You don't want the signature to be public data.  If you do that, anyone can
copy it.  Might as well use magstripe.

You want to store a private key in a non-readable file.  Then you send a
challenge to the card, and the card uses the challenge in some way to
produce a proof that it has the private key.  Any beginning text on
smartcards, or crypto in general, will describe example protocols.
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