Hello Blue Sky! Check out first post in my series on transparency systems. educatedguesswork.org/posts/transp...
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Finally, when you go to the Web server, it proves that it's certificate matches the summary. What this means technically is that it gives you a Merkle inclusion proof that goes back to the root.
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Of course, actually deploying this in practice turns out to be a lot harder than it sounds, which I'll get to in the next post. educatedguesswork.org/posts/transp...