CRC Polynomial Division - algorithm

Question

I am trying to use polynomial division to find the CRC check bits, but I am struggling with the last stage of the calculation.
I am believe the below conversions are correct:
Pattern = 1010
= x^3 + x
Dataword = 9 8 7
= 1001 1000 0111
= x^11 + x^8 + x^7 + x^2 + x + 1
And finally the polynomial long division I am attempted is:
x^8 + x^6 + x^5 + x^3 + x
_______________________________________
x^3 + x | x^11 + x^8 + x^7 + x^2 + x + 1
x^11 + x^9
....
x^4 + x^2 + x + 1
x^4 + x^2
= x + 1
My question is, is the remainder / answer x + 1 or do I take it a step further and remove the x leaving the remainder as just 1?
Thank you for your help!

Answer 1

We can check by mod 2 division (XOR) too, the following code shows a python implementation of CRC checking, we need to follow the steps listed below:
Convert CRC / data polynomials to corresponding binary equivalents.
if the CRC key (binary representation obtained from the polynomial) has k bits, we need to pad an additional k-1 bits with the data to check for errors. In the example given, the bits 011 should be appended to the data, not 0011, since k=4.
At the transmitter end,
The binary data is to be augmented first by adding k-1 zeros in the end of the data.
Use modulo-2 binary division to divide binary data by the CRC key and store remainder of division.
Append the remainder at the end of the data to form the encoded data and send the same
At the receiver end (Check if there are errors introduced in transmission)
Perform modulo-2 division again on the sent data with the CRC key and if the remainder is 0, then there are no errors.
Now let's implement the above:
def CRC_polynomial_to_bin_code(pol):
return bin(eval(pol.replace('^', '**').replace('x','2')))[2:]
def get_remainder(data_bin, gen_bin):
ng = len(gen_bin)
data_bin += '0'*(ng-1)
nd = len(data_bin)
divisor = gen_bin
i = 0
remainder = ''
print('\nmod 2 division steps:')
print('divisor dividend remainder')
while i < nd:
j = i + ng - len(remainder)
if j > nd:
remainder += data_bin[i:]
break
dividend = remainder + data_bin[i:j]
remainder = ''.join(['1' if dividend[k] != gen_bin[k] else '0' for k in range(ng)])
print('{:8s} {:8s} {:8s}'.format(divisor, dividend, remainder[1:]))
remainder = remainder.lstrip('0')
i = j
return remainder.zfill(ng-1)
gen_bin = CRC_polynomial_to_bin_code('x^3+x')
data_bin = CRC_polynomial_to_bin_code('x^11 + x^8 + x^7 + x^2 + x + 1')
print('transmitter end:\n\nCRC key: {}, data: {}'.format(gen_bin, data_bin))
r = get_remainder(data_bin, gen_bin)
data_crc = data_bin + r
print('\nencoded data: {}'.format(data_crc))
print('\nreceiver end:')
r = get_remainder(data_crc, gen_bin)
print('\nremainder {}'.format(r))
if eval(r) == 0:
print('data received at the receiver end has no errors')
# ---------------------------------
# transmitter end:
#
# CRC key: 1010, data: 100110000111
#
# mod 2 division steps:
# divisor dividend remainder
# 1010 1001 011
# 1010 1110 100
# 1010 1000 010
# 1010 1000 010
# 1010 1011 001
# 1010 1100 110
# 1010 1100 110
#
# encoded data: 100110000111110
# ---------------------------------
# receiver end:
#
# mod 2 division steps:
# divisor dividend remainder
# 1010 1001 011
# 1010 1110 100
# 1010 1000 010
# 1010 1000 010
# 1010 1011 001
# 1010 1111 101
# 1010 1010 000
#
# remainder 000
# data received at the receiver end has no errors
# ---------------------------------