Some transforms in cyclic redundancy check (CRC) computation

Some transforms in cyclic redundancy check (CRC) computation, such as the multiplication of two messages' polynomials, shift, complement, and different initial (defaulted) remainder of a message, are studied. The relationships of the CRCs between the transformed message and the original one are also addressed from the mathematical point of view. Some examples are given to validate such transforms. We find that these transforms are useful in CRC computation implementations of both software and hardware.