

Example:
(x^{2} + x + 1) × (x + 1) = (x^{3} + x^{2} + x) + (x^{2} + x + 1) = x^{3} + 2x^{2} + 2x + 1 = x^{3} + 1 (mod 2) 


Example:
Generator polynomial: 1001 has the factor 11 

Message: 10011 (has an odd # 1 bits) CRC polynomial: 1001 (1001 has 11 as factor) Compute CRC code:  1001 / 10011000 1001  01000 1001  001 < Remainder CRC protected message: 10011001 (has an even number 1 bits !!!) 


This makes sense, because:


Error polynomial representing kconsecutive errors: 1111..1 <> k bits Generator polynomial = 1.........1 <> n+1 bits The division:  1.........1 / 1111..1 <> <> n+1 bits k bits will have a nonzero remainder if: k ≤ n 

Name  Generator polynomial 

CRCCCITT  10001000000100001 
CRC16  11000000000000101 
CRC32  100000100110000010001110110110111 