# What is the error-detection-probability of CRC

I have a table which stores all serial numbers of devices in my system:

|------------------------|--------------------------|
|            #           |  Serial number (4 bytes) |
|------------------------|--------------------------|
|            1           |        0x????????        |
|------------------------|--------------------------|
|            2           |        0x????????        |
|------------------------|--------------------------|
|           ...          |            ...           |
|------------------------|--------------------------|
|           31           |        0x????????        |
|------------------------|--------------------------|


If I add a device to the system the position in the table is not predictable. If something in the system changes (e.g. add one or more devices, remove one or more devices, replace a device) I want to detect with a "high" propability that the system change. Therefor I want to use a CRC-32. In my system I have at least always 2 devices but max. 31 devcies. Meaning that the input of the CRC can vary between $2^{64}$ to $2^{992}$ different input streams.

As far as I understand:

• The minimum input should be at least 32 bits
• The most important is to use the right generator polynom. E.g.:
• CRC-32
• CRC-32-IEEE
• CRC-32C
• The selection of generator polynom depends on:
• Input stream length
• Error-detection-feasability
• Collision probablility
• Detection of a single bit error: All CRC's can do this
• Detection of burst errors: All $n$-bit CRC's can detect burst errors up to $n$ consecutive bits. Also meaning appending new $n$-bit to the end of the input stream.

My concerns are the collision probability of the CRC32 is to high and due to that I don't detect a change in the configuration. Therefor I want to calcualte this probabilities and my questions are:

1. Is CRC the right method to use for my use case? If not what would be the best?

2. What is the "best" generator polynom for my use case?

3. How to cacluate the error-detection-feasability or collision probability if more then 32 consecutive bits change?

4. How to cacluate the error-detection-feasability or collision probability if $x$ not consecutive (different locations in the complete input stream) bits change?