Let's assume that I have 512-byte packets plus a 16-bit CRC at the end. I would like to determine what the CRC parameters are.
It's a Fujitsu chip, where I'm writing the the flash with a programmer, the programmer calculates the CRC for me, and I read out the CRC with an oscilloscope. I have the ability to check every possible combination.
My test messages are 512 zeros except for one byte that I set to the values 0 to 17 in decimal. The one byte is one of the first four or last two in the packet. Here are the resulting CRCs in hexadecimal, where the rows are the value of the byte, and the columns are which byte is set:
00 01 02 03 510 511
00 00 00 00 00 00 00
01 0x8108 0x0100 0x3020 0xC6B0 0xF1F0 0x8108
02 0x8318 0x0200 0x6040 0x0C68 0x62E8 0x8318
03 0x0210 0x0300 0x5060 0xCAD8 0x9318 0x0210
04 0x8738 0x0400 0xC080 0x18D0 0xC5D0 0x8738
05 0x0630 0x0500 0xF0A0 0xDE60 0x3420 0x0630
06 0x0420 0x0600 0xA0C0 0x14B8 0xA738 0x0420
07 0x8528 0x0700 0x90E0 0xD208 0x56C8 0x8528
08 0x8F78 0x0800 0x0008 0x31A0 0x0AA8 0x8F78
09 0x0E70 0x0900 0x3028 0xF710 0xFB58 0x0E70
10 0x0C60 0x0A00 0x6048 0x3DC8 0x6840 0x0C60
11 0x8D68 0x0B00 0x5068 0xFB78 0x99B0 0x8D68
12 0x0840 0x0C00 0xC088 0x2970 0xCF78 0x0840
13 0x8948 0x0D00 0xF0A8 0xEFC0 0x3E88 0x8948
14 0x8B58 0x0E00 0xA0C8 0x2518 0xAD90 0x8B58
15 0x0A50 0x0F00 0x90E8 0xE3A8 0x5C60 0x0A50
16 0x9FF8 0x1000 0x0010 0x6340 0x1550 0x9FF8
17 0x1EF0 0x1100 0x3030 0xA5F0 0xE4A0 0x1EF0
As you can see the first and last bytes give the same value. I tried several variations of CRC-16, but without much luck. The closet one was CRC-16 with polynomial 0x1021 and initial value 0.
The fact that every single CRC ends in 0 or 8 strongly suggests that it is not a 16-bit CRC, but rather a 13-bit CRC. Indeed, all of the sequences check against a 13-bit CRC with polynomial 0x1021 not reflected, initial value zero, and final exclusive-or zero.
We can't be sure about the initial value and final exclusive-or unless you can provide at least one packet with a length other than 512. With only examples of a single length, there are 8,191 other combinations of initial values and final exclusive-ors that would produce the exact same CRCs.