Why is the CRC32 generating polynomial 33 bits long?

Question

First off, if there's a better site to ask this question then please do migrate this or close it and let me know where to go.

Secondly, we're discussing CRC in one of my classes, and neither us nor the professor understand why CRC polynomials are one bit longer than the name (or resulting checksum) suggest. I've done some searching, but nothing seems to discuss why it's one bit longer.

Answer 1

A CRC is the remainder after dividing the message by the polynomial. By definition, the remainder has to be less than the length of the polynomial. Hence the CRC for a "33-bit" polynomial is 32 bits.

Note that the largest exponent of a "33-bit" polynomial is 32 (the lowest term has exponent zero), so the degree of the polynomial, as well as the length of the CRC is 32.