As an example, let's say that Alice is an attacker trying to learn Bob's wallet using this method. Alice sends 1 satoshi to some address that Bob owns.
At this point in time, Alice does not know anything about Bob's wallet, except that he has 1 sat that she sent him. But the problem is when Bob wants to send Bitcoin.
Now Bob sends some Bitcoin to a merchant, Carol. When he does this, his wallet will look at what UTXOs it can spend, construct a transaction, sign it, and send it. For this example, let's say that his wallet chooses the UTXO for the 1 sat that Alice sent him. Of course he is paying Carol more than 1 sat and he also has to pay transaction fees. So Bob's wallet chooses several other UTXOs to cover the amount being spent and the transaction fee.
When Bob broadcasts his transaction, Alice can see it too. This is where the privacy comes in. Alice can see the 1 sat she sent Bob in this transaction he made. Since that transaction includes other UTXOs from Bob's wallet, Alice learns that Bob had those UTXOs in his wallet. She can now check the addresses for those UTXOs and learn some addresses in Bob's wallet. If Bob was reusing addresses, she can now learn a lower bound to the amount in Bob's wallet.
Furthermore, because she knows those addresses belong to Bob, she can watch for other transactions that spend those address's UTXOs. With those transactions, she can do the same process of examining the inputs and learning additional addresses in Bob's wallet. Because Alice can see many of the transactions Bob makes, she can see who Bob is paying and who is paying Bob.
Bob can mitigate this part by not reusing addresses. If Bob doesn't reuse addresses, then Alice will learn less information because once she learns about some of Bob's addresses in a transaction, Bob will never use those again so there isn't any additional tracing that can be done.
However, there is still the change outputs that Bob's wallet likely made in his transaction. If Alice is able to identify the change outputs in any of Bob's transactions, she will also be able to do the above.
Now this kind of tracking is not unique to the dust spam attack. Anyone can do this kind of tracing with any amount of Bitcoin. It can be done by someone being malicious after having done a normal Bitcoin transaction.
The 1 satoshi spam attack becomes more interesting when Alice does it en masse. If Alice sends 1 satoshi to millions of addresses, Alice can learn a lot more.
Suppose Alice looks up every address that has ever been used and sends 1 sat to every single one of them. This will include Bob's addresses. She also sends 1 sat to an address ,
Bobs_addr, that she knows belongs to Bob.
Bob has never reused an address. But Alice has now forced him to reuse addresses because she has sent 1 sat to all of his addresses. When Bob makes a transaction, his wallet may be unaware of this attack and may choose to include a couple of Alice's 1 sat UTXOs in a transaction, in addition to the 1 sat at
Now Alice has a transaction that links
Bobs_addr with several of Bob's other addresses. Furthermore, because the 1 sat UTXOs belong to addresses that have now been reused, Alice can look up what transactions those addresses were involved in and do the same analysis that I described earlier with the address reuse problem.
There are several mitigations to these privacy revealing attacks. These privacy revealing attacks all rely an assumption that all of the inputs in a transaction belong to the same person. This is known as the common input heuristic. To break this assumption, a sender can join up with other people sending money and they all create a transaction together. This is known as a Coinjoin. A Coinjoin transaction has inputs from multiple people, and outputs which go to multiple people. So this breaks the common input heuristic.
Wallets can also be made aware of such dust spam attacks. If the wallet developers know that this is something that occurs, they can program the wallet such that it will never spend reused addresses unless absolutely necessary. If Bob never reused an address and his wallet avoided address reuse, he could prevent Alice from learning anything more about his wallet with her spam attack because his wallet would choose to ignore Alice's 1 sat UTXOs sent to addresses he already used.
In addition to avoiding reuse, if reuse has occurred, the wallet can also choose to spend all UTXOs to the same address at the same time. This prevents being able to link addresses to additional transactions because all of the reuse is spent at the same time.
To prevent Alice from being able to trace Bob's wallet by trying to figure out his change addresses, Bob's wallet can also choose to not make change outputs. His wallet could be using a coin selection algorithm which searches for an input set which exactly matches the amount that Bob is trying to send. Combined with not reusing addresses, avoiding spending reused addresses, and spending reused addresses all at once, this could effectively prevent Alice from learning anything about Bob because the only lead she has into his wallet is where those coins never lead back to Bob.