are CRC generator bits same for all?

Question

In CRC if sender has to send data, It will divide the data bits with g(x) which will give some remainder bits. Those remainder bits are appended to the data bits. Now when this code word is sent to the receiver, the receiver will divide the code word with same g(x) giving some remainder. If this remainder is zero that means data is correct.

Now, If all the systems can communicate with each other does that mean every system in the world have the same g(x). because sender and receiver must have common g(x).

(please answer only if have correct knowledge with some valid proof)

Answer 1

It depends on the protocol. CRC by itself can work for different polynomials, the protocols that use it define the g(x) polynomial to use.

There is a list of examples on https://en.wikipedia.org/wiki/Cyclic_redundancy_check#Standards_and_common_use

This is not an issue since systems cannot communicate using different protocols on the sending and receiving end, obviously. Potentially, a protocol could also use a variable polynomial, somehow decided at the start of the communication, but I can't see why that would be useful

Answer 2

That's a big no. Furthermore, there are several other variations besides just g(x).

If someone tells you to compute a CRC, you have many questions to ask. You need to ask: what is g(x)? What is the initial value of the CRC? Is it exclusive-or'ed with a constant at the end? In what order are bits fed into the CRC, least-significant or most-significant first? In what order are the CRC bits put into the message? In what order are the CRC bytes put into the message?

Here is a catalog of CRCs, with (today) 107 entries. It is not all of the CRCs in use. There are many lengths of CRCs (the degree of g(x)), many polynomials (g(x)), and among those, many choices for the bit orderings, initial value, and final exclusive-or.

The person telling you to compute the CRC might not even know! "Isn't there just one?" they might naively ask (as you have). You then have to either find a definition of the CRC for the protocol you are using, and be able to interpret that, or find examples of correct CRC calculations for your message, and attempt to deduce the parameters.

By the way, not all CRCs will give zero when computing the CRC of a message with the CRC appended. Depending on the initial value and final exclusive-or, you will get a constant for all correct messages, but that constant is not necessarily zero. Even then, you will get a constant only if you compute the bits from the CRC in the proper order. It's actually easier and faster to ignore that property and compute the CRC on just the received message and then simply compare that to the CRC received with the message.