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Validation operations can be accomplished using homomorphic encryption without decrypting, because credit card validation is performed by way of mathematical operations.

In addition, it seems likely that credit card transactions can be completed entirely using credit card data subject to homomorphic encryption without ever decrypting it, but only if every step of the process is using the same encryption technologies -- including the card processing entity at the far end of the process. That kind of buy-in from all parties for the same process is likely to be very difficult to achieve at this point, unfortunately.