I am learning CRC theory following a document of CRC16. The document
mentioned that CRC of B(X) is the remainder of B(X)*2^^16 / G(X). In
my undersdanding, G(X) is a constant. The document gives C source code
of the algorithm calculating CRC 16 bit by bit, and also gives the
code of byte by byte. The programs are given at bottom of this post. I
have some questions about CRC, see here below.

1. From the values of crc_ta[0:14] in the algorithm byte by byte, I
figured out that G(X) in this sample is 0xEFDF. So that 1*2^^16/G(X)
would have the remainder 0x1021, and until 0xE*2^^16/G(X) would have
the remainder 0xE1CE.

But 0xF*2^^16/G(X) would have the remainder 0x210. However, the table
in the program shows that the remaider is 0xF1EF. Notice that 0xF1EF >
0xEFDF, so it cannot be the remainder of division by 0xEFDF. How come?

Further elements are even more surprising. CRC of 17 (0x11) should be
( 0x11*2^^16 mod 0xEFDF = 0x2252 = 8786 ). I found that 0x2252 is not
crc_ta[17], but crc_ta[19] in this table. When I continue the
calculation, I found that many elements have different location from
what I expected. So what's the correct way to interpret this table?

2. The document gives following standards:
CRC-16:		G(X) = X16 + X15 + X2 + 1
CRC-CCITT:	G(X) = X16 + X12 + X5 + 1

In my understanding,
CRC-16: 	G(X) = X16 + X15 + X2 + 1
	means using 11000000000000101 = 0x18005 as G(X), and
CRC-CCITT:	G(X) = X16 + X12 + X5 + 1
	means using 10001000000100001 0 0x11021 as G(X).
Is this correct?

And G(X)=0xEFDF in this sample means that G(X)=1110111111011111
= X15 + X14 + X13 + X11 + X10 + X9 + X8 + X7 + X6 + X4 + X3 + X2 + X1
+ 1

Is this correct?

Thanks.


Source code in the document:

- - - - - - -
 bit by bit
- - - - - - -

unsigned int cal_crc(unsigned char *ptr, unsigned char len) {
unsigned char i;
unsigned int crc=0;
while(len--!=0) {
 for(i=0x80; i!=0; i/=2) {
  if((crc&0x8000)!=0) {crc*=2; crc^=0x1021;}
   else crc*=2;
  if((*ptr&i)!=0) crc^=0x1021;
 }
 ptr++;
}
return(crc);
}


- - - - - - - -
 byte by byte
- - - - - - - -

unsigned int cal_crc(unsigned char *ptr, unsigned char len) {
unsigned int crc;
unsigned char da;
unsigned int crc_ta[256]={
0x0000, 0x1021, 0x2042, 0x3063, 0x4084, 0x50a5, 0x60c6, 0x70e7,
0x8108, 0x9129, 0xa14a, 0xb16b, 0xc18c, 0xd1ad, 0xe1ce, 0xf1ef,
0x1231, 0x0210, 0x3273, 0x2252, 0x52b5, 0x4294, 0x72f7, 0x62d6,
0x9339, 0x8318, 0xb37b, 0xa35a, 0xd3bd, 0xc39c, 0xf3ff, 0xe3de,
0x2462, 0x3443, 0x0420, 0x1401, 0x64e6, 0x74c7, 0x44a4, 0x5485,
0xa56a, 0xb54b, 0x8528, 0x9509, 0xe5ee, 0xf5cf, 0xc5ac, 0xd58d,
0x3653, 0x2672, 0x1611, 0x0630, 0x76d7, 0x66f6, 0x5695, 0x46b4,
0xb75b, 0xa77a, 0x9719, 0x8738, 0xf7df, 0xe7fe, 0xd79d, 0xc7bc,
0x48c4, 0x58e5, 0x6886, 0x78a7, 0x0840, 0x1861, 0x2802, 0x3823,
0xc9cc, 0xd9ed, 0xe98e, 0xf9af, 0x8948, 0x9969, 0xa90a, 0xb92b,
0x5af5, 0x4ad4, 0x7ab7, 0x6a96, 0x1a71, 0x0a50, 0x3a33, 0x2a12,
0xdbfd, 0xcbdc, 0xfbbf, 0xeb9e, 0x9b79, 0x8b58, 0xbb3b, 0xab1a,
0x6ca6, 0x7c87, 0x4ce4, 0x5cc5, 0x2c22, 0x3c03, 0x0c60, 0x1c41,
0xedae, 0xfd8f, 0xcdec, 0xddcd, 0xad2a, 0xbd0b, 0x8d68, 0x9d49,
0x7e97, 0x6eb6, 0x5ed5, 0x4ef4, 0x3e13, 0x2e32, 0x1e51, 0x0e70,
0xff9f, 0xefbe, 0xdfdd, 0xcffc, 0xbf1b, 0xaf3a, 0x9f59, 0x8f78,
0x9188, 0x81a9, 0xb1ca, 0xa1eb, 0xd10c, 0xc12d, 0xf14e, 0xe16f,
0x1080, 0x00a1, 0x30c2, 0x20e3, 0x5004, 0x4025, 0x7046, 0x6067,
0x83b9, 0x9398, 0xa3fb, 0xb3da, 0xc33d, 0xd31c, 0xe37f, 0xf35e,
0x02b1, 0x1290, 0x22f3, 0x32d2, 0x4235, 0x5214, 0x6277, 0x7256,
0xb5ea, 0xa5cb, 0x95a8, 0x8589, 0xf56e, 0xe54f, 0xd52c, 0xc50d,
0x34e2, 0x24c3, 0x14a0, 0x0481, 0x7466, 0x6447, 0x5424, 0x4405,
0xa7db, 0xb7fa, 0x8799, 0x97b8, 0xe75f, 0xf77e, 0xc71d, 0xd73c,
0x26d3, 0x36f2, 0x0691, 0x16b0, 0x6657, 0x7676, 0x4615, 0x5634,
0xd94c, 0xc96d, 0xf90e, 0xe92f, 0x99c8, 0x89e9, 0xb98a, 0xa9ab,
0x5844, 0x4865, 0x7806, 0x6827, 0x18c0, 0x08e1, 0x3882, 0x28a3,
0xcb7d, 0xdb5c, 0xeb3f, 0xfb1e, 0x8bf9, 0x9bd8, 0xabbb, 0xbb9a,
0x4a75, 0x5a54, 0x6a37, 0x7a16, 0x0af1, 0x1ad0, 0x2ab3, 0x3a92,
0xfd2e, 0xed0f, 0xdd6c, 0xcd4d, 0xbdaa, 0xad8b, 0x9de8, 0x8dc9,
0x7c26, 0x6c07, 0x5c64, 0x4c45, 0x3ca2, 0x2c83, 0x1ce0, 0x0cc1,
0xef1f, 0xff3e, 0xcf5d, 0xdf7c, 0xaf9b, 0xbfba, 0x8fd9, 0x9ff8,
0x6e17, 0x7e36, 0x4e55, 0x5e74, 0x2e93, 0x3eb2, 0x0ed1, 0x1ef0
};
crc=0;
while(len--!=0) {
 da=(uchar) (crc/256);
 crc<<=8;
 crc^=crc_ta[da^*ptr];
 ptr++;
}
return(crc);
}