The bits of data that are the message to be checked by the CRC can be represented by a polynomial of degree . As an example the one octet message is represented by the polynomial . In general, leading zero bits do not affect whereas trailing zero bits each add a factor of . The generating polynomial is chosen by very clever mathematicians and is essentially fixed. The polynomial is used to represent 32 successive ones.
The CRC is defined as the ones complement of a remainder obtained from the modulo two division of by the generating polynomial . I.e. the CRC is .
The multiplication by corresponds to leaving space at the end of the message for the generated CRC. The addition of (which is equivalent to inverting the first 32 bits of the message) precludes erroneous addition or deletion of leading zeros. The inversion of the CRC before sending it precludes erroneous addition of trailing zeros.
for some quotient and remainder . The CRC is the complement.
The transmitted message denoted is followed by the CRC.
At the receiver the operation is again repeated. This time however the value used is rather than also the message is 32 bits longer so is replaced by . We consider:
Factoring out we get:
Substituting (3) and thence (2) gives:
Taking the divisor inside we can now substitute (1).
From which the terms cancel in our modulo two arithmetic. We are, or course, only interested in the remainder of the division which is therefore.
Which is a known constant value (called the residual or residue) for all messages .