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There is an problem shown in bitcoinwiki:

Someone runs both a money exchanger and a site meant to trap people. When Mr. Doe buys from the exchanger and uses those same coins to buy something from the trap site, the attacker can prove that these two transactions were made by the same person. The block chain would show:

Import coins from address A. Send 100 to B. Authorized by (signature).

Import coins from address B. Send 100 to C. Authorized by (signature).

Bitcoin transactions do not have a "from" address, but if the attacker believes that address B is controlled by Mr. Doe because the attacker received $5 from Mr. Doe's PayPal account and then sent 100 BTC to that address then they can infer the identity of the party sending to C. This assumption is not always correct because address B may have been an address held on behalf of Mr. Doe by a third party and the transaction to C may have been unrelated.

I can't understand several issues. Is "someone who runs both a money exchanger and a site meant to trap people" attacker? If so, than I (as attacker) already know the owner of address B, don't I? why I need trap site?

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Maybe a more specific example would help:

Senator Doe buys 100 BTC at Ed's Exchange, where his ID is required. He then withdraws those coins to his own address 1qwerty. Wearing a trench coat and dark glasses, he uses those 100 BTC to make a purchase at Frank's Embarrassing Merchandise Emporium, sending them to Frank's freshly generated address 1abcxyz.

He thinks he's safe:

  • Ed knows that 1qwerty belongs to Senator Doe, and that Doe transferred them to the address 1abcxyz, but only Doe and Frank know that 1abcxyz belongs to Frank. It was randomly generated, and just looking at the address doesn't reveal anything about whom it belongs to.

  • Frank knows that he sold something embarrassing to the owner of the address 1qwerty, but doesn't know the true identity of the disguised person who came into his shop.

So, thinks Doe, nobody knows that in fact it was Doe who made the embarrassing purchase.

But if Ed and Frank are cooperating, or are secretly the same person, then they can share their knowledge and learn that it was Doe that made the embarrassing purchase. Now they can inform the newspapers and/or blackmail Doe.

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I (as attacker) already know the owner of address B, don't I? why I need trap site?

You're right, if you sold the bitcoin and sent it to an address, and you know the identity of the person to whom it was sold (from KYC, for example), then you can associate that person that address. Since it is publicly verifiable on the blockchain, you don't need a trap site in order to see when that bitcoin is spent, but if you run a trap site, you can determine what they spent the Bitcoin on.

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