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Why would you want to do that with a blockchain? Is it necessary to store this logging data to thousands of nodes? What about 5 servers, geographically distributed?

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Why do you assume a blockchain has to consist of thousands of nodes? A private blockchain works equally well across those 5 geographically distributed nodes.

So why use blockchain? Whats the benefit? The only thing I can think of is proof of authentication attempts, a kind of a irrefutable log of some sort. But whats the benefit its providing, that we are lacking today?

That irrefutable log is highly valuable in environments where there is a need to audit who has requested access to what resources, and when.

I think this is a really good idea. Would it be worthwhile to have a neutral party also maintain a node? Would that give the evidence more weight in court for example?

While I am not allowed to talk too much about it, this is exactly what I'm currently working on for certain high profile companies that wish to share information (query results) between themselves, without ever actually sharing the data itself, in a way that the country-wide regulator can verify and check for anticompetitive behaviour.

I'm currently having fun experimenting with homomorphic encryption, to see if I can't add another layer of "trustlessness"

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