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I'm concern about uniqueness of the onion sites in terms of finding the right onion hostname and publishing it. How I can be guaranteed that my onion site will be the only one in Tor network with the same address?

So I'm not sure if it's possible, but since there is the same script used by everybody (scallion) to generate onion hostname for certain pattern, like mytextxyz and multiple people are looking for same pattern, isn't they end up with the exactly same private key (or pattern)?

Or another possibility can include copying the private key, then what happens when few independent servers running the website with the same private key?

For example when somebody finds the private key for the same pattern like facebookcorewwwi.onion (which is nearly impossible, but somehow they manage to find the pattern which they wanted, so hypothetically we can assume it can be possible) and publishes the fake site for phishing attacks (or multiple of them in different countries with the same copied key for a better results). Which website users will be seeing? The official, or the fake once?

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Let's assume that two separate sites have identical keys. It is possible, though highly unlikely for two servers to generate the same keys, but you can also consider someone moving servers for the same hidden service, but leaving both instances running and accessible.

When you visit a .onion site, you are essentially retrieving a service descriptor for how to access it. If you have the private key that corresponds to this service descriptor, you can update it.

If two people have the keys and update it, whichever publishes most recently will be believed as genuine and clients will be directed there.

This is bad if the second site to publish is a fake, but the network has no way of knowing if you are just upgrading servers, taking a site back from a scammer, or "stealing" one yourself.

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It depends on the exponent, if the people will use the same exponent - yes, they will have an identical keys. And - as far as I know - as soon you publish your HS descriptor, no descriptor with the same name is possible to exist in network.
So - its a good and bad thing at the same time.

A Very good thing is that no duping/hijacking possible in the way you've mentioned in your question.

A bad thing is that, for example, if it would be possible on first publishing a HS descriptor to opt-in a fact that multiple hosts may run it with the same key - it would give us a very handy load-ballancing and failsafe feature for existing HS.

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