位元組；
2)訊息以1個或多個連續的0位元組開始，丟掉1個或多個0；
3)在訊息(包括校驗碼)之後，增加1個或多個0位元組；
4)訊息(包括校驗碼)以1個或多個連續的0位元組結尾，丟掉1個或多個0；
顯然，這幾種差錯都檢測不出來，其原因就是如果暫存器值為0，處理0訊息位元組(或位)，寄
存器值不變。為了解決前2個問題，只需暫存器的初值非0即可，對do_crc作以下改進：
unsigned short do_crc(unsigned short reg_init, unsigned char *message, unsigned int len)
{
unsigned short crc_reg = reg_init;
while (len--)
crc_reg = (crc_reg >> 8) ^ crc16_ccitt_table[(crc_reg ^ *message++) & 0xff];
return crc_reg;
}在CRC16-CCITT標準中reg_init = 0xffff，為了解決後2個問題，在CRC16-CCITT標準中將計
算出的校驗碼與0xffff進行異或，即：
unsigned short code = do_crc(0xffff, message, len);
code ^= 0xffff;
message[len] = code & 0x00ff;
message[len + 1] = (code >> 8) & 0x00ff;
顯然，現在接收端對收到的所有位元組執行do_crc，如果沒有差錯發生則結果應為某一常值
GOOD_CRC。其滿足以下關係：
unsigned char p[]= {0xff, 0xff};
GOOD_CRC = do_crc(0, p, 2);
其結果為GOOD_CRC = 0xf0b8。