First off, you don't want to use a checksum for authentication.
A checksum can be thought of as a hash with a very small size. The smaller the size of a hash the less unique it is. The less unique it is the more likely it is that incorrect data will generate the same hash and authenticate someone it shouldn't.
For instance, a 32 bit checksum (hash) of 1kB data (8192 bits of information) is a 256-fold reduction in the uniqueness of the value. At 32 bits you get 232 unique values. With 8192 bits you get 28192 possible values. I can't calculate how many values that actually is, but as you can imagine there will be many billions of billions of combinations that would all equate to the same checksum.
If you must use a checksum then make it an actual recognised hash. For instance an MD5 (considered weak by today's standards) has 128 bits. SHA-256 has, as you would imagine, 256 bits.
However calculating these hashes on an Arduino is not a simple matter owing to a lack of memory. Not impossible, but not that easy.
If you're going as far as to use such large amounts of data for authentication and using heavy mathematical routines to create a hash you may as well go the whole hog and use a proper cryptographic key pair.
One method would be to encrypt the UUID of the card off-line and store the encrypted result on the card - that data would then be unique to that card and tied to it - you wouldn't be able to clone the card unless you could clone the UUID as well.
Then either the UUID is read by the Arduino and also encrypted using the same key and if the results match authentication succeeds. Or the encrypted value is decrypted by the Arduino and compared to the UUID to see if they match. Which you choose depends on if you have asymmetric or symmetric encryption (the former is more like a hash but requires the encryption key to replicate it whereas a hash doesn't).
You can also include other information with the UUID, such as the name of the card owner (for identification purposes), etc - as long as you are able to already know that data to create the cryptographic has to compare with what is on the card.