A magnetic strip on a card encodes the same information that is printed on the card itself (the issuing bank, the card number, expiration date). When it’s swiped and the information is read it’s effectively like they just took a photocopy of your card. Just in a way that is more convenient for computers to read.
The chip on a card is a small computer that can digitally sign things using a secret cryptographic key. The key is stored in the chip in a way that card readers can’t read the key directly, they can only ask the chip to sign things.
So when you use a magnetic strip to pay, the card reader essentially ends up with a copy of your card and you just have to hope that they’re not going to do anything sketchy with it. With a chip they only end up with a digital signature that authorizes a specific transaction, so there is nothing there for them to steal (it’s even done in a way that you can’t reuse the signature to authorize another identical transaction).
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