Find a message given a CRC and generating polynomial

[JJBladester]

I am working on a circuit that inputs a 31-bit pseudo-random binary string into a CCIT CRC-16 block which generates a 16-bit CRC output.

I know that M(x)/G(x) = Q(x) + R(x) and the transmitted code will be R(x) appended to M(x).

When I simulated the circuit, I got a CRC of 1 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0. From the help file of the software (VisSim/Comm), I know that the generating polynomial is G(x) = 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1.

How do I determine the 31-bit input sequence that generated the CRC result?

Since M(x)/G(x) = Q(x) + R(x), we can say that M(x) = [Q(x) + R(x)]G(x). But we don't know Q(x). Assuming the 16-bit CRC is equal to R(x) and the G(x) is given, how do I go about finding M(x) without knowing Q(x)?

 

[Baluncore]

Because the generating polynomial is symmetrical = 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 you should be able to reverse the process in time without rewiring the generator.

First reverse the order of bits in your final CRC and load it into the generator. Then feed the data stream bits backward in time through the generator until you reach the original starting position.