Question

Could someone please show me the steps in calculating this CRC code?
Generator polynomial G(x)=10
data:110
message with crc appended: 110

When the data is divided by G(x) then the remainder is 0. So do that mean the CRC is 0 or nothing or is that even possible?

Answer 1

The division is done almost the same manner as in school:

10|110
   10  (xor)
------
   010
    10  (xor)
-------
     0 <- remainder

The only difference is that one does have to only check the msb of each row
for the divisor to fit into the partial remainder:
The appended message is 1100

To check the crc one can perform the division again:

10|1100
   10    (xor)
-------
    100
    10   (xor)
--------
     00
     10  (no xor)
-------
      0  <-- remainder is 0, so the message is valid

There’s something suspicious though: generally crc polynomials have a term +1 and if the highest order term is x^2 (e.g.) then the complete polynomial has 3 terms, out of which only 2 are traditionally explicitly declared.
Then it would only make sense that your polynomial actually was x^2 + 1 and that would be represented in reverse order as 1 + 0*x (+ 1* x^2 term omitted) and then the remainder would be two bits

101|110
    101
--------
     11  <-- final remainder; concatenated message = 11011

checking message integrity:

101|11011
    101
 --------
     1111
     101
  --------
      101
      101
  --------
       00 <-- checked!