"but also huge amount of permutations"
By this do you think they mean they encrypt the same piece of data many times and say validate a random selection of them.
That would certainly make it harder to break. But only by increasing the workload, not by increasing the strength of the encryption. Replacing one static key with 100 doesn't remove the fact they are static keys and once broken useless.
If the keys are not static (ie each card has its own keys) then since there is no way for the card to communicate with the server at the other end, then cards must contain the keys, which means you just need to break the method used to encrypt the keys on the card.
Not trying to talk down the stock, just interested in how you could possibly create a barcode for drivers licenses or passports that was truely secure and not just secure until someone breaks into the office and steals the keys.
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