In my privacy class I was asked this question related to privacy in Bitcoin:
Which is the computational hard problem that is used to guarantee the privacy of users?
Can anyone help me? The only two computationally hard problems in Bitcoin I know of are the proof of work and breaking the hash collision resistant function, but I cannot quite see how these issues link to privacy.
Bitcoin does not guarantee privacy. Everyone can see exactly how much Bitcoin is associated with any address. The privacy comes from people using multiple addresses so as to hide how much Bitcoin they actually have (and this is also a generally recommended practice) and the fact that there is no way to find out the person behind an address from just the blockchain data itself. To deanonymize a person, they need to have leaked information about themselves and any addresses they control via some other medium such as forum posts, tweets, reddit posts, etc. There are also ways to figure out what addresses are likely to be in the same wallet and thus owned by the same person.
tl;dr privacy in Bitcoin comes down to the fact that users' real life identities are not directly linked to their addresses.
You are right, these problems are not related to privacy.
Keeping your privacy while using Bitcoin is a conscious and constant effort. You need to not reuse addresses, use widely available wallets to try to limit the amount of metadata you're leaking forever on the blockchain.
What Bitcoin provides (and most people confound with anonymity) is pseudonymity: your bitcoin identity is not linked to your legal identity. But it is still possible to link together bitcoin payments to a single-but-unknown identity.
You can think of a bitcoin address as an email address: I may not know who it is, but it's most likely a single person behind it.
For more information about Bitcoin privacy, you can visit the Open Bitcoin Privacy Project.
Technically, the hardness of the elliptic curve discrete log problem keeps your private key private! But we would normally say that this is what protects your Bitcoins from being stolen, since that is what happens if your private key gets leaked. So I think this is not really a good answer except to a confused or trick question.
When people talk about privacy in the context of Bitcoin they are normally talking about the keeping the identities of users privacy, and in Bitcoin this isn't provided by cryptography but only by the use of pseudonyms: single or few use, randomly generated addresses.
