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On many cards there's the instruction to, if found, cut it in two pieces and mail them to some address. I'm wondering a bit about the security of this and if I could recover some data.

It's not necessarily a perfect cut through the exact middle, since people usually aren't that careful.

First off, if I cut the magnet stripe in half, can I repair it, or get the data off of it to make a copy?

Secondly, if I cut the chip in half, how likely is it that it's too broken to recover data from?

Thirdly, contactless payments. Same question.

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    Well, if the chip is well-designed, cutting it in half will probably count as a "tampering attempt" and will zeroize everything on the chip. – SEJPM May 21 '16 at 14:17
  • By the time you get round to making a copy or extracting data, it's more than likely the cardholder will have liased with the bank and the card deactivated. So any data you collect off the card will be somewhat irrelevant. – Little Code May 21 '16 at 15:12
  • I typically run over the strip with a neodymium magnet, cut it into several (> 10) pieces, and then keep a couple of the strip pieces. – user64273 May 23 '16 at 18:26
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For mag stripe, as answered by Thomas. But note that most countries implement EMV differently and may allow mag stripe purchase if the chip fails to read.

The actual chip area of an IC card is very small. (If you look carefully at a well-used card, you may be able to see the location of the actual chip.)

chip

If the cut is not right through the chip, the chip can probably still be salvaged (e.g. by attaching wires to it).

The antenna for contactless payment will certainly be damanged by cutting, but it can normally be fixed if the chip is not damanged.

Obviously, the latter two require professional equipments and expertise, but criminals increasingly invest in these areas unfortunately.

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I'm quite sure you will be able to recover "some data" from the magnet stripe - likely even all the data that's on there. However, you won't be able to withdraw any cash with a copy of that stripe because nowadays use the information on the chip, which is much safer as it is much harder - close to impossible, I would guess - to copy that. The only thing you could hope for is a store with an old terminal, but then you have to give your copy of the card to a person...

As for the chip, if you don't have the means of an intelligence agency or something like that, you won't be able to repair it. (The chip is not just a "data store" that is to be read but a device that can answer cryptographic challenges.) Same thing should apply for contactless payment.

Thanks for the comment: I'm referring to the situation in the European Union, namely EC-cards, which are most common over here.

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    Americans aren't really using chip and pin yet, and even on cards that do use it you can downgrade to magnet stripe. Here's one way. – Filip Haglund May 21 '16 at 14:36

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