A simple solution to credit card fraud

It seems to me there is one way to prevent fraudulent credit card transactions, such as from those whose data was stolen recently.

This would initially be a large project for the credit card companies but would be well worth it in terms of effectiveness.

It is simply this: confirm all credit card transactions via mobile phone messages. And do so on a number known only to the credit card company and cardholder, so that even if the card and/or its data was stolen, the thieves would not know the phone that would be called for confirmation and so could make no transactions. As a card holder there would be a slight increase in bureaucracy but well worth it for assurance purposes. So if I am in a restaurant signing a receipt for a meal, I immediately get a text message on my (primitive) mobile phone saying ‘Do you want to pay €x to restaurant Y ?’ Only by pressing ‘1’ on my phone will the transaction actually go through.